Random Walks, Boundaries and Spectra

Progress in Probability Volume 64

Series Editors Charles Newman Sidney I. Resnick

For other volumes published in this series, go to www.springer.com/series/4839

Random Walks, Boundaries and Spectra Daniel Lenz Florian Sobieczky Wolfgang Woess Editors

Editors Daniel Lenz Mathematisches Institut Universität Jena Ernst-Abbe-Platz 2 D-07737 Jena Germany [email protected]

Florian Sobieczky Institut für Mathematik C TU Graz Steyrergasse 30 A-8010 Graz Austria [email protected]

Wolfgang Woess Institut für Mathematik C TU Graz Steyrergasse 30 A-8010 Graz Austria [email protected]

2010 Mathematics Subject Classification: 60-06

ISBN 978-3-0346-0243-3 DOI 10.1007/978-3-0346-0244-0

e-ISBN 978-3-0346-0244-0

© Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the right of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use, permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com

Austrian Science Fund (FWF) Project P18703 Random Walks on random subgraphs of transitive graphs

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Programme of the Workshop on “Boundaries” . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Programme of the Alp-Workshop 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Publications of D.I. Cartwright . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Publications of M.A. Picardello . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviii Publications of V.A. Kaimanovich . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii M.J. Dunwoody An Inaccessible Graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

J. Parkinson and B. Schapira A Local Limit Theorem for Random Walks on the Chambers of A˜2 Buildings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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A. Erschler On Continuity of Range, Entropy and Drift for Random Walks on Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

55

Y. Guivarc’h and C.R.E. Raja Polynomial Growth, Recurrence and Ergodicity for Random Walks on Locally Compact Groups and Homogeneous Spaces . . . . . . . . . . . . . .

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M. Bj¨ orklund Ergodic Theorems for Homogeneous Dilations . . . . . . . . . . . . . . . . . . . . . . .

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A. Gnedin Boundaries from Inhomogeneous Bernoulli Trials . . . . . . . . . . . . . . . . . . . .

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P.E.T. Jorgensen and E.P.J. Pearse Resistance Boundaries of Infinite Networks . . . . . . . . . . . . . . . . . . . . . . . . . . 111 M. Arnaudon and A. Thalmaier Brownian Motion and Negative Curvature . . . . . . . . . . . . . . . . . . . . . . . . . .

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R.K. Wojciechowski Stochastically Incomplete Manifolds and Graphs . . . . . . . . . . . . . . . . . . . .

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S. Haeseler and M. Keller Generalized Solutions and Spectrum for Dirichlet Forms on Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 R. Froese, D. Hasler and W. Spitzer A Geometric Approach to Absolutely Continuous Spectrum for Discrete Schr¨ odinger Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 A. Bendikov, B. Bobikau and C. Pittet Some Spectral and Geometric Aspects of Countable Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 P. M¨ uller and P. Stollmann Percolation Hamiltonians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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T.S. Turova Survey of Scalings for the Largest Connected Component in Inhomogeneous Random Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259 D. D’Angeli, A. Donno and T. Nagnibeda Partition Functions of the Ising Model on Some Self-similar Schreier Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 I. Krasovsky Aspects of Toeplitz Determinants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

Preface This book contains the joint proceedings of the workshop on Boundaries that took place in Graz, from June 29–July 3, and the Alp-Workshop that was held immediately afterwards in Sankt Kathrein am Offenegg, on the weekend July 4–5, 2009. The two events were dedicated to related subjects. The aim of the Boundaries workshop was to bring together mathematicians working on groups, graphs, manifolds, etc., in the context of probability (random walks, Brownian motion), harmonic analysis, potential theory, ergodic theory, geometric group theory and related topics. The title indicates a central topic but was not to be considered the exclusive theme. The scientific committee of the meeting consisted of Tatiana NagnibedaSmirnova (Geneva), Christophe Pittet (Marseille), Hamish Short (Marseille), and Wolfgang Woess (Graz). The local organisation rested on the shoulders of Ecaterina Sava and Wolfgang Woess at Graz University of Technology in the capital of Styria, southeastern province of Austria. Three special guests were particularly featured in view of their “milestone birthdays” taking place in 2009: • Donald I. Cartwright (Sydney; 60th birthday) • Vadim A. Kaimanovich (Bremen; 50th birthday) • Massimo Picardello (Rome; 60th birthday) Each of these three has given substantial contributions to the mathematical subject of the workshop, and to each of them, a half-day session was dedicated, featuring in particular their own (respective) invited talks. In the present volume, we display their lists of publications (state of September, 2010). The Alp-Workshop 2009 was devoted to “Spectral and probabilistic properties of random walks on random graphs”. The aim was a discussion between experts from spectral theory, ergodic theory and probability theory about the special topics of random walk theory in which the methods from group theory and harmonic analysis fail: Discrete structures with much irregularity, such as Percolation, Random Graphs, or Branching Processes were the main focus. Instead of a detailed discussion of each talk we refer to the attached programme. During the

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first afternoon-session, there were six twenty-minutes talks by young researchers of whom several have contributed to the proceedings. The Alp-Workshop was organised by Florian Sobieczky with the budget of project P18703 (“Random Subgraphs of Transitive Graphs”) of the Austrian Science Foundation (FWF). Furthermore, the main part of the publication cost of these proceedings was carried by the budget of this research project. The “Almenland” in the mountains east of Graz provided a picturesque environment for the interdisciplinary discussion about random walks. Its remoteness allowed inviting more people with the given budget while keeping a high standard of the venue. The editing of the proceedings contributed by the Alp-Workshop’s participants was undertaken by Daniel Lenz and Florian Sobieczky. The contributions from the Boundaries-Workshop were edited by Wolgang Woess. All articles underwent anonymous refereeing by experts from the respective field. We would like to thank everyone who was directly or indirectly involved in helping to organise these meetings. This volume is dedicated to

Donald I. Cartwright

October 2010,

Massimo A. Picardello

Vadim A. Kaimanovich

Daniel Lenz Florian Sobieczky Wolfgang Woess

Programme of the Workshop on “Boundaries” June 29th (Mon.) 09:00–09:10 09:10–10:10 10:10–10:40 10:40–11:10 11:20–1150 12:00–12:20

12:20–14:30 14:30–15:20 15:30–16:00 16:10–16:40 16:40–17:30

17:40–18:00

Opening Francois Ledrappier, University of Notre Dame Linear drift for the Brownian motion on covers Coffee & Registration Martin Dunwoody, University of Southampton An inaccessible graph Panos Papazoglou, University of Athens Topology of boundaries and splittings Barbara Bobikau, University of Wroclaw Spectral properties of a class of random walks on locally finite groups Lunch Massimo Picardello, Tor Vergata University in Rome Harmonic functions on homogeneous trees and buildings Sara Brofferio, University of Paris-Sud 11 Poisson boundary of matrix groups with rational coefficients Coffee Yves Guivarc’h, University of Rennes Random walk in a random medium on Z, and random walks on homogeneous spaces Daniele D’Angeli, University of Geneva The boundary action of the Basilica group

June 30th (Tue.) 09:30–09:50 10:00–10:30 10:40–11:11

11:20–11:50

12:00–12:20

12:20–14:30 14:30–15:20 15:30–16:00

Tim Riley, Cornell University How wild can a group with a quadratic Dehn function be? Coffee Anton Thalmaier, University of Luxembourg The Poisson boundary of certain Cartan-Hadamard manifolds of unbounded curvature Alexander Gnedin, Utrecht University Boundaries of the generalised Pascal triangles and larger graded graphs Jeremy Macdonald, McGill University Compressed words and automorphisms in fully residually free groups Lunch Tim Steger, University of Sassari Background on fake planes Jean L´ecureux, Claude Bernard University Lyon 1 Combinatorial boundaries of buildings

xii 16:10–16:40 16:40–17:30 17:40–18:00

Programme: Workshop on “Boundaries” Coffee Donald Cartwright, University of Sidney The 50 fake projective planes Bernhard Kr¨on, University of Vienna Vertex cuts, ends and group splittings

July 1st (Wed.) 09:00–09:50 10:00–10:50

10:50–11:20 11:30–12:00 Afternoon

Anna Erschler, University of Paris-Sud 11 Boundaries of amenable groups Poster Session & Coffee Poster: Elisabetta Candellero, Lorenz Gilch, Motoko Kotani, Jeremy Macdonald, Sebastian M¨ uller, Svetla Vassileva Matthias Keller, Universit¨ at Jena Heat transfer to the boundary on discrete graphs Erin Pearse, University of Iowa & University of Oklahoma Resistance analysis of infinite networks Excursion

July 2nd (Thu.) 09:00–09:50 10:00–10:30 10:40–11:10

11:20–11:50

12:00–12:20 12:20–14:30 14:30–15:20

15:30–16:00

16:00–16:40 16:40–17:30 17:40–18:00

James Parkinson, University of Sydney Random walks on p-adic groups and affine buildings Coffee Agelos Georgakopoulos, Graz University of Technology Uniqueness of currents in an electrical network of finite total resistance J¨org Schmeling, Lund University Large dimension of limit sets of Kleinian groups and transience of critical random walks Riddhi Shah, Jawaharlal Nehru University Distal actions on locally compact groups Lunch Vadim Kaimanovich, University of Ottawa Random graphs, stochastic homogenization and equivalence relations Alexander Bendikov, University of Wroclaw On a class of random walks on groups with infinite number of generators Coffee Volodymyr Nekrashevych, Texas A& M University Hyperbolic duality Fr´ederic Math´eus, LMAM University of South-Brittany Poisson boundary of free-by-cyclic groups

Programme: Alp-Workshop 2009 July 3rd, (Fri.) 09:00–09:50 10:00–10:40 10:40–11:10 11:20–11:50 11:50–13:30 13:30–14:00 14:10–15:00

Klaus Schmidt, University of Vienna Sandpiles and the harmonic model Coffee Tatiana Smirnova-Nagnibeda, University of Geneva Sandpiles and self-similar groups Markus Neuhauser, RWTH Aachen Further examples to a question of Atiyah Lunch Michael Bj¨ orklund, Hebrew University Sharp sumset inequalities for Bohr sets Anatoly Vershik, St.Petersburg State University Adjoint dynamics to a question of Atiyah

Programme of the Alp-Workshop 2009 July 4th (Sat.) 09:15–09:30 09:30–10:15

10:20–11:05 11:05–11:20 11:20–12:05

12:10–12:55 12:55–14:00 14:00–16:30

Evening

Welcome Christoph Pittet, University of Aix-Marseille 1 Return probabilities and spectral distribution of Laplace operators Peter M¨ uller, Ludwigs Maximilians University Munich Ergodic properties of randomly coloured aperiodic point sets Coffee Tatyana Turova, Lund University Asymptotic size of the largest cluster in inhomogeneous random graphs: sub-critical and critical phases Vadim Kaimanovich, Jacobs University Bremen Stochastic homogenization of graphs: case studies Lunch Short Talks-Session & Coffee Wolfgang Spitzer, Bernt Metzger, Radoslaw Wojciechowski, Matthias Keller, Sebastian M¨ uller, Erin Pearse Hike and Dinner at Mountain Cabin

July 5th (Sun.) 10:00–10:45

10:45–11:00 11:00–11:45

Daniel Lenz, Universit¨ at Jena Amenability of Horocyclic Products of uniformly growing trees Coffee Tatiana Smirnova-Nagnibeda, Geneva University Amenability and percolation

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Programme: Alp-Workshop 2009

11:50–12:35

12:35–14:00 14:00–14:45

14:50–15:55

16:00–16:45

17:00–17:45 18:00–18:45

J¨org Schmeling, Lund University Random trees generated by a dynamical system and the structure of typical orbits Lunch Franz Lehner, Graz University of Technology On the Eigenspaces of Lamplighter Random Walks and Percolation Clusters on Graphs Poster-Session & Coffee Erin Pearse, Lorenz Gilch, Ecaterina Sava, Wilfried Huss, Seon Hee Lim, Michael Matter, Uta Freiberg, Elisabetta Candellero Peter M¨orters, University of Bath Simultaneous multifractal analysis of branching and visibility measure on a Galton-Watson tree Ivan Veseli´c, TU Chemnitz Percolation clusters on Caley graphs and their spectra Tyll Kr¨ uger, Rainer Siegmund-Schultze, TU Berlin Epidemic processes on networks and generalisations

A Steyr 480a “Postbus” waiting for its passengers to board before taking them to St. Kathrein am Offenegg, the venue of the AlpWorkshop 2009.

Donald I. Cartwright Research Publications [1] The order completeness of some spaces of vector-valued functions. Bull. Austral. Math. Soc. 11 (1974), 57–61. MR50#14207. [2] Extensions of positive operators between Banach lattices. Mem. Amer. Math. Soc. 3 (1975), no. 164, iv + 48 pp. MR52#3913. [3] (with Lotz, Heinrich P.) Some characterizations of AM - and AL-spaces. Math. Z. 142 (1975), 97–103. MR52#3912. [4] (with Lotz, Heinrich P.) Disjunkte Folgen in Banachverb¨ anden und Kegel-absolutsummierende Operatoren. Arch. Math. (Basel) 28 (1977), 525–532. MR58#2442. [5] (with McMullen, John R.) A note on the fractional calculus. Proc. Edinburgh Math. Soc. (2) 21 (1978/79), 79–80. MR57#16488. [6] (with Field, M.J.) A refinement of the arithmetic mean–geometric mean inequality. Proc. Amer. Math. Soc. 71 (1978), 36–38. MR57#16516. [7] (with Howlett, Robert B.; McMullen, John R.) Extreme values for the Sidon constant. Proc. Amer. Math. Soc. 81 (1981), 531–537. MR#82c:43005. [8] (with McMullen, John R.) A structural criterion for the existence of infinite Sidon sets. Pacific J. Math. 96 (1981), 301–317. MR#83c:43009. [9] (with McMullen, John R.) A generalized universal complexification for compact groups. J. Reine Angew. Math. 331 (1982), 1–15. MR#84d:22009. [10] Lp -norms of characters on the exceptional compact Lie groups. Boll. Un. Mat. Ital. B (6) 2 (1983), 339–351. MR#84i:22014. [11] (with Soardi, Paolo M.) Best conditions for the norm convergence of Fourier series. J. Approx. Theory 38 (1983), 344–353. MR#85a:42017. [12] Lebesgue constants for Jacobi series. Proc. Amer. Math. Soc. 87 (1983), 427–433. MR#84b:42019. [13] (with Brown, Timothy C.; Eagleson, G.K.) Characterizations of invariant distributions. Math. Proc. Cambridge Philos. Soc. 97 (1985), 349–355. MR#86i:60023. [14] (with Barbour, A.D.; Donnelly, J.B.; Eagleson, G.K.) A new rank test for the ksample problem. Comm. Statist. A – Theory Methods. 14 (1985), 1471–1484. [15] (with Brown, Timothy C.; Eagleson, G.K.) Correlations and characterizations of the uniform distribution. Austral. J. Statist. 28 (1986), 89–96. MR#87i:62032. [16] (with Soardi, Paolo M.) Harmonic analysis on the free product of two cyclic groups. J. Funct. Anal. 65 (1986), 147–171. MR#87m:22015. [17] (with Soardi, Paolo M.) Random walks on free products, quotients and amalgams. Nagoya Math. J. 102 (1986), 163–180. MR#88i:60120a. [18] (with Soardi, Paolo M.) A local limit theorem for random walks on the cartesian product of discrete groups. Boll. Un. Mat. Ital. (7) 1-A (1987), 107–115. MR#89a:60159. [19] Some examples of random walks on free products of discrete groups. Annali di Matematica pura ed applicata 106 (1988), 1–15. MR#90f:60018. [20] (with Kucharski, Krzysztof) Jackson’s theorem for compact connected Lie groups. J. Approx. Theory 55 (1988), 352–359. MR#89j:43008.

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[21] Random walks on direct sums of discrete groups. J. Theoretical Probability 1 (1988), 341–356. MR#89j:60013. [22] (with P.M. Soardi) Convergence to ends for random walks on the automorphism group of a tree. Proc. Amer. Math. Soc. 107 (1989), 817–823. MR#90f:60137. [23] On the asymptotic behaviour of convolution powers of probabilities on discrete groups. Monatshefte f¨ ur Mathematik 107 (1989), 287–290. MR#91a:60024. [24] (with S. Sawyer) The Martin boundary for general isotropic random walks on a tree. J. Theoretical Probability 4 (1991), 111–136. [25] (with Wolfgang Woess) Infinite graphs with nonconstant Dirichlet finite harmonic functions. SIAM J. Discrete Math. 5 (1992), 380–385. [26] Singularities of the Green function of a random walk on a discrete group. Monatshefte f¨ ur Mathematik 113 (1992), 183–188. [27] (with P.M. Soardi, Wolfgang Woess) Martin and end compactifications of non locally finite graphs. Trans. Amer. Math. Soc. 338 (1993), 679–693. [28] (with Anna Maria Mantero, Tim Steger and Anna Zappa) Groups acting simply ˜2 I, Geom. Ded. 47 (1993), 143– transitively on the vertices of a building of type A 166. [29] (with Anna Maria Mantero, Tim Steger and Anna Zappa) Groups acting simply ˜2 II: the cases q = 2 and q = 3, transitively on the vertices of a building of type A Geom. Ded. 47 (1993), 167–226.
2 -groups. Annales [30] (with Wojciech Mlotkowski and Tim Steger) Property (T ) and A de l’Institut Fourier 44 (1994), 213–248. [31] (with Wojciech Mlotkowski) Harmonic analysis for groups acting on triangle buildings. J. Aust. Math. Soc. 56 (1994), 345–383. [32] (with Vadim A. Kaimanovich and Wolfgang Woess) Random walks on the affine group of local fields and homogeneous trees. Annales de l’Institut Fourier 44 (1994), 1243–1288. [33] Groups acting simply transitively on the vertices of a building of type A˜n . Proceedings of the 1993 Como conference “Groups of Lie type and their geometries”, pp. 43–76, W.M. Kantor, L. Di Martino, editors, London Mathematical Society Lecture Note Series 207, Cambridge University Press, 1995. [34] (with Michael Shapiro) Hyperbolic buildings, affine buildings and automatic groups. Mich. Math. J., 42 (1995), 511–523. [35] A brief introduction to buildings, Contemp. Math. 206 (1997), 45–77. ˜ n -groups. Israel J. Math., 103 (1998), 125–140. [36] (with Tim Steger) A family of A ˜2 groups. [37] (with Tim Steger) Application of the Bruhat-Tits tree of SU3 (h) to some A J. Aust. Math. Soc., 64 (1998), 329–344. ˜n . Proceedings of the 1997 Cortona [38] Harmonic functions on buildings of type A conference “Random Walks and Discrete Potential Theory”, pp. 104–138, Massimo Picardello and Wolfgang Woess, editors, Symposia Mathematica, vol XXXIX, Cambridge University Press, 1999. [39] (with Gabriella Kuhn and Paolo M. Soardi) A product formula for spherical representations of a group of automorphisms of a homogeneous tree, I. Trans. Amer. Math. Soc., 353 (2001), 349–364.

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[40] (with Gabriella Kuhn) A product formula for spherical representations of a group of automorphisms of a homogeneous tree, II. Trans. Amer. Math. Soc. 353 (2001), 2073–2090. ˜n . Monatshefte f¨ [41] Spherical harmonic analysis on buildings of type A ur Mathematik 133 (2001), 93–109. [42] (with Tim Steger) Elementary symmetric polynomials in numbers of modulus 1. Canadian J. Math. 54 (2002), 239–262. [43] (with Joseph Kupka) When factorial quotients are integers. Gazette Aust. Math. Soc. 29 (2002), 19–26. [44] (with Gabriella Kuhn) Restricting cuspidal representations of the group of automorphisms of a homogeneous tree. Boll. Un. Mat. Ital. (8) 6-B (2003), 353–379. ˙ ˜n . Discrete [45] (with Patrick Sol´e and Andrzej Zuk) Ramanujan geometries of type A Mathematics 269 (2003), 35–43. [46] (with Wolfgang Woess) Isotropic random walks in a building of type A˜d . Math. Zeitschrift. 247 (2004), 101–135. [47] (with Bernhard Kr¨ on) On Stallings’ unique factorization groups. Bulletin Austral. Math. Soc. 73 (2006), 27–36. [48] (with Wolfgang Woess) The spectrum of the averaging operator on a network (metric graph). Illinois J. Math. 51 (2007), 805–830. [49] (with Tim Steger) Enumeration of the 50 fake projective planes. C. R. Acad. Sci. Paris, Ser. I 348 (2010), 11–13.

Massimo A. Picardello Research Publications [1] A. Fig` a-Talamanca, M.A. Picardello, Multiplicateurs de A(G) qui ne sont pas dans B(G), C. R. Acad. Sci. Paris 277 (1973), 117–119. [2] M.A. Picardello, Lacunary sets in discrete noncommutative groups, Boll. Un. Mat. It. 8 (1973), 494–508. [3] M.A. Picardello, Random Fourier series on compact noncommutative groups, Canad. J. Math. 27 (1975), 1400–1407. [4] A. Fig` a-Talamanca, M.A. Picardello, Functions that operate on the algebra B0 (G), Pacific J. Math. 74 (1978), 57–61. [5] M.A. Picardello, Locally compact unimodular groups with atomic dual, Rend. Sem. Mat. Fis. Milano 48 (1978), 197–216. [6] M.A. Picardello, A unimodular non-type I group with purely atomic regular representation, Boll. Un. Mat. It. 16-A (1979), 331–334. [7] G. Mauceri, M.A. Picardello, Noncompact unimodular groups with purely atomic Plancherel measures, Proc. Amer. Math. Soc. 78 (1980), 77–84. [8] M.A. Picardello, Unimodular Lie groups without discrete series, Boll. Un. Mat. It. 1-C (1980), 61–80. [9] G. Mauceri, M.A. Picardello, F. Ricci, Hardy spaces associated with twisted convolution, Advances Math. 39 (1981), 270–288. [10] G. Mauceri, M.A. Picardello, F. Ricci, Twisted convolution, Hardy spaces and H¨ ormander multipliers, Rend. Circ. Mat. Palermo (Suppl. 1) (1981), 191–203. [11] A. Fig` a-Talamanca, M.A. Picardello, Spherical functions and harmonic analysis on free groups, J. Functional Anal. 47 (1982), 281–304. [12] M.A. Picardello, Spherical functions and local limit theorems on free groups, Ann. Mat. Pura Appl. 133 (1983), 177–191. [13] A. Iozzi, M.A. Picardello, Graphs and convolution operators, in “Topics in Modern Harmonic Analysis” 1, Ist. Naz. Alta Matem., Roma (1983), 187–208. [14] A. Iozzi, M.A. Picardello, Spherical functions on symmetric graphs, Lecture Notes in Math. 993, Springer, New York–Berlin (1983), 344–386. [15] A. Fig` a-Talamanca, M.A. Picardello, Restriction of spherical representations of P GL2 (Qp ) to a discrete subgroup, Proc. Amer. Math. Soc. 91 (1984), 405–408. [16] J. Faraut, M.A. Picardello, The Plancherel measures for symmetric graphs, Ann. Mat. Pura Appl. 138 (1984), 151–155. [17] M.A. Picardello, W. Woess, Random walks on amalgams, Monatshefte Math. 100 (1985), 21–33. [18] M.A. Picardello, Positive definite functions and Lp -convolution operators on amalgams, Pacific J. Math. 123 (1986), 209–221. [19] A. Kor´ anyi, M.A. Picardello, Boundary behaviour of eigenfunctions of the Laplace operator on trees, Ann. Sci. Sc. Norm. Sup. Pisa 13 (1986), 389–399. [20] M.A. Picardello, W. Woess, Martin boundaries of random walks: ends of trees and groups, Trans. Amer. Math. Soc. 302 (1987), 285–305.

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[21] A. Kor´ anyi, M.A. Picardello, M.H. Taibleson, Hardy spaces on non–homogeneous trees, Symp. Math. 29 ( 1987), 205–265. [22] M.A. Picardello, P. Sj¨ ogren, The minimal Martin boundary of a cartesian product of trees, Proc. Centre Math. Anal. Austral. Nat. Univ. 16 (1988), 226–246. [23] M.A. Picardello, W. Woess, Harmonic functions and ends of graphs, Proc. Edinburgh Math. Soc. 31 (1988), 457–461. [24] M.A. Picardello, T. Pytlik, Norms of free operators, Proc. Amer. Math. Soc. 104 (1988), 257–261. [25] J.M. Cohen, M.A. Picardello, The 2-circles and 2-discs problems on trees, Israel J. Math. 64 (1988), 73–86. [26] M.A. Picardello, W. Woess, A converse to the mean value property on homogeneous trees, Trans. Amer. Math. Soc. 311 (1989), 209–225. [27] M.A. Picardello, W. Woess, Ends of graphs, potential theory and electric networks, in “Cycles and Rays”, NATO ASI Ser. C, Kluwer Academic Publishers, Dordrecht (1990), 181–196. [28] C.A. Berenstein, E. Casadio Tarabusi, J.M. Cohen, M.A. Picardello, Integral geometry on trees, Amer. J. Math. 113 (1991), 441–470. [29] M.A. Picardello, M.H. Taibleson, Substochastic transition operators on trees and their associated Poisson integrals, Coll. Math. 59 (1990), 279–296. [30] M.A. Picardello, M.H. Taibleson, Degeneracy of Hardy spaces on a two-sheeted graph: a sandwich of trees, Ars Combinatoria 29B (1990), 161–174. [31] M.A. Picardello, W. Woess, Examples of stable Martin boundaries of Markov chains in “Potential Theory”, De Gruyter & Co., Berlin – New York (1991), 261–270. [32] M.A. Picardello, M.H. Taibleson, W. Woess, Harmonic functions on cartesian products of trees with finite graphs, J. Functional Anal 102 (1991), 379–400. [33] M.A. Picardello, P. Sj¨ ogren, Boundary behaviour of eigenfunctions of the Laplacian in a bi-tree, J. Reine Angew. Math. 424 (1992), 133–144. [34] M.A. Picardello, M.H. Taibleson, W. Woess, Harmonic measure on the planar Cantor set from the viewpoint of graph theory, Discrete Math. 109 (1992), 193–202. [35] C.A. Berenstein, E. Casadio Tarabusi, M.A. Picardello, Radon transforms on hyperbolic spaces and their discrete counterparts, in “Proceedings of the Conference in Radon Transforms”, Rende (1991). [36] M.A. Picardello, W. Woess, Martin boundaries of Cartesian products of Markov chains, Nagoya Math. J. 128 (1992), 153–169. [37] E. Casadio Tarabusi, J.M. Cohen, M.A. Picardello, The horocyclical Radon transform on trees, Israel J. Math. 78 (1992), 363–380. [38] M. Bozejko, M.A. Picardello, Weakly amenable groups and amalgamated products, Proc. Amer. Math. Soc. 117 (1993), 1039–1046. [39] E. Casadio Tarabusi, J.M. Cohen, F. Colonna, M.A. Picardello, Characterization of the range and functional analysis of the X-ray transform on trees, C. R. Acad. Sci. Paris 316 (1993), 559–564. [40] E. Casadio Tarabusi, J.M. Cohen, M.A. Picardello, The range of the X-ray transform on trees, Adv. Math 109 (1994), 143–156.

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[41] M.A. Picardello, W. Woess, The full Martin boundary of the bi-tree, Ann. Prob. 22 (1994), 2203–2222. [42] F. Di Biase, M.A. Picardello, The Green formula and H p spaces on trees, Math. Zeitsch. 218 (1995), 253–272. [43] M. Pagliacci, M.A. Picardello, Heat diffusion on homogeneous trees, Adv. Math 100 (1995), 175–190. [44] J. Cohen, F. Colonna, M.A. Picardello, Image reconstruction from exponential blurring, Circuits, Systems, Signal Process. 15 (1996), 261–274. [45] M.A. Picardello, Characterizing harmonic functions by mean value properties on trees and symmetric spaces, Contemp. Math. 206 (1997), 161–163. [46] E. Casadio-Tarabusi, J.M. Cohen, A. Kor´ anyi, M.A. Picardello, Converse mean value theorems on trees and symmetric spaces, Jour. Lie Theory 8 (1998), 229–254. [47] M.A. Picardello, The geodesic Radon transform on trees, in “Harmonic Analysis and Integral Geometry”, CRC/Chapman Hall (2000). [48] E. Casadio-Tarabusi, S.G. Gindikin, M.A. Picardello, The circle Radon transform on trees, Diff. Geom. and Applications 19 (2003), 295–305. [49] N. Arcozzi, E. Casadio-Tarabusi, F. Di Biase, M.A. Picardello, A potential theoretic approach to twisting, in “New Trends in Potential Theory”, The Theta Foundation, Bucharest (2005), 3–15. [50] N. Arcozzi, E. Casadio-Tarabusi, F. Di Biase, M.A. Picardello, Twist points of planar domains, Trans. Amer. Math. Soc. 358 (2006), 2781–2798. [51] E. Casadio-Tarabusi, M.A. Picardello, The algebras generated by the Laplace operators in a semi-homogeneous tree, preprint. [52] L. Atanasi, M.A. Picardello, The Lusin area function and local admissible convergence of harmonic functions on homogeneous trees, Trans. Amer. Math. Soc. 360 (2008), 3327–3343. [53] J.M. Cohen, M. Pagliacci, M.A. Picardello, Radial heat diffusion from the root of a semi-homogeneous tree and the combinatorics of paths, Boll. Un. Mat. It. 1 (3) (2008), 619–628. [54] F. Andreano, M.A. Picardello, Approximate identities on some homogeneous Banach spaces, Monashefte Math. 158 (2009), 235–246. [55] M.A. Picardello, Local admissible convergence of harmonic functions on non-homogeneous trees, in print in Colloquium Math. Books [1] A. Fig` a-Talamanca, M.A. Picardello, “Harmonic Analysis on Free Groups ”, Lecture Notes in Pure and Appl. Math. 87, M. Dekker, New York–Basel, 1983. [2] S. Campi, M.A. Picardello, G. Talenti, “ Analisi Matematica e Calcolatori”, Boringhieri, Torino, 1990. [3] M.A. Picardello (ed.), “Harmonic Analysis and Discrete Potential Theory”, Plenum Publishing Co. 1992. [4] W. Baldoni, M.A. Picardello (eds.), “Representation Theory of Lie Groups and Quantum Groups”, Pitman Research Notes in Math. 311, Longman, Harlow, Essex, 1994.

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[5] E. Casadio Tarabusi, M.A. Picardello, G. Zampieri (eds.), “Integral Geometry, Radon Transforms and Complex Analysis”, Lecture Notes in Math. 1684, Springer, Berlin, Heidelberg, New York, 1998. [6] M.A. Picardello, W. Woess (eds.), “Random Walks and Discrete Potential Theory”, Cambridge University Press Symp. Math., Cambridge University Press, Cambridge, 1999. [7] M.A. Picardello (ed.), “Harmonic Analysis and Integral Geometry ”, CRC/Chapman Hall, 2000. [8] A. D’Agnolo, E. Casadio Tarabusi, M.A. Picardello (eds.), “ Representation Theory and Complex Analysis”, Lecture Notes in Math. 1931 (2006), Springer, Berlin, Heidelberg, New York. [9] M.A. Picardello, “Analisi di Fourier e trattamento numerico dei segnali”, www.mat.uniroma2.it/ picard/SMC/ didattica/materiali did/An.Arm./LIBRO.pdf [10] M.A. Picardello, L. Zsid´ o, “Appunti di Algebra Lineare”, http://www.mat.uniroma2.it/ picard/SMC/ didattica/materiali did/Alg.Lin./AlgLin.pdf [11] M.A. Picardello, “Algoritmi e metodi numerici, analitici e statistici in Computer Graphics”, www.mat.uniroma2.it/∼picard/SMC/ didattica/materiali did/Comp.Graph./Note di Computer Graphics.pdf [12] M.A. Picardello, “Elaborazione digitale di immagini con Adobe Photoshop”, www.mat.uniroma2.it/∼picard/SMC/didattica/materiali did/Photoshop/ Libro Photoshop.pdf [13] M.A. Picardello, “Il linguaggio Java”, www.mat.uniroma2.it/∼picard/SMC/ didattica/materiali did/Java/Matematica Computazionale/ Matem Computazionale.pdf [14] A. Pantano, M.A. Picardello, “Rappresentazioni di SL2 (R)”, in preparation.

Vadim A. Kaimanovich Research Publications [1] A.M. Vershik, V.A. Kaimanovich, Random walks on groups: boundary, entropy and uniform distribution, Dokl. Akad. Nauk SSSR, 249 (1979), 15–18 (Russian); English translation: Soviet Math. Dokl., 20 (1979), 1170–1173. [2] V.A. Kaimanovich, Spectral measure of the transition operator and harmonic functions connected with random walks on discrete groups, Zapiski Nauchn. Sem. LOMI, 97 (1980), 102–109 (Russian); English translation; J. Soviet Math., 24 (1984), 550– 555. [3] V.A. Kaimanovich, Boundaries of random walks on discrete groups, Diploma (MSc) Thesis, Leningrad University, 1980 (Russian). [4] V.A. Kaimanovich, Boundaries of random walks on discrete groups, Teoriya Veroyatn. i ee Prim., 26:3(1981), 637–639 (Russian); English translation: Theory Probab. Appl., 26:3(1981), 624–625. [5] V.A. Kaimanovich, A topological model of the boundary for random walks on groups, VINITI publ. 5052–81, 1981 (Russian). [6] V.A. Dymshits, V.A. Kaimanovich, On the problem of the genetic code structure, Proceedings of the Annual Conference of Young Scientists, Tartu University, 1981 (Russian). [7] V.A. Kaimanovich, Examples of non-commutative groups with non-trivial exit boundary, Zapiski Nauchn. Sem. LOMI, 123 (1983), 167–184 (Russian); English translation: J. Soviet Math., 28 (1985), 579–591. [8] V.A. Kaimanovich, The differential entropy of the boundary of a random walk on a group, Uspekhi Mat. Nauk, 38:5(1983), 187–188 (Russian); English translation: Russian Math. Surveys, 38:5(1983), 142–143. [9] V.A. Kaimanovich, A.M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Probab., 11 (1983), 457–490. [10] V.A. Kaimanovich, A complete description of the tail sigma-algebra of random walks and related problems, Teoriya Veroyatn. i ee Prim., 30:1(1985), 189–190 (Russian); English translation: Theory Probab. Appl., 30:1(1985), 207–208. [11] V.A. Kaimanovich, An entropy criterion for maximality of the boundary of random walks on discrete groups, Dokl. Akad. Nauk SSSR, 280 (1985), 1051–1054 (Russian); English translation: Soviet Math. Dokl., 31 (1985), 193–197. [12] V.A. Kaimanovich, The uniform distribution on compact homogeneous spaces and the Kantorovich-Rubinshtein metric, Teoriya Veroyatn. i ee Prim., 30:4(1985), 779– 782 (Russian); English translation: Theory Probab. Appl., 30:4(1985), 828–831. [13] V.A. Kaimanovich, A global law of large numbers for the Lie groups, Fourth International Vilnius Conference on Probability Theory and Mathematical Statistics, Abstracts of Communications, Akad. Nauk Litovsk. SSR, Vilnius, 1985, 2, 9–11 (Russian). [14] V.A. Kaimanovich, Boundaries of random walks on discrete groups, Candidate of Sciences (Ph. D.) Thesis, Leningrad University, 1985 (Russian). [15] V.A. Kaimanovich, Brownian motion and harmonic functions on covering manifolds. An entropy approach, Dokl. Akad. Nauk SSSR, 288 (1986), 1045–1049 (Russian); English translation: Soviet Math. Dokl., 33 (1986), 812–816.

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[16] V.A. Kaimanovich, Boundaries of random walks on polycyclic groups and the law of large numbers for solvable Lie groups, Vestnik Leningrad. Univ. Mat. Mekh. Astronom., 1987, vyp. 4, 93–95 (Russian); English translation: Vestnik Leningrad University: Mathematics, 20:4(1987), 49–52. [17] V.A. Kaimanovich, Lyapunov exponents, symmetric spaces and a multiplicative ergodic theorem for semi-simple Lie groups, Zapiski Nauchn. Sem. LOMI, 164 (1987), 30–46 (Russian); English translation: J. Soviet Math., 47 (1989), 2387–2398. [18] V.A. Kaimanovich, Brownian motion on manifolds and Markov chains, Abstracts of Communications at the Leningrad Probability Seminar, 1987 (Russian). [19] V.A. Kaimanovich, Brownian motion on foliations: entropy, invariant measures, mixing, Funktsional. Anal. i Prilozhen., 22:4(1988), 82–83 (Russian); English translation: Funct. Anal. Appl., 22:4(1988), 326–328. [20] V.A. Kaimanovich, Boundary and entropy of random walks in random environment, Fifth International Vilnius Conference on Probability Theory and Mathematical Statistics, Abstracts of Communications, Akad. Nauk Litovsk. SSR, Vilnius, 1989, 1, 234–235. [21] V.A. Kaimanovich, The entropy and the Liouville property of Riemannian manifolds, Uspekhi Mat. Nauk, 44:4(1989), 225–226 (Russian); English translation: Russian Math. Surveys, 44:4(1989), 195–196. [22] V.A. Kaimanovich, Harmonic and holomorphic functions on coverings of complex manifolds, Mat. Zametki, 46:5(1989), 94–96 (Russian). [23] V.A. Kaimanovich, E.M. Krupitski, A.V. Spirov, The possible contribution of intracellular electric fields to oriented assemblage of microtubules, Journal of Bioelectricity, 8 (1989), 243–245. [24] V.A. Kaimanovich, Boundary and entropy of random walks in random environment, Probability Theory and Mathematical Statistics, Fifth International Conference, Vilnius, 1989 (B. Grigelionis, Yu.V. Prohorov, V.V. Sazonov, V. Statulivicius eds.), Mokslas-VSP, Vilnius-Utrecht, 1990, 1, 573–579. [25] V.A. Kaimanovich, Invariant measures of the geodesic flow and measures at infinity on negatively curved manifolds, Ann. Inst. H. Poincar´e, Phys. Th´eor., 53 (1990), 361–393. [26] V.A. Kaimanovich, E.M. Krupitski, A.V. Spirov, Possible role of intracellular electric fields in microtubule assembly orientation, Biofizika, 35 (1990), 603–604 (Russian). [27] V.A. Kaimanovich, Bowen-Margulis and Patterson measures on negatively curved compact manifolds, Dynamical Systems and Related Topics, Nagoya, 1990 (K. Shiraiwa ed.), World Sci. Publishing, River Edge, NJ, 1991, 223–232. [28] V.A. Kaimanovich, Poisson boundaries of random walks on discrete solvable groups, Probability Measures on Groups X, Oberwolfach, 1990 (H. Heyer ed.), Plenum, New York, 1991, 205–238. [29] V.A. Kaimanovich, Dirichlet norms, capacities and generalized isoperimetric inequalities for Markov operators, Potential Anal., 1 (1992), 61–82. [30] V.A. Kaimanovich, W. Woess, The Dirichlet problem at infinity for random walks on graphs with a strong isoperimetric inequality, Probab. Theory Related Fields, 91 (1992), 445–466.

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[31] V.A. Kaimanovich, Bi-harmonic functions on groups, C. R. Acad. Sci. Paris S´er. I Math., 314 (1992), 259–264. [32] V.A. Kaimanovich, Discretization of bounded harmonic functions on Riemannian manifolds and entropy, Potential Theory, Nagoya, 1990 (M. Kishi ed.), de Gruyter, Berlin, 1992, 213–223. [33] V.A. Kaimanovich, Measure-theoretic boundaries of Markov chains, 0-2 laws and entropy, Harmonic Analysis and Discrete Potential Theory, Frascati, 1991 (M.A. Picardello ed.), Plenum, New York, 1992, 145–180. [34] V.A. Kaimanovich, O.V. Narvskaya, V.V. Babkov, L.A. Kaftyreva, Computer-aided statistical analysis of the biological properties of Salmonella Typhimurium, J. Microbiol., 1992, no. 1, 70 (Russian). [35] V.A. Kaimanovich, The Poisson boundary of hyperbolic groups, C. R. Acad. Sci. Paris S´er. I Math., 318 (1994), 59–64. [36] V.A. Kaimanovich, E.M. Krupitski, A.V. Spirov, Electrical activity of biomembranes and vectorization of intracellular processes, Electro- and Magnetobiology, 13 (1994), 149–158. [37] V.A. Kaimanovich, Ergodicity of harmonic invariant measures for the geodesic flow on hyperbolic spaces, J. Reine Angew. Math., 455 (1994), 57–103. [38] D. Cartwright, V.A. Kaimanovich, W. Woess, Random walks on the affine group of local fields and of homogeneous trees, Ann. Inst. Fourier (Grenoble), 44 (1994), 1243–1288. [39] V.A. Kaimanovich, The Poisson boundary of covering Markov operators, Israel J. Math., 89 (1995), 77–134. [40] V.A. Kaimanovich, The Poisson boundary of polycyclic groups, Probability measures on groups and related structures, XI, Oberwolfach, 1994 (H. Heyer ed.), World Sci. Publishing, River Edge, NJ, 1995, 182–195. [41] V.A. Kaimanovich, W. Woess, Construction of discrete, non-unimodular hypergroups, Probability measures on groups and related structures, XI, Oberwolfach, 1994 (H. Heyer ed.), World Sci. Publishing, River Edge, NJ, 1995, 196–209. [42] V.A. Kaimanovich, Boundaries of invariant Markov operators: the identification problem, Ergodic Theory of Zd actions, Warwick, 1993–1994 (M. Pollicott, K. Schmidt eds.), London Math. Soc. Lecture Note Ser. 228 (1996), 127–176. [43] V.A. Kaimanovich, H. Masur, The Poisson boundary of the mapping class group, Invent. Math., 125 (1996), 221–264. [44] V.A. Kaimanovich, E.M. Krupitski, A.V. Spirov, Electrical activity of biomembranes and oriented assemblage of microtubules in neurones, Suppl. “Consciousness Research Abstracts”, J. Consciousness Studies, 3 (1996), 73. [45] V.A. Kaimanovich, Harmonic functions on discrete subgroups of semi-simple Lie groups, Contemp. Math., 206 (1997), 133–136. [46] V.A. Kaimanovich, Hopf-Tsuji-Sullivan theorem, Encyclopedia of Mathematics, Kluwer, Dordrecht, 1997, 300–301. [47] V.A. Kaimanovich, Gromov hyperbolic space, Encyclopedia of Mathematics, Kluwer, Dordrecht, 1997, 277–278. [48] V.A. Kaimanovich, Hopf alternative, Encyclopedia of Mathematics, Kluwer, Dordrecht, 1997, 294–296.

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[49] V.A. Kaimanovich, Amenability, hyperfiniteness and isoperimetric inequalities, C. R. Acad. Sci. Paris, S´er. I 325 (1997), 999–1004. [50] V.A. Kaimanovich, H. Masur, The Poisson boundary of Teichm¨ uller space, J. Funct. Anal. 156 (1998), 301–332. [51] V.A. Kaimanovich, Hausdorff dimension of the harmonic measure on trees, Ergodic Theory Dynam. Systems 18 (1998), 631–660. [52] V.A. Kaimanovich, A. Fisher, A Poisson formula for harmonic projections, Ann. Inst. H. Poincar´e Prob. Stat. 34 (1998), 209–216. [53] V.A. Kaimanovich, A discrete time Harnack inequality and its applications, Random Walks and Discrete Potential Theory, Cortona, 1997 (M. Picardello, W. Woess eds.), Cambridge Univ. Press, Symposia Mathematica 29 (1999), 214–230. [54] V.A. Kaimanovich, Ergodicity of the horocycle flow, Dynamical Systems, LuminyMarseille, 1998, World Sci. Publishing River Edge, NJ, 2000, 274–286. [55] V.A. Kaimanovich, Ergodic properties of the horocycle flow and classification of Fuchsian groups, J. Dynam. Control Systems 6 (2000), 21–56. [56] V.A. Kaimanovich, The Poisson formula for groups with hyperbolic properties, Ann. of Maths. 152 (2000), 659–692. [57] V.A. Kaimanovich, Equivalence relations with amenable leaves need not be amenable, Topology, Ergodic Theory, Real Algebraic Geometry, Amer. Math. Soc. Transl. (Ser. 2) 202 (2001), 151–166. [58] V.A. Kaimanovich, W. Woess, Boundary and entropy of space homogeneous Markov chains, Ann. Probab. 30 (2002), 323–363. [59] V.A. Kaimanovich, Non-Euclidean affine laminations, Foliations: geometry and dynamics (Warsaw, 2000), World Sci. Publishing, River Edge, NJ, 2002, 333–349. [60] V.A. Kaimanovich, K. Schmidt, Ergodicity of cocycles. I: General theory, preprint, 2000. [61] V.A. Kaimanovich, Y. Kifer, B.-Z. Rubshtein, Boundaries and harmonic functions for random walks with random transition probabilities, J. Theoret. Probab. 17 (2004), 605–646. [62] V.A. Kaimanovich, SAT actions and ergodic properties of the horosphere foliation, Rigidity in Dynamics and Geometry (Cambridge, 2000), Springer, Berlin, 2002, 261–282. [63] V.A. Kaimanovich, The Poisson boundary of amenable extensions, Monatsh. Math. 136 (2002), 9–15. [64] V.A. Kaimanovich, Random walks on Sierpinski graphs: hyperbolicity and stochastic homogenization, in: Fractals in Graz 2001, Birkh¨ auser, Basel, 2002, 145–183 [65] S. Kh. Aranson, V.Z. Grines, V.A. Kaimanovich, Classification of supertransitive 2-webs on surfaces, J. Dynam. Control Systems 9 (2003), 455–468. [66] V.A. Kaimanovich, Double ergodicity of the Poisson boundary and applications to bounded cohomology, GAFA, 13 (2003), 852–861. [67] V.A. Kaimanovich, Boundary amenability of hyperbolic spaces, Contemp. Math., 347 (2004), 83–111. [68] V.A. Kaimanovich, Amenability and the Liouville property, Israel J. Math., 149 (2005), 45–85. [69] V.A. Kaimanovich, “M¨ unchhausen trick” and amenability of self-similar groups, Internat. J. Algebra Comput. 15 (2005), 907–937.

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[70] V.A. Kaimanovich, I. Kapovich, P. Schupp, Generic stretching factors for free group automorphisms, Israel J. Math. 157 (2007), 1–46. [71] V.A. Kaimanovich, Self-similarity and random walks. In: Fractal Geometry and Stochastics IV. Progress in Probability 61, Birkh¨ auser, 2009, pp. 45–70. [72] V.A. Kaimanovich, F. Sobieczky, Stochastic homogenization of horospheric tree products. In: Probabilistic Approach to Geometry. Advanced Studies in Pure Mathematics 57, Mathematical Society of Japan, 2010, pp. 199–229. [73] L. Bartholdi, V.A. Kaimanovich, V. Nekrashevych, On amenability of automata groups, Duke Math. J., to appear (2010); available at arXiv:0802.2837 (February 2008). [74] V.A. Kaimanovich, Hopf decomposition and horospheric limit sets, Ann. Acad. Sci. Fenn. Math., to appear (2010); available at arXiv:0807.0995 (July 2008). [75] V.A. Kaimanovich, V. Le Prince, Matrix random products with singular harmonic measure, Geom. Dedicata, to appear (2010); available at arXiv:0807.1015 (July 2008). [76] R.I. Grigorchuk, V.A. Kaimanovich, T. Nagnibeda, Ergodic properties of boundary actions and Nielsen–Schreier theory, arXiv:0901.4734 (January 2009). In preparation: [77] T. B¨ uhler, V.A. Kaimanovich, Markov operators on groupoids and amenability. [78] P. Freitas, V.A. Kaimanovich, Compactifications of symmetric spaces. [79] V.A. Kaimanovich, Boundary behaviour of Thompson’s group. [80] V.A. Kaimanovich, Differential properties of Gibbs measures on negatively curved manifolds. [81] V.A. Kaimanovich, Poisson boundary of discrete groups: a survey. [82] V.A. Kaimanovich, V. Le Prince, Random walks with maximal entropy on free products. [83] V.A. Kaimanovich, K. Schmidt, Ergodicity of cocycles. II. Geometric applications. [84] V.A. Kaimanovich, F. Sobieczky, Horospheric products of random trees. Books [1] N. Martin, J. England, Mathematical Theory of Entropy, Mir, Moscow, 1988, translation into Russian and editorial comments. [2] V.A. Kaimanovich (ed.) Random walks and geometry (Vienna, 2001), de Gruyter, 2004. [3] V.A. Kaimanovich, M. Lyubich, Conformal and harmonic measures on laminations associated with rational maps, AMS, 2005. [4] V.A. Kaimanovich, A.A. Lodkin (eds.), Representation Theory, Dynamical Systems, and Asymptotic Combinatorics, AMS, 2006. In preparation: [5] V.A. Kaimanovich, B.-Z. Rubshtein, Partitions in ergodic theory and probability. [6] V.A. Kaimanovich, Boundary and entropy of random walks on countable groups. [7] V.A. Kaimanovich, Amenability beyond groups.

Progress in Probability, Vol. 64, 1–14 c 2011 Springer Basel AG


An Inaccessible Graph M.J. Dunwoody Abstract. An inaccessible, vertex transitive, locally finite graph is described. This graph is not quasi-isometric to a Cayley graph. Mathematics Subject Classification (2000). Primary 05C63; Secondary 05E18. Keywords. Ends of graphs, quasi-isometry.

1. Introduction Let X be a locally finite connected graph. A ray is a sequence of distinct vertices v0 , v1 , . . . such that vi is adjacent to vi+1 for each i = 1, 2, . . . . Obviously for a ray to exist, the graph X has to be infinite. For any two vertices u, v ∈ V X let d(u, v) be the length of a shortest path joining u, v. We say that two rays R, R
belong to the same end ω, if for no finite subset F of V X or EX do R1 and R2 eventually lie in distinct components of X \ F . We define E(X) to be the set of ends of X. We say that ω is thin if it does not contain infinitely many vertex disjoint rays. As in [16] the end ω is said to be thick if it is not thin. In their nice paper [16] Thomassen and Woess define an accessible graph. A graph X is accessible if there is some natural number k such that for any two ends ω1 and ω2 of X, there is a set F of at most k vertices in X such that F separates ω1 and ω2 , i.e., removing F from X disconnects the graph in such a way that rays R1 , R2 of ω1 , ω2 respectively eventually lie in distinct components of X \ F . A finitely generated group G is said to have more than one end (e(G) > 1) if its Cayley graph X(G, S) with respect to a finite generating set S has more than one end. This property is independent of the generating set S chosen. Stallings [14] showed that if e(G) > 1 then G splits over a finite subgroup, i.e., either G = A ∗C B where C is finite, C = A, C = B or G is an HNN extension G = A∗C = A, t|t−1 ct = θ(c), where C is finite, C ≤ A and θ : C → A is an injective homomorphism. A group is accessible if the process of successively factorizing factors that split in a decomposition of G eventually terminates with factors that are finite or one ended.

2

M.J. Dunwoody

Thomassen and Woess show that the Cayley graph of a finitely generated group G is accessible if and only if G is accessible. In [5, 6] I have given examples of inaccessible groups, and so not every locally finite connected graph is accessible. Let ω be an end of X. As in [16], p. 259 define k(ω) to be the smallest integer k such that ω can be separated from any other end by at most k vertices. If this number does not exist, put k(ω) = ∞. Thomassen and Woess show that X is accessible if and only if k(ω) < ∞ for every end ω. We say that an end ω is special if k(ω) = ∞. In this paper we construct a locally finite, connected, inaccessible, vertex transitive graph X. The property of being inaccessible is invariant under quasiisometry. If X, Y are graphs, then a quasi-isometry θ : X → Y induces a bijection E(θ) : E(X) → E(Y ) which takes thick ends to thick ends, and special ends to special ends. One can put a topology on E(X) in a natural way. The map E(θ) is then a homeomorphism. Woess asked in [17, 15] if every vertex transitive, locally finite graph is quasiisometric to a Cayley graph. It was shown in [11, 12] that the Diestel-Leader graph DL(m, n), m = n (see [3] or [17]) is not quasi-isometric to a Cayley graph, answering the question of Woess. It is shown here that the graph X is another example. I originally thought that X was hyperbolic, and the fact that X was not quasi-isometric to a Cayley graph then followed because a hyperbolic group is finitely presented, and would therefore have an accessible Cayley graph by [4]. However there are arbitrarily large cycles in X for which the distance apart of two vertices in the cycle is the same as that in X. This cannot happen in a hyperbolic graph. It seems likely that a hyperbolic graph must be accessible. The vertex transitive graph X we construct is based on a construction in [7]. In that paper, Mary Jones and I construct a finitely generated group G for which G∼ = A ∗C G where C is infinite cyclic. The vertex set of the graph X is the set of left cosets of D in G, where D has index 2 in C. One could take the vertex set of X to be the left cosets of A or C as they are commensurable with D. In fact it is easier to work with a G-graph Y quasi-isometric to X, in which there are two orbits of vertices for the action of G on Y . In general, if a group G is the commensurizer of a subgroup H, and G is generated by H ∪ S, then one can construct a vertex transitive, connected graph, in which the vertices are the cosets of H, and there are edges (H, sH) for each s ∈ S. If G actually normalizes H, then this graph is a Cayley graph for G/H. Conversely if X is a connected, vertex transitive, locally finite graph and H is the stabilizer of a vertex v, then G is the commensurizer of H and G is generated by H ∪ S, where S is any subset of G with the property that for each u adjacent to v there is an s ∈ S such that sv = u. The graph Y has an orbit of cut points, i.e., vertices whose removal disconnects the graph. It is well known that cut points in a graph give rise to a tree decomposition. This is described – for example – in [10], in which the theory of structure trees is extended to graphs that can be disconnected by removing finitely many vertices rather than finitely many edges. The cut point tree T for Y has two

An Inaccessible Graph

3

orbits of vertices under G. One orbit corresponds to the set of 2-blocks, where each 2-block is a maximal 2-connected subgraph, and the other orbit corresponds to the cut points. It is then shown that after a subdivision and two folding operations, each of which is a quasi-isometry, and removing spikes (a spike is an edge with a vertex of degree one) each 2-block becomes a graph isomorphic to Y . Thus the graph Y has a self-similarity property that comes from the fact that G ∼ = A∗C G where C is infinite cyclic. One would not expect this to happen in a Cayley graph, as it is not possible that for a finitely generated group G to be isomorphic to A ∗C G where C is finite. This follows from a result of Linnell [13], which indicates that in a process of successively factorizing factors that split in a decomposition of an inaccessilbe group G, the size of the finite groups over which the factors split must increase. Thus after carrying out the subdivision and folding operations, the graph Y = Y1 becomes a graph Y2 which has a single orbit of disconnecting edges. Removing (the interior of) all these edges will give a single orbit of points each with stabilizer a conjugate of A, and a second orbit, consisting of 2-blocks each of which is isomorphic to Y , with stabilizer conjugate to the subgroup of G which is the second factor in the decomposition G ∼ = A ∗C G. If we repeat this process n − 1 times, then we a obtain a graph Yn which has n − 1 orbits of disconnecting edges. Removing these edges produces n − 1 orbits of vertices each of which has finite stabilizer, isomorphic to A, and a single orbit of 2-blocks each of which is isomorphic to Y . Let Bn be one of these blocks. The graph Y has an orbit of subgraphs each of which is a trivalent tree. Let Z be a particular trivalent subtree of Y . Although the folding operations do involve folding Z, the result of the operations is another trivalent tree. We will see that any two rays in Z represent a particular special end ω of Y . There will also be uncountably many special ends that do not correspond to a translate of Z. A ray representing a special end must eventually lie in a translate of Bn , since otherwise it will represent a thin end. However the initial number xn of points in the ray outside a translate of Bn may tend to infinity with n. There will be uncountably many such special ends. If the ray eventually ends up in a translate of Z, then xn is bounded, since each translate of Z lies in a translate of Bn . Since each translate of Bn contains a translate of Z, the orbit of ω is dense in the space of special ends. We will show that in a Cayley graph, if there is a countable set of special ends which is dense in the subspace of all special ends, then there must be a special end corresponding to a 1-ended subgraph. There is no special end of Y corresponding to a 1-ended subgraph, and so the graph Y cannot be quasi-isometric to a Cayley graph. As it is important in our construction, we repeat the description of G below. In another paper [8], Mary Jones and I went on to construct a finitely generated group G1 for which G1 ∼ = G1 ∗C1 G1 with C1 infinite cyclic. It might be expected that the coset graph X1 of C1 in G1 has similar properties to X. This will not be the case. Although X1 is inaccessible and locally finite, it is quasi-isometric to a Cayley graph. This is because C1 contains a central subgroup Z as a subgroup of finite index. Then X1 is quasi-isometric to the Cayley graph of G1 /Z.

4

M.J. Dunwoody A

3 a = a1

6

3

a2 b = a0

a 3 = d1

3 2 a 6 = d0 Figure 1. Subgroup lattice in A

2. The graph We recall the group G constructed in [7]. Let A = a, b|b3 = 1, a−1 ba = b−1 . As noted in [7], a2 is in the centre of A and A/a2  ∼ = S3 . Also A is generated by a3 2 −3 2 3 2 −1 2 and a b since a (a b)a = a b , and so b = b−1 ∈ a3 , a2 b. The group A has a lattice of subgroups as in Fig. 1. Put x = a3 , y = a2 b. Then, since a2 is central x2 = y 3 . Also y −1 x = y 2 x−1 = −1 b a and (y −1 x)2 = b−1 ab−1 a = a2 , so (y −1 x)6 = x2 . We have y −1 xy = a2 b−1 , and so y −1 xyx−1 = b and (y −1 xyx−1 )3 = 1. Also a = a3 a−2 = x(y −1 x)−2 = yx−1 y. Note – we use it later – that (xy)6 = (y −1 x−1 )−6 = (y −1 xx−2 )−6 = (y −1 x)−6 x12 = x10 . The group G is generated by four elements a, b, c and d, subject to an infinite set of defining relations as follows. Firstly c−1 dc = d2 , so that c, d generate a subgroup B isomorphic to the soluble Baumslag-Solitar group BS(1, 2). Also a3 = d, together with the relations of A, b3 = 1, a−1 ba = b−1 . The remaining relations are defined inductively. Put d = d1 , a = a1 and di+1 = cdi c−1 so that d2i+1 = di . Put d0 = d21 and a0 = a2 b. Then, as above, the subgroup A = a, b = a0 , d1 . Now −1 −1 define inductively ai+1 = ai d−1 i+1 ai , bi+1 = ai di+1 ai di+1 and add the relations

An Inaccessible Graph

5

A2

A = A1

a2

a = a1 2 a2 b = a0 2 3

a = d1 2 a 6 = d0 Figure 2. Subgroup lattice in G

−1 ∼ b3i+1 = 1, a−1 i+1 bi+1 ai+1 = bi+1 for each i to make Ai+1 = ai+1 , bi+1  = A. Note −1 −1 that for i = 1 we have a = a1 = a0 d1 a0 = yx y as above. The group G is best understood in terms of the subgroup lattice shown in Fig. 2 and the folding sequence shown in Fig. 3. Folding operations are described in [7]. The sequence here only involves Type II folds and vertex morphisms. In a Type II fold, edges in the same orbit are folded together. The stabilizer of a representative edge in the orbit is increased from E to E, g, and the stabilizer of the orbit of the terminal vertex is increased from U to U, g. Here g is an element of the representative vertex group V of the initial vertex. It is possible that the initial vertex and terminal vertex are in the same orbit, i.e., U = V . A vertex morphism involves a homomorphism of a particular vertex group that restricts to an isomorphism on any incident edge group. Such a homomorphism induces a morphism of the trees associated with the graph of groups and a homomorphism of the corresponding fundamental groups. These morphisms are described in detail in [9]. In fact we do not use vertex morphisms in our construction as explained below. In [7] it is shown that G ∼ = A ∗C G where A = a, b = a0 , d1  and C = a1 . Let D = d1 . Let Y be the G-graph with two orbits of vertices V Y = {gA, gD|g ∈ G} and two orbits of edges EY = {(gA, gD), (gD, gcD)|g ∈ G}.

6

M.J. Dunwoody a31 = d1

A = A1

B

(subdivision) A1

B

(Type II folds) A1

a1

a1 , d1 

d2

B

a32 = d2

B

(vertex morphism) A1

a1 = a22 b2

A2

(repeating process) A1

a1

A2

a2

A3

a33 = d3

B

(repeating process infinitely many times) A1

A2

A3

B

Figure 3. Folding sequence of graph of groups In Y the vertex A is incident with [A, D] = 9 edges, as is every vertex in the same orbit. The vertex D is incident with 4 edges. One edge in one edge orbit connects D to A and there are three edges in the other orbit connecting D to cD, c−1 D and dc−1 D. Note that d = d1 fixes the edge (D, cD) and transposes the edges (D, c−1 D), (D, dc−1 D). If one removes the edges of Y in the first orbit one is left with a set of 3-regular trees. If one directs these subgraphs by putting an arrow from D to cD, then every vertex has one edge pointing away from it and two pointing towards it. The graph Y is connected because G is generated by A, D and c. One obtains a vertex transitive G-graph X from Y by taking the orbit of vertices containing D and joining two vertices by an edge if they are joined by an edge in Y , or they are not joined by an edge in Y but are distance two apart in Y . In X each vertex will have degree 10. Thus D is a vertex in X. It has 2 vertices adjacent to it which were already adjacent to it in Y . The one vertex in Y adjacent to D in Y which is not in X has 9 adjacent vertices including D itself, the 8 other vertices will be adjacent to D in X. It is easier to work with the graph Y , which

An Inaccessible Graph A = A1

a31 = d1

7 d1

d1

(Type II fold) A1

A1

A1

A1

d2 (subdivision and Type II folds) A2 a1 = a22 b2 a32 = d2

d2

(subdivision and Type II folds) a1 A2 a2 A3 a33 = d3 d3 (repeating infinitely many times) A2 A3

d2

d2

d3

d1 , d2 , d3 , . . . 

Figure 4. Folding sequence of graphs is quasi-isometric to X. In Fig. 4 a sequence of folding operations is described for the graph Y . These are similar to those of Fig. 3. However in Fig. 3 the operations are for trees. Vertex morphisms are included which change the group acting. In Fig. 4 the operations are on G-graphs in which the group acting remains the same throughout. Thus we are assuming that all the vertex morphisms have been carried out before we start. The first diagram in Fig. 4 shows the graph G\Y . Each edge of the quotient graph is labelled by its stabilizer in a lift to Y , as in Bass-Serre theory. The first folding operations results in the edges at D (labelled with a • in Fig. 4) in the same d orbit being folded together. The stabilizer of D is increased, as are the stabilizers of all the edges in the orbit of (D, cD). A new stabilizer includes the original stabilizer as a subgroup of index two. The degree of D changes to 5 as D is identified with d2 D and the two edges (A, D), (d2 A, D) are now incident with the new vertex. The graph still contains 3-regular trees as before. The next operation, which is subdivision, inserts a ◦ vertex on each edge of the orbit containing (A, D). The next operation comprises folding from the ◦ vertex labelled A1 and the • vertex labelled D towards the new ◦ vertex just created. In the folding from A, edges corresponding to the three cosets of D which belong to a are folded together. The vertex A will then have degree 3. In the folding from D, the edges in the same d2  are folded together, so that the degree of D again becomes 4. The defining relations for G ensure that the created vertex has stabilizer A2 . In the graph now obtained, the vertices in the orbit of A = A1 are cut points. Removing one of these vertices gives three components. Removing all the vertices in this orbit with all their incident edges gives an infinite set of

8

M.J. Dunwoody

component graphs, each of which is isomorphic to Y . Thus one can repeat this process on each of these graphs as indicated in Fig. 4. In fact in Y , before any folding operation, each of the ◦ vertices is a cut point. There is a tree decomposition of Y as in [10], in which the G-tree T has two orbits of vertices. One orbit corresponds to the 2-blocks, where each 2-block corresponds to a maximal 2-connected subgraph, and the other orbit corresponds to the cut points. Removing a particular ◦ vertex from Y results in 3 components. In each 2-block a ◦-vertex has degree 3. If we consider a cycle in Y , then under the successive subdivision and folding operations, the cycle will eventually be a subtree. But there will have to be at least one fold at a vertex at each stage in the iteration. Thus if we start with a cycle with k edges then after k iterations, the image of the cycle will be a subtree. We give a more precise explanation of how this happens after considering an example. To illustrate the remarks above, in Fig. 5 and Fig. 6 the effect of the folding sequence is shown on a particular cycle in Y . This particular cycle is reduced to a subtree after one iteration of the folding sequence. Probably it is a shortest cycle in Y . Vertices in the orbit of A in Y are indicated by a small •, the other vertices are indicated by a larger •. Vertices created in the process by subdivision are indicated with a ◦. There are two orbits of edges. The ones in the orbit of edges incident with an A-orbit vertex are indicated with a continuous line. These are called solid edges. The others are indicated with a dashed line are called dashed edges. As the cycle passes through an A-orbit vertex, the different directions one can proceed correspond to the nine cosets of a3  in A. As folding takes place at each vertex in the cycle, the direction taken must correspond to one of the two cosets containing either a or a2 . The diagram shows the choice at each such vertex. (We always make the same choice a.) At a • vertex, if one is proceeding from a solid edge to a dashed edge, one proceeds along the only edge directed away from the vertex. (Recall that every • vertex has one dashed edge directed away from it and two directed towards it.) If one arrives at a • vertex along a dashed edge and leaves along a dashed edge, then one leaves along the other dashed edge directed towards the vertex. The first diagram indicates a uniqe path in Y which in fact turns out to be a cycle. The fact that one has a cycle is because (xy)6 = x10 ∈ D fixes an edge of Y . In general if one starts with a cycle in Y , then after one stage of the iteration the cycle will have become a closed path and folding will have taken place at at least one vertex. If the image is not already a subtree (as in the example above) then further folding must take place at the next stage. This folding must take place at a point that is at the end of a fold of the previous stage. Thus it is either at a • vertex which is at the end of two dashed edges which have been folded together and the new fold will also be between two dashed edges, or it will be at a ◦ and it will be between edges which come from two distinct folds at • vertices. It can be seen that the number of points where folding can take place is strictly less than at the previous stage. Thus if there are original cycle has k edges (or

An Inaccessible Graph

9

a

a

a

a

a

a

d2

d2

d2

d2

d2

d2

Figure 5. Folding a cycle I vertices) then its image is a subtree after k stages. In fact it will become a subtree after many less stages. In Fig. 7 we show a 60-cycle in Y that reduces to a tree after two stages of the iteration. After one stage it will be the like the first cycle of Fig. 5 with spikes attached. In Fig. 7 we preserve the previous convention that edges in the 3-regular subtrees are dashed, and the other edges are shown with continuous lines. There is a sequence Cn of cycles of increasing size such that Cn

10

M.J. Dunwoody

d2

d2 a

a

a

d2

d2 a

a a d2

x

d2

y

x

y

y

x

x y

y x

y

x

Figure 6. Folding a cycle II

An Inaccessible Graph

11

Figure 7. A 60-cycle folds to Cn−1 with spikes after one iteration of the folding sequence. If cn is the number of vertices of Cn , then c1 = 24, c2 = 60, c3 = 132. Each of these cycles is such that the distance between two vertices in the cycle is the same as that in Y . Let Z be a particular subgraph which is 3-regular tree consisting of dashed edges. The way the edges are oriented gives a height function φ : V Z → Z by defining φ(v0 ) = 0 for some fixed vertex v0 ∈ V Z, and such that if e is an oriented edge of Z with initial vertex ιe and terminal vertex τ e, then ∂φ(e) = φ(τ e) − φ(ιe) = 1. Two vertices of Z are joined by a path in Y in which the only vertices in Z are the end vertices if and only if the two vertices have the same height. The shortest such path will be much longer than the shortest path joining them

12

M.J. Dunwoody A¯1

A¯2

A¯3

A¯k

A¯k+1 , H

Figure 8. Structure tree Sn in Z. Thus, from Fig. 5, two vertices at the same height in Z that are distance two apart, are joined by a path in Y internally disjoint from Z of length 22 . And, from Fig. 7 two vertices at the same height in Z that are distance 4 apart, are joined by a path in Y of length 56 which is internally disjoint from Z. The fact that any two vertices at the same height are joined by a path outside Z means that any two rays in Z represent the same end. Consider the effect of a quasi-isometry on a graph U . Let θ : U → W be a quasi-isometry. Then θ induces a bijection E(θ) : E(U ) → E(V ) and E takes special ends to special ends. This is because if ω1 and ω2 are ends of U that are separated by a set of s vertices then E(θ)(ω1 ) and E(θ)(ω2 ) are separated by a set of f (s) vertices where f is a function of the form f (x) = cx + d. Thus k(ω) = ∞ if and only if k(E(θ)(ω)) = ∞. To clarify why the graph Y is not quasi-isometric to a Cayley graph, we construct an inaccessible Cayley graph with similar properties to Y , but point out the significant difference. We construct an inaccessible group using the lattice of Fig. 2. Let P be the subgroup of G generated by all the Ai ’s, i = 1, 2, . . . . It can be seen that P is the fundamental group of a graph of groups (G, N ) in which the underlying graph N has vertex set which is the natural numbers {1, 2, . . . } and there are edges (i, i + 1) for each i ∈ V N . The vertex group G(i) = Ai and the edge group corresponding to (i, i + 1) is generated by ai . In P the element d0 = a6 is central. If we form the quotient group P¯ = P/d0  then P¯ also has a graph of groups decomposition with the same underlying graph and in which the vertex group corresponding to i is A¯1 = Ai /d0 . Consider the subgroup D of P generated by d1 , d2 , . . . . This will be locally cyclic. After factoring out d0 we get a ¯ which is locally finite cyclic. In fact it is isomorphic to the additive group group D of dyadic rationals, i.e rationals of the form m/2n , where m, n are integers. Note ¯ and A¯1 . Let H be a that P is generated by D and A1 , so P¯ is generated by D ¯ Such finitely generated one-ended group that contains a subgroup isomorphic to D. a group certainly exists. Any countable group is contained in a finitely generated group and the direct product of a finitely generated group with a free abelian group ¯ = P¯ ∗D¯ H. This will be of rank two creates a one ended group. Form the group G an inaccessible group. The sequence Sn of structure trees for a Cayley graph of ¯ will be very similar to the sequence Tn of structure trees for Y . The structure G tree Sn is the fundamental group of the graph of groups shown in Fig. 8. An edge group of Sn will be a conjugate of the finite cyclic group ai /d0  for some i. Note ¯ so that it includes a generating however that we can choose a generating set for G ¯ will have a locally set for H, and then the corresponding Cayley graph W for G finite one-ended subgraph. This is the important difference with the graph Y .

An Inaccessible Graph

13

The graph Y has countably many subgraphs which are 3-regular trees. These subgraphs are a single orbit under the action of G. Let Z be one of these subgraphs. Let W be a Cayley graph, and suppose there are quasi-isometries θ : Y → W and φ : W → Y . Any two rays in Z represent the same end ω, and this end will be special. Since G is countable, the orbit containing this end is countable. It is also dense in the subspace of special ends. Let ω
= E(θ)(ω). Then ω
will be a special end of W , which is the Cayley graph of an inaccessible group Q. Since Q is inaccessible, there will be an infinite sequence Q = Q1 , Q2 , . . . where Qi has a decomposition as a free product with amalgamation over a finite subgroup in which Qi+1 is one of the factors, or Qi is an HNN-group with vertex group Qi+1 and finite edge group. At least one factor in each decomposition is inaccessible. If there is an infinite sequence of factorizations in which there is more than one inaccessible factor infinitely many times (as can in fact happen in some inaccessible groups) then there will be no countable orbit of special ends that is dense in the space of all special ends. Thus for any sequence of decompositions of factors of Q we will eventually obtain a term Qj that for each i > j we have that Qi = Qi+1 ∗Fi+1 Q
i+1 and Q
i+1 is accessible. In fact if Q
i+1 has an infinite one-ended factor, then Q would contain a thick end ω1 with k(ω1 ) finite. But Y contains no such thick end and so Q has no such end. We are then, very much, as in the situation of the example above, in which all the Q
i+1 factors are finite. Let Q
be the subgroup of Q generated by all the Q
i ’s. This will have a graph of groups decomposition with infinitely many factors, in which the Q
i ’s are the vertex groups. This group is not finitely generated. ˆ = ∩{Qi | i = 1, 2, . . . }. Then Q = Q ˆ ∗ ˆ Q
, where Fˆ is a locally Now put Q F finite subgroup of Q which is a union of an increasing sequence of finite subgroups ˆ must be finitely generated Fi
where Fi
≤ Fi . As in the example above, the group Q and one ended. It is finitely generated because Q is finitely generated, and when we write a generating set for Q as words given by the finite graph of groups ˆ by decomposition just described then we will get a finite set of generators for Q writing each generator of Q as a word in the elements of the vertex groups of ˆ It will have to the tree product and then taking those elements that are in Q. be one ended because if it split over a finite subgroup, then this decomposition will be induced by a similar decomposition of Q, since a locally finite subgroup ˆ must lie in a conjugate of one of the factors of the splitting. The one ended of Q subgraph of W must, under a quasi-isometry, correspond to a one-ended subgraph of Y which determines a special end. The graph Y has no such subgraph. We have a contradiction. We have proved that the locally finite graph Y is quasi-isometric to a vertex transitive graph, but it is not quasi-isometric to a Cayley graph.

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References [1] M.R. Bridson and A. Haefliger, Metric spaces of non-positive curvature. Springer 1999. [2] Warren Dicks and M.J. Dunwoody, Groups acting on graphs, Cambridge University Press, 1989. [3] R. Diestel and I. Leader, A conjecture concerning a limit of non-Cayley graphs. J. Algebraic Combin. 14 (2001) 17–25. [4] M.J. Dunwoody, The accessibility of finitely presented groups, Invent. Math. 81 (1985) 449–457. [5] M.J.Dunwoody, An inaccessible group, in: Geometric Group Theory Vol. 1 (ed. G.A. Niblo and M.A. Roller) LMS Lecture Notes 181 (1993) 75–78. [6] M.J. Dunwoody, Inaccessible groups and protrees, J. Pure Appl. Alg. 88 (1993) 63–78. [7] M.J. Dunwoody and J.M. Jones, A group with strange decomposition properties, J. Graph Theory 1 (1998) 301–305. [8] M.J. Dunwoody and J.M. Jones, A group with very strange decomposition properties, J. Austral. Math. Soc. 67 (1999) 185–190. [9] M.J. Dunwoody, Folding sequences. In The Epstein birthday schrift, pages 139–158 (electronic). Geom. Topol., Coventry, 1998. [10] M.J. Dunwoody and B. Kr¨ on, Vertex cuts, arXiv:0905.0064. [11] A. Eskin, D. Fisher, K. Whyte, Coarse differentiation of quasi-isometries I: spaces not quasi-isometric to Cayley graphs. arXiv math/0607207. [12] A. Eskin, D. Fisher, K. Whyte, Coarse differentiation of quasi-isometries II: spaces not quasi-isometric to Cayley graphs. arXiv math/0706.0940. [13] P.A. Linnell, On accessiblility of groups. J. Pure appl. Algebra 30 (1983) 39–46. [14] J.R. Stallings, Group theory and three-dimensional manifolds, Yale University Press (1971). [15] P.M. Soardi and W. Woess, Amenability, unimodularity, and the spectral radius of random walks on infinite graphs. Math. Z. 205 (1990), 471–486. [16] C. Thomassen and W. Woess, Vertex-transitive graphs and accessibility, J. of Comb. Theory, Series B 58 (1991) 248–268. [17] W. Woess, Topological groups and infinite graphs. Directions in infinite graph theory and combinatorics, (Cambridge, 1989). Discrete Math. 95 (1991), 373–384. [18] W. Woess Lamplighters, Diestel-Leader graphs, random walks, and harmonic functions, Combinatorics, Probability and Computing 14 (2005) 415–433. M.J. Dunwoody University of Southampton Southampton SO16 7GR, UK e-mail: [email protected]

Progress in Probability, Vol. 64, 15–53 c 2011 Springer Basel AG


A Local Limit Theorem for Random Walks ˜2 Buildings on the Chambers of A James Parkinson and Bruno Schapira Abstract. In this paper we outline an approach for analysing random walks on the chambers of buildings. The types of walks that we consider are those which are well adapted to the structure of the building: Namely walks with transition probabilities p(c, d) depending only on the Weyl distance δ(c, d). We carry through the computations for thick locally finite affine buildings ˜2 to prove a local limit theorem for these buildings. The technique of type A centres around the representation theory of the associated Hecke algebra. This representation theory is particularly well developed for affine Hecke algebras, with elegant harmonic analysis developed by Opdam ([28], [29]). We give an introductory account of this theory in the second half of this paper. Mathematics Subject Classification (2000). Primary 20E42; Secondary 60G50. Keywords. Random walks, affine buildings, Hecke algebras, harmonic analysis, Plancherel theorem, p-adic Lie groups.

Introduction Probability theory on real Lie groups and symmetric spaces has a long and rich history (see [4], [16], [17] and [40] for example). A landmark work in the theory is Bougerol’s 1981 paper [4] where the Plancherel Theorem of Harish-Chandra [18] is applied to prove a local limit theorem for real semisimple Lie groups. There has also been considerable work done for Lie groups over local fields, such as SLn (Qp ) (see [9], [22], [31], [37] and [38] for example). In this case the group acts on a beautiful geometric object; the affine building, and probability theory on the group can be analysed by studying probability theory on the building. It is this approach that we take here – we develop a general setup for studying radial random walks on arbitrary buildings, and explicitly carry out the technique for A˜2 buildings to prove a local limit theorem for random walks on the chambers of these buildings. A building is a geometric/combinatorial object that can be defined axiomatically (see Definition 1.4). It is a set C of chambers (the rooms of the building) glued

16

J. Parkinson and B. Schapira

together in a highly structured way. The chambers can be visualized as simplices (all of the same dimension) and the gluing occurs along their codimension 1 faces, called panels. Panels are the ‘doors’ of the chambers – one moves from chamber c to chamber d via the panel common to c and d (see Figure 2). Each panel π has a type type(π) (in some index set I) such that each chamber has exactly one panel of each type. If chambers c and d are glued together along their type i panels then they are said to be i-adjacent. There is a relative position function δ(c, d) between any two chambers c and d. This function takes values in a Coxeter group W associated to the building, and it encodes the types of walks (or galleries) in the building: If there is a minimal length walk from c to d passing through panels of types i1 , . . . , i
then δ(c, d) = si1 · · · si
where si , i ∈ I, are the generators of the Coxeter group W . We will be considering random walks on the set C of chambers of a building. Let A = (p(c, d))c,d∈C be the transition operator of the walk, where p(c, d) is the probability that the walker moves from c to d in one step. A local limit theorem is an asymptotic estimate for the n-step transition probability p(n) (c, d) as n → ∞, with c and d fixed. Let us give a rough summary of the results and techniques of this paper. Let (C, δ) be a building with Coxeter group W . We will assume that (C, δ) satisfies a mild regularity condition (Definition 1.6). Under this assumption the cardinality of the w-sphere |{d ∈ C | δ(c, d) = w}| = qw is independent of the centre c ∈ C (for each w ∈ W ). A random walk with transition operator A = (p(c, d))c,d∈C is radial if p(c, d) = p(c
, d
) whenever δ(c, d) = δ(c
, d
). It is elementary that a random walk with operator A = (p(c, d))c,d∈C is radial if and only if   A= aw Aw where aw ≥ 0 and aw = 1, w∈W

w∈W

where Aw = (pw (c, d))c,d∈C is the transition matrix for the random walk with transition probabilities  1 if δ(c, d) = w pw (c, d) = qw 0 otherwise. This naturally leads us to consider the linear span A over C of the operators Aw , w ∈ W . It is well known that A is an algebra under convolution (see Proposition 1.13). This algebra is the Hecke algebra of the building; it is a noncommutative associative unital algebra. It is not difficult to see that if A = (p(c, d))c,d∈C is the transition operator of a radial random walk then −2 p(n) (c, d) = qw Tr(An Aw−1 )

if δ(c, d) = w,

(0.1)

where Tr : A → C is the canonical trace functional given by linearly extending Tr(Aw ) = δw,1 . One can complete A into a C ∗ -algebra A . Then Tr extends to a trace on A . Under certain conditions on the representation theory of A (for example, liminality) there is general machinery on the decomposition of a

Random Walks on Buildings

17

trace that guarantees the existence of a unique Borel probability measure µ (the Plancherel measure) such that (see [13, §8.8])  Tr(A) = χπ (A) dµ(π) for all A ∈ A (0.2) spec(A )

where spec(A ) is the spectrum of A , and χπ is the character of the representation π ∈ spec(A ) (we will be working in the situation where the irreducible representations are finite dimensional, and so χπ (A) = tr(π(A)) where tr is the usual matrix trace). The usefulness of (0.2) for random walk theory is clear: If A = (p(c, d))c,d∈C is the transition operator of a radial random walk, then by (0.1) we have  (n) −2 p (c, d) = qw χπ (An Aw−1 ) dµ(π) if δ(c, d) = w. (0.3) spec(A )

Therefore if we have a good understanding of µ and the representations π in spec(A ) then it should be possible to extract the leading behaviour of the integral as n → ∞, thereby proving a local limit theorem. This is delicate work: The representation theory of Hecke algebras is a beautiful and rich subject with many subtleties. The representation theory is only really well developed in the cases where W is finite or affine. It is for this reason that in the end we will restrict ourselves to the affine case – here we have at our disposal the elegant harmonic analysis of Opdam [29]. In fact we will restrict our specific computations to the A˜2 case (see Figures 1 and 3). The general affine case will appear elsewhere, where we also provide central limit theorems and rate of escape theorems. This paper is divided into Parts I and II, which can be more or less read independently. The local limit theorem appears in Part I, and the derivation of the Plancherel formula for type A˜2 is given in Part II. Part I also includes relevant background on Coxeter groups, buildings and the Hecke algebra of a building. Part II contains relevant structure theory and representation theory of affine Hecke algebras, and an account of the harmonic analysis on affine Hecke algebras. The structural and representation theoretic results are well known, and the harmonic analysis results are from [28] and [29], with some minor modifications. We make no claim of originality in Part II, however we believe that this part is a nice contribution to the literature because it gives an introduction to the quite profound general analysis undertaken by Opdam ([28] and [29]). Let us conclude this introduction by mentioning some related work. Brown and Diaconis [6] and Billera, Brown and Diaconis [7] have studied random walks on hyperplane arrangements. This elegant theory is ‘just around the corner’ from random walks on spherical (finite) buildings. Diaconis and Ram [12] apply the representation theory of finite-dimensional Hecke algebras to prove mixing time theorems for random walks on spherical buildings (see also [11]). In the context of affine buildings initial results came from the theory of homogeneous and semihomogeneous trees (these are the A˜1 buildings, arising from groups like SL2 (Qp )). See [37]. The next simplest (irreducible) affine buildings are the A˜2 buildings.

18

J. Parkinson and B. Schapira

Random walks on the vertices of these buildings are studied in [22] by Lindlbauer and Voit. Cartwright and Woess [9] study walks on the vertices of A˜d buildings and Parkinson [31] generalised this to walks on the vertices of arbitrary (regular) affine buildings. This work applies harmonic analysis from Macdonald [24] and Matsumoto [26]. We note that the analysis on the vertices of an affine building is somewhat simpler than the chamber case, because the underlying Hecke algebra in the vertex case is commutative. Finally, Tolli [38] has proved a local limit theorem for SLd (Qp ), which gives results for random walks on the associated building.

Part I: The local limit theorem 1. Buildings and random walks Morally a building is a way of organising the flag variety G/B of a Lie group or Kac-Moody group into a geometric object that reflects the combinatorics of the Bruhat decomposition and the internal structure of the double cosets BgB. Remarkably buildings can be defined axiomatically, without any reference to the underlying connections with Lie groups and Kac-Moody groups. In this section we recall one of the axiomatic definitions of buildings. We define radial random walks on buildings, and write down the Hecke algebra of the building. Standard references for this section include [1], [5], [19], [35] and [41]. 1.1. Coxeter groups The notion of a Coxeter group is at the heart of building theory. Definition 1.1. A Coxeter system is a pair (W, S) where W is a group generated by a finite set S = {s0 , . . . , sn } subject to relations (si sj )mij = 1

for all i, j = 0, 1, . . . , n,

where (i) mii = 1 for all i, (ii) mij = mji for all i, j, and (iii) mij ≥ 2 is an integer or ∞ for i = j. We usually simply call W a Coxeter group. Coxeter groups are “abstract reflection groups”. Indeed one can build a vector space on which W acts by reflections (the reflection representation). The relations s2i = 1 say that W is generated by reflections, and the relations (si sj )mij = 1 say that the product of the reflections si and sj is a rotation by 2π/mij . Definition 1.2. The length (w) of w ∈ W is

(w) = min{ ≥ 0 | w can be written as a product of generators in S}. If (w) = then an expression w = si1 · · · si
is a reduced expression for w. It is easy to see that if si ∈ S and w ∈ W then (wsi ) = (w) ± 1. Example 1.3. Consider the Coxeter system (W, S) with S = {s0 , s1 , s2 } and s20 = s21 = s22 = (s0 s1 )3 = (s1 s2 )3 = (s0 s2 )3 = 1.

Random Walks on Buildings I 4 ,  I 4

19

I 4
 I 4
4 I 4
5 I 4
6 I 4
7

I(
6 I(
5 (` t2 t4 `3

"3 t4 t2

3

t4 t3
`3

"4 t3 t2

t2 t4

I(
4

t2 t3 `4

I 2

t3 I(

t3 t4

x2 x2 t2


`4

I(,  I(,4


(`

I(,5 I(,6

I 3
 I 3

I 3 ,  I 3 ,4 I 3 ,5 I 3 ,6 I 3 ,7

Figure 1. The A˜2 Coxeter group This is the Coxeter group of type A˜2 , and it is the main example that we will consider in this work. This group can be realised nicely as a reflection group in R2 . The elements s0 , s1 , s2 are the reflections in the hyperplanes labeled by Hα0 , Hα1 and Hα2 . Then W acts simply transitively on the set of triangles. In the building language these triangles are called chambers, and in some other aspects of Lie theory they are called alcoves. The remainder of the details are explained later. 1.2. Buildings We adopt the following modern definition of a building, from [1]. Definition 1.4. A building of type (W, S) is a pair (C, δ) consisting of a nonempty set C of chambers, together with a map δ : C × C → W such that for all a, b, c ∈ C: (B1) δ(a, b) = 1 if and only if a = b. (B2) If δ(a, b) = w and δ(b, c) = si , then δ(a, c) ∈ {w, wsi }. If (wsi ) = (w) + 1 then δ(a, c) = wsi . (B3) If δ(a, b) = w then for each si there is a chamber c
∈ C with δ(b, c
) = si such that δ(a, c
) = wsi . This chamber is unique if (wsi ) = (w) − 1. The function δ : C × C → W is the Weyl distance function. It follows from the axioms that δ(a, b) = δ(b, a)−1 .

20

J. Parkinson and B. Schapira

One can visualize the building geometrically as follows. For simplicity, let us suppose that S = {s0 , s1 , s2 } (like in the A˜2 example). Then each chamber of the building is imagined as a triangle, with the sides (codimension 1 faces) being called panels. Each panel π is assigned a type type(π) ∈ {0, 1, 2} such that every chamber has exactly one panel of each type. If chambers c, d ∈ C have δ(c, d) = si then we glue the chambers c and d together along the type i panels. Therefore the local picture of the building looks like Figure 2. .. ....... . ....... . ........... ... ... ... ....... ... ..... . ... ....... . .. .. ...... ... ...... .... ............................................... ... . . . . .. . . ... ........................................ ... .................. .... .......... . . . . . . . . . . . . . . . . . . . . . . . . . . ...... .... .............. .... .... .. .... .... ... ............. .. .... . .... c .. .... .. ... ........... .... ...... .... .. i .... .. ... ..... .... .. ... .. . .... . . . ....... . . . . .... .. ... .. . . . . .... . ... .. .. ............ .... . .. .. ....... ...... .... . ....... .... ... ........ .......

Figure 2. The local view of a rank 3 building One calls chambers c and d i-adjacent if δ(c, d) = si or c = d. This is an equivalence relation, and we write c ∼i d if c and d are i-adjacent. Figure 2 shows the set of all chambers i-adjacent to c. A gallery of type i1 · · · i
from c to d is a sequence (c0 , c1 , . . . , c
) of chambers with c = c0 ∼i1 c1 ∼i2 · · · ∼i
c
= d,

with

ck−1 = ck

for k = 1, . . . , .

So a gallery is a “walk” from chamber to chamber through the building. One can show that if w = si1 · · · si
is a reduced expression then δ(c, d) = w

⇐⇒

there is a minimal length gallery of type i1 · · · i
from c to d.

So the Weyl distance encodes the types of the minimal length galleries from c to d. Definition 1.5. Let w ∈ W and c ∈ C. The w-sphere centred at c is Cw (c) = {d ∈ C | δ(c, d) = w}. In particular if si ∈ S then Csi (c) = {d ∈ C | c ∼i d}\{c}. Therefore if w = si1 · · · si
is a reduced expression then Cw (c) is the set of all chambers in the building that are connected to c by a gallery of type i1 · · · i
. Definition 1.6. A building (C, δ) with Coxeter system (W, S) is: • thin if |Cs (c)| = 1 for all s ∈ S and c ∈ C, • thick if |Cs (c)| ≥ 2 for all s ∈ S and c ∈ C,

Random Walks on Buildings

21

• locally finite if |Cs (c)| < ∞ for all s ∈ S and c ∈ C, • regular if for each s ∈ S, |Cs (c)| = |Cs (d)| for all chambers c, d ∈ C. If (C, δ) is a locally finite regular building then we define q0 , . . . , qn ∈ Z>0 by qi = |Csi (c)| for any c ∈ C. The integers q0 , . . . , qn are called the parameters of the building. For example if Figure 2 represents part of a locally finite regular building then qi = 4 (there are 5 = 4 + 1 chambers on each i-panel). Henceforth we will assume that our buildings are locally finite and regular. Remark 1.7. If (C, δ) is thick and locally finite and if mij < ∞ for each i, j then by [30, Theorem 2.4] (C, δ) is regular. So regularity is a very weak hypothesis. A simple induction shows that if w = si1 · · · si
is a reduced expression then |Cw (c)| = qi1 · · · qi


for all c ∈ C.

(1.1)

Thus we can define qw = qi1 · · · qi
(equation (1.1) shows that this is independent of the particular reduced expression for w). Since si sj si · · · = sj si sj · · · (mij factors on each side) are both reduced expressions it follows that qi = qj whenever mij is finite and odd. Then it follows that if sj = wsi w−1 for some w ∈ W then qi = qj (see [5, IV, §1, No. 3, Proposition 3]). Remark 1.8. Given a Coxeter system (W, S), define a building (W, δW ) where δW (u, v) = u−1 v. Figure 1 shows the A˜2 case. The building (W, δW ) is thin, and all thin buildings arise in this way. A general building of type (W, S) contains many thin sub-buildings of type (W, S). These sub-buildings are called the apartments of the building. The apartments fit together in a highly structured way: (A1) Given chambers c, d ∈ C there exists an apartment containing both. (A2) If A and A
are apartments with A ∩ A
= ∅ then there is an isomorphism ψ : A → A
fixing each chamber of the intersection A ∩ A
. These facts give a global picture of a building (see Figure 3). ......... ................. .... ........................................................................ ....... . . . . . . . . . . . . . . . . ..................................................... .... ...... ............................... .................... .... .. .. . .................................................................................................................. .... .... .... ............................................................. ........................... .... ....... ....... ........ ....... ....... ....... ........................................................................................ ....... ....... ....... ................................ ....... . . . ....................... .... ........ ............ .................................................................................................................... .... .... . . . . . . . .. ...... ............................ .................................... ......... ...... ....................... .... ...... .... ..... .. . . .... .... . .... ....... .... . ... .... ...... .. . . .... .... ............... ..... .... ..... .... ..... .... ................................................................. .... ..... .... ...................... ..... .... ................................................................................................................................................. ....... ........ ................................................................................................ ................................... . . ................. ...... ..... ....... ..... ...... ..... .............................................................................. ..... ....... ..... ....................... ...... ...... .. .. ....... .. . . .. . .. . ... .. .. . . .. ....... .. . . .... ............................................................................................................................ ..... ...... ..... .................................................................... ............................... ................. ....... ....... ....... ....... ........ ........ ....................................................................................... ........ ....... ....... ............................. ....... . . . ...... ... .. .. .. .. ................ .. ... ..... .............. .. .. .. ...... .. .... ........................................................................................................................................ ....... ...... ............................................................................... ...................................................... . .... ... .... ... ....... ... .... .. ............ .............. .............. ............................................................................................... .............. ........ ........ . . . . ................................................................................... ............................. ..................... ............................ ............................ ............................ ............................. . .. ...... . ... .... .............................................. ............ .... ....... ....... .............................. . . . .........................................................

Figure 3. The global view of an A˜2 building Note that the apartments are as in Figure 1; there are 6 apartments shown in Figure 3. However if the building is thick then the “branching” is actually happening

22

J. Parkinson and B. Schapira

along all of the walls of the building. Therefore a thick A˜2 building has infinitely many apartments. To understand buildings it is useful to have both the local and global pictures in mind. Remark 1.9. A locally finite regular A˜2 building necessarily has q0 = q1 = q2 = q because m0,1 = m1,2 = m2,0 = 3 are odd. An A˜2 building is not determined by its thickness parameter q. For example the buildings constructed from SL3 (Qp ) and SL3 (Fp ((t))) are non-isomorphic and both have thickness q = p. Furthermore it is unknown which parameters q can occur as the thickness of an A˜2 building. By [34] this is closely related to the famous unsolved problem of classifying finite projective planes. Remark 1.10. The definition of buildings is driven by the combinatorics of KacMoody groups, which are infinite-dimensional generalisations of semisimple Lie groups. If G is a Kac-Moody group with Borel subgroup B and Weyl group W then the flag variety G/B is a building with δ(gB, hB) = w if and only if g −1 hB ⊆ BwB. 1.3. Random walks and the Hecke algebra A random walk consists of a finite or countable state space X  and a transition operator A = (p(x, y))x,y∈X where p(x, y) ≥ 0 for all x and y and y∈X p(x, y) = 1 for all x ∈ X. As an operator acting on the space of functions f : X → C we have  (Af )(x) = p(x, y)f (y) for all f : X → C and x ∈ X. y∈X

The numbers p(x, y) are the transition probabilities of the walk. The natural interpretation of a random walk is that of a walker taking discrete steps in the space X, with p(x, y) being the probability that the walker, having started at x, moves to y in one step. The n-step transition probability p(n) (x, y) is the probability that the walker, having started at x, is at y after n steps. Then An = (p(n) (x, y))x,y∈X . A local limit theorem is a theorem giving an asymptotic estimate for p(n) (x, y) as n → ∞ (with x, y ∈ X fixed). Here we consider random walks with state space C (the set of chambers of a building). We consider random walks which are well adapted to the structure of the building: Definition 1.11. A random walk with operator A = (p(c, d))c,d∈C on the chambers of a building (C, δ) is radial if p(c, d) = p(c
, d
) whenever δ(c, d) = δ(c
, d
). Recall that we assume our buildings are locally finite and regular, and so |Cw (c)| = qw . For each w ∈ W , the w-averaging operator is (Aw f )(c) =

1 qw

 d∈Cw (c)

f (d)

for f : C → C and c ∈ C.

Random Walks on Buildings

23

Then Aw = (pw (c, d))c,d∈C is the transition operator of the radial walk with  −1 qw if δ(c, d) = w pw (c, d) = 0 otherwise. The following proposition is elementary (c.f. [41, §19.C]). Proposition 1.12. A random walk with operator A = (p(c, d))c,d∈C is radial if and only if   A= aw Aw where aw ≥ 0 and aw = 1, w∈W

in which case p(c, d) =

w∈W −1 aw qw

if δ(c, d) = w.

Therefore we are naturally lead to consider linear combinations of the (linearly independent) operators Aw , w ∈ W . Let A be the vector space over C with basis {Aw | w ∈ W }. The following simple proposition tells us how to compose the averaging operators, and shows that A is an algebra. Proposition 1.13. Let w ∈ W and si ∈ S. The averaging operators satisfy  Awsi if (wsi ) = (w) + 1   Aw Asi = −1 −1 qi Awsi + 1 − qi Aw if (wsi ) = (w) − 1. Therefore A is an algebra. Proof. Using the definition of the operators we see that   1 1  (Aw Asi f )(c) = f (e) = |Cw (c) ∩ Csi (e)|f (e). qw qi qw qi d∈Cw (c) e∈Csi (d)

e∈C

If Cw (c) ∩ Csi (e) = ∅ then (B2) implies that e ∈ Cw (c) or e ∈ Cwsi (c). Then:  0 if (wsi ) = (w) + 1 (by (B2)) If e ∈ Cw (c), |Cw (c) ∩ Csi (e)| = qi − 1 if (wsi ) = (w) − 1 (by (B3))  1 if (wsi ) = (w) + 1 (by (B3)) If e ∈ Cwsi (c), |Cw (c) ∩ Csi (e)| = qi if (wsi ) = (w) − 1 (by (B2)). Therefore if (wsi ) = (w) − 1 we have qwsi (Aw Asi f )(c) = (Awsi f )(c) + (1 − qi−1 )(Aw f )(c). qw Since (wsi ) = (w) − 1 we have qw = q(wsi )si = qwsi qi . This completes the proof when (wsi ) = (w) − 1, and the case (wsi ) = (w) + 1 is similar. Now a simple induction on (v) shows that Au Av is a linear combination of terms Aw , w ∈ W . Therefore A is an algebra.
Definition 1.14. The algebra A is the Hecke algebra of the building (C, δ).

24

J. Parkinson and B. Schapira

 If a radial walk with operator A = (p(c, d))c,d∈C is written as A = aw Aw as in Proposition 1.12 then the n-step transition probabilities p(n) (c, d) can be found from the following calculation:   n  −1 n p(n) (c, d) = a(n) q , where A = a A = a(n) (1.2) w w w w w Aw . w∈W

w∈W (n)

So finding p(n) (c, d) is the equivalent to finding the coefficient aw of Aw in An .

2. The Plancherel Theorem In this section we discuss how the representation theory of the Hecke algebra can be used to achieve the goal of computing p(n) (c, d). The representation theory of Hecke algebras is particularly well developed in two important cases: When the underlying Coxeter group is a finite Weyl group or an affine Weyl group. In the case of a finite Weyl group the building is a finite object, and the types of questions one asks are quite different to what we do here (see [12]). We will focus on the affine case (but our initial setup will remain rather general). 2.1. The Hecke algebra as a C ∗ -algebra Let 2 (C) be the space of square summable functions f : C → C, with inner product  f, g = f (c)g(c). Each A ∈ A maps 2 (C) into itself (c.f. [8, Lemma 4.1]): Lemma 2.1. Let w ∈ W . If f ∈ 2 (C) then Aw f ∈ 2 (C) and Aw  ≤ 1, where A = sup{Af 2 : f ∈ 2 (C), f 2 ≤ 1} is the 2 -operator norm of A ∈ A . Therefore A is a subalgebra of the C ∗ -algebra B( 2 (C)) of bounded linear operators on 2 (C), and since A∗w = Aw−1 we see that A is closed under the adjoint involution. Let A denote the completion of A with respect to the 2 -operator norm. Therefore A is a (non-commutative) C ∗ -algebra. Let o ∈ C be a fixed chamber. Let δo be the indicator function of {o}, and −1 let 1Cw (o) be the indicator function of Cw (o). Since Aw δo = qw 1Cw−1 (o) it follows that Au δo , Av δo  = δu,v qu−1 . Let (A, B) := Aδo , Bδo 

for A, B ∈ A .

The value of (A, B) does not depend on the particular fixed chamber o ∈ C. Moreover, (·, ·) defines an inner product on A . The only thing to check is: Lemma 2.2. Let A ∈ A . If Aδo = 0 then A = 0. Proof. This is easily checked for A ∈ A , and thus is true for A ∈ A by density.




It is routine to verify the following properties: (AB, C) = (B, A∗ C) and (A, B) = (B ∗ , A∗ ) for all A, B, C ∈ A .

(2.1)

Random Walks on Buildings

25

2.2. The trace functional Let o ∈ C be a fixed chamber. The linear map Tr : A → C

with

Tr(A) = (Aδo )(o) = (A, I)

defines a trace on A , because by (2.1) we have Tr(AB) = (AB, I) = (B, A∗ ) = (A, B ∗ ) = (BA, I) = Tr(BA).  Note that Tr( aw Aw ) = a1 and Tr(A∗u Av ) = (Av , Au ) = qu−1 δu,v , and so (0.1) follows from (1.2). There is a general theory centred around decomposing a trace on a liminal C ∗ -algebra into an integral over irreducible ∗-representations of the algebra (the spectral decomposition). For an elegant account see [13, §8.8]. A C ∗ -algebra A is liminal if for every irreducible representation π of A and for each x ∈ A the operator π(x) is compact. Suppose that A is liminal; indeed this is true if (C, δ) is affine because all of the irreducible representations are finite dimensional (see Proposition 5.11). Then by [13, §8.8] there exists a unique Borel probability measure µ (the Plancherel measure) such that (0.2) holds. The way we plan to apply (0.2) was explained in the introduction. ˜2 2.3. Statement of the Plancherel Theorem for type A By the Plancherel Theorem we mean the computation of the measure µ and the spectrum spec(A ) in (0.2). Let us state the Plancherel Theorem for Hecke algebras of type A˜2 . Since this is a representation theoretic statement we first write down some representations of A . See Section 8 for the details. Recall that in type A˜2 we have q0 = q1 = q2 = q. A 6-dimensional representation: For each t = (t1 , t2 ) with t1 , t2 ∈ C× there is a 6-dimensional representation πt : A → M6 (C) given on the generators of A by the matrices   q 0 0 0 0 t1 t2  0 q 0 t2 0 0     1  0 0 q 0 t1 0  , πt (A0 ) = √  t−1 0 0 0 0  q 0 2   0 0 t−1 0 0 0  1 −1 t−1 0 0 0 0 0 1 t2     0 1 0 0 0 0 0 0 1 0 0 0 1 q 0 0 0 0 0 0 0 0 1 0       1 0 q 0 0 0 1  1 0 0 0 1 0 0    , , πt (A2 ) = √  πt (A1 ) = √  q 0 0 1 q 0 0 q 0 0 0 0 0 1   0 0 0 0 0 1  0 1 0 0 q 0 0 0 0 0 1 q 0 0 0 1 0 q 1

1

where for type-setting convenience q = q 2 − q − 2 . This representation is the principal series representation of A with central character t = (t1 , t2 ). It is irreducible if

26

J. Parkinson and B. Schapira

and only if t1 , t2 = q ±1 , and every irreducible representation of A is a composition factor of a principal series representation for some central character t. A 3-dimensional representation: For each u ∈ C× there is a 3-dimensional repre(1) sentation πu : A → M3 (C) given on the generators of A by the matrices   q 0 −u 1 1 πu(1) (A0 ) = √  0 −q − 2 0 , q −1 −u 0 0    −1  0 1 0 −q 2 0 0 1 1 0 . πu(1) (A1 ) = √  0 πu(1) (A2 ) = √ 1 q 0 1 , 1 q q 0 1 q 0 0 −q − 2 This representation is an induced representation, constructed by lifting a representation of a parabolic subalgebra of A to the full algebra. A 1-dimensional representation: There is a 1-dimensional representation of A π (2) : A → C given on the generators of A by π (2) (A0 ) = π (2) (A1 ) = π (2) (A2 ) = −q −1 . It can be shown that all of the above representations of A extend to A if and only if t1 , t2 , u ∈ T. The details will be provided elsewhere in a more general setting. We can now state the Plancherel Theorem for type A˜2 . Let T be the circle group T = {t ∈ C | |t| = 1}, (1)

and let dt be normalised Haar measure on T. Let χt , χu and χ(2) be the characters (1) of πt , πu and π (2) respectively. For example, χt (A) = tr(πt (A)), where tr is the usual matrix trace on M6 (C). Theorem 2.3. Let (C, δ) be a thick locally finite regular A˜2 building. Then   (1) 1 χt (A) (q − 1)2 χu (A) (q − 1)3 (2) Tr(A) = 3 dt dt + du + 3 χ (A) 1 2 2 2 2 2 6q T2 |c(t)| q (q − 1) T |c1 (u)| q −1 for all A ∈ A , where −1 −1 −1 (1 − q −1 t−1 t2 )(1 − q −1 t−1 1 )(1 − q 1 t2 ) −1 −1 −1 (1 − t−1 1 )(1 − t2 )(1 − t1 t2 ) Proof. See Section 7.

c(t) =

3

and

c1 (u) =

1 − q − 2 u−1 1 2

1 − q u−1

.


3. The local limit theorem Let (C, δ) be a locally finite thick A˜2 building. Therefore (C, δ) is necessarily regular, and q0 = q1 = q2 = q ≥ 2. Let P = 13 (A0 + A1 + A2 ) be the transition operator for the simple random walk on (C, δ). That is P = (p(c, d))c,d∈C with  1 if c ∼ d and c = d p(c, d) = 3q 0 otherwise.

Random Walks on Buildings

27

This walk is irreducible (because {s0 , s1 , s2 } generates W ) and aperiodic (this follows from A2i = q −1 + (1 − q −1 )Ai ). Our techniques will work for general radial random walks (not just the simple random walk), but the additional generality requires a more careful study of the representation theory of affine Hecke algebras to obtain the bounds and estimates required to make the analysis work. We have chosen to deal with this in a later work, where walks on general affine buildings are studied. If θ = (θ1 , θ2 ) ∈ R2 we write eiθ = (eiθ1 , eiθ2 ) ∈ T2 . For θ ∈ R2 and ϕ ∈ R (1) the matrices πeiθ (P ), πeiϕ (P ) and π (2) (P ) are given by   q 1 1 0 0 ei(θ1 +θ2 )   1 2q 0 eiθ2 1 0   iθ1  1  1 0 2q 1 e 0  , πeiθ (P ) = √   0 e−iθ2 1 q 0 1 3 q    0 1 e−iθ1 0 q 1 e−i(θ1 +θ2 ) 0 0 1 1 2q  1  1 q 2 − 2q − 2 1 −eiϕ 1  1 1 (1)  and π (2) (P ) = −q − 32 πeiϕ (P ) = √ 1 q 2 − 2q − 2 1 3 q 1 1 −e−iϕ 1 q 2 − 2q − 2 1

1

where as before q = q 2 − q − 2 . The local limit theorem requires a careful study of the eigenvalues of πeiθ (P ) for (θ1 , θ2 ) close to (0, 0). Let λ1 (θ) ≥ · · · ≥ λ6 (θ) be the eigenvalues of πeiθ (P ). (1) Let λi = λi (0). Let µ1 (ϕ) ≥ µ2 (ϕ) ≥ µ3 (ϕ) be the eigenvalues of πeiϕ (P ), where ϕ ∈ R. All of these eigenvalues are real, because the matrices are Hermitian. Explicit formulae for the eigenvalues are not feasible, and so we turn to techniques from perturbation theory. Standard references include [3] and [21]. For perturbation theory to work nicely one wants to have complete eigenvalue and eigenvector information for π1 (P ). The eigenvalues of π1 (P ) are easily computed. In decreasing order of magnitude they are 1 > λ1 > λ2 = λ3 > λ4 = λ5 > λ6 > 0 with λ1 , λ2 , λ4 and λ6 given by   3(q − 1) + q 2 + 34q + 1 2(q − 1) q − 1 3(q − 1) − q 2 + 34q + 1 , , , 6q 3q 3q 6q respectively. The eigenspaces e(λ) are e(λ1 ) = C(a, 1, 1, a, a, 1),

e(λ2 ) = C(−1, 0, 0, 1, 0, 0) + C(−1, 0, 0, 0, 1, 0),

e(λ6 ) = C(−b, 1, 1, −b, −b, 1), e(λ4 ) = C(0, −1, 1, 0, 0, 0) + C(0, −1, 0, 0, 0, 1) √2 √ q +34q+1−(q−1) q−1+ q2 +34q+1 √ √ where a = and b = . Let v 1 (θ), . . . , v 6 (θ) be an 6 q 6 q 6 orthonormal basis of C with v i (θ) a λi (θ)-eigenvector. Remark 3.1. Perron-Frobenius guarantees that the largest eigenvalue of π1 (P ) is simple with a positive eigenvector (note that π1 (P )2 has all entries positive).

28

J. Parkinson and B. Schapira 1

1

1 The eigenvalues of π1 (P ) are 3√ (q 2 + 2q − 2 − 1) and q with the first eigenvalue repeated.  Lemma 3.2. Let A = aw Aw ∈ A with aw ≥ 0. Then (1)

|χeiθ (A)| ≤ χ1 (A)

1 1 √ (q 2 3 q

1

− 2q − 2 − 2)

for all θ ∈ R2 .

Proof. The proof uses some of the general representation theory from Section 5. It follows from Theorem 5.16 that χeiθ (Aw ) is a linear combination of terms {eikθ1 ei
θ2 | k, ∈ Z} with nonnegative coefficients. Therefore χeiθ (A) also has this property, and the result follows.
In the proof of the following lemma we will use some well-known inequalities between the eigenvalues of the sum of Hermitian matrices (see the interesting survey [14]). In particular, if X and Y are arbitrary d × d Hermitian matrices with eigenvalues x1 ≥ · · · ≥ xd and y1 ≥ · · · ≥ yd and if z1 ≥ · · · ≥ zd are the eigenvalues of Z = X + Y then z1 + · · · + zr ≤ x1 + · · · + xr + y1 + · · · + yr

for each 1 ≤ r ≤ d.

It follows that z r ≤ x1 + y 1

and

zr ≥ xd + yd

for all 1 ≤ r ≤ d

(3.1)

(for the second inequality use the trace identity tr(Z) = tr(X) + tr(Y )). Lemma 3.3. We have the following. 1. |λi (θ)| ≤ λ1 with equality if and only if i = 1 and θ1 , θ2 ∈ 2πZ. 2. |µi (ϕ)| < λ1 for all i = 1, 2, 3 and all ϕ ∈ R. 3. |χ(2) (P )| < λ1 . Proof. 1. If |λi (θ)| > λ1 then |χeiθ (P k )| > χ1 (P k ) for sufficiently large k, contradicting Lemma 3.2. Therefore |λi (θ)| ≤ λ1 for all i = 1, . . . , 6 and all θ ∈ R2 . Suppose that |λi (θ)| = λ1 . Writing πeiθ (P ) = π1 (P ) + E(θ) we see that E(θ) has 2 2 2 2 eigenvalues ± 3√ | sin θ21 |, ± 3√ | sin θ22 | and ± 3√ | sin θ1 +θ |, and so by (3.1) q q q 2 2 λi (θ) ≥ λ6 − √ > −λ1 3 q

for all i = 1, . . . , 6 and all θ ∈ R2 .

Hence |λi (θ)| = λ1 implies that λi (θ) = λ1 . Then λ1 (θ) = · · · = λi (θ) and so if i > 1 then |χeiθ (P k )| > χ1 (P k ) for sufficiently large k, contradicting Lemma 3.2. Therefore if i > 1 then we have |λi (θ)| < λ1 for all θ ∈ R2 . Finally we need to show that |λ1 (θ)| < λ1 unless θ1 and θ2 are multiples of 2π. For this we observe the (rather remarkable) identity: √ 3 q det(πeiθ (P ) − λ1 I) = 150 − 48(cos θ1 + cos θ2 + cos(θ1 + θ2 )) − 2(cos(θ1 + 2θ2 ) + cos(2θ1 + θ2 ) + cos(θ1 − θ2 )), from which the result follows.

Random Walks on Buildings

29

2. Write πeiϕ (P ) = π1 (P ) + E
(ϕ). Then the eigenvalues of E
(ϕ) are 0 and 2 2 √ Therefore by (3.1) we have µ3 (0) − 3√ q ≤ µi (ϕ) ≤ µ1 (0) + 3 q , and (1)

(1)

ϕ 2 ± 3√ q | sin 2 |.

so   1  1 1 1  1 1 √ q 2 − 2q − 2 − 4 ≤ µi (ϕ) ≤ √ q 2 + 2q − 2 + 1 for each i = 1, 2, 3. 3 q 3 q It follows that |µi (ϕ)| < λ1 for all i = 1, 2, 3 and all ϕ ∈ R. 3 3. This is obvious since |χ(2) (P )| = q − 2 .




Lemma 3.4. For θ1 , θ2 ∈ R we have   1 q6 = θ12 θ22 (θ1 + θ2 )2 1 + O(θ3 ) iθ 2 6 |c(e )| (q − 1) Proof. Since q > 1 we have    1 − e−ix 2  q 2 x2  3    1 − q −1 e−ix  = (q − 1)2 1 + O(|x| )

for all x ∈ R


and the result follows from the definition of c(eiθ ). Lemma 3.5. Let w ∈ W and n ∈ Z≥0 . Then χeiθ (P n A∗w ) = Cw λ1 (θ)n (1 + O(θ)) + o(λn1 )

where

Cw = v T1 π1 (A∗w )v 1 ,

where v 1 = v 1 (0) is a unit eigenvector of π1 (P ) for λ1 . T Proof. Let X and Y be d × d matrices with X Hermitian. Let X  = P DP be  an orthogonal diagonalisation with D = diag(ν1 , . . . , νd ) and P = u1 · · · ud . Then

tr(X n Y ) = tr(P Dn P T Y ) = tr(Dn P T Y P ) =

d 

[P T Y P ]i,i νin =

i=1

d 

(uTi Y ui )νin .

i=1

πeiθ (A∗w )

Applying this to X = πeiθ (P ) and Y = and using Lemma 3.3 gives   χeiθ (P n A∗w ) = v 1 (θ)T πeiθ (A∗w )v 1 (θ) λ1 (θ)n + o(λn1 ). General perturbation theory gives v 1 (θ) = v 1 + O(θ). Since the entries of the matrix πeiθ (A∗w ) satisfy [πeiθ (A∗w )]ij = [π1 (A∗w )]ij + O(θ) (see Theorem 5.16) it follows that        v 1 (θ)T πeiθ (A∗w )v 1 (θ) = v T1 π1 (A∗w )v 1 1 + O(θ) = Cw 1 + O(θ) .
Lemma 3.6. We have     λ1 (θ) = λ1 1 − β θ12 + θ22 + θ1 θ2 + O(θ3 )

where

β=

2  . 2 9λ1 q + 34q + 1

Proof. Since λ1 has multiplicity 1, general results from perturbation theory imply that there is a neighbourhood of (0, 0) in which λ1 (θ) and v 1 (θ) are represented

30

J. Parkinson and B. Schapira

by convergent power series in the variables θ1 and θ2 (see [3, Supplement, §1]). The first few terms in these series can be computed in a few ways, for example by adapting the analysis of [3, §3.1.2] to the 2-variable setting. The details are omitted.
Theorem 3.7. For the simple random walk on the chambers of a thick A˜2 building with thickness 1 < q < ∞ we have    Cw q 3−2
(w) p(n) (c, d) = √ λn1 n−4 1 + O n−1/2 if δ(c, d) = w, 27 3β 4 π(q − 1)6 where β is as in Lemma 3.6 and Cw is as in Lemma 3.5. Proof. By (0.3), Theorem 2.3, and Lemma 3.3 we have  π  π q −2 χeiθ (P n A∗w ) p(n) (c, d) = 3 w 2 dθ1 dθ2 + o(λn1 ) 6q (2π) −π −π |c(eiθ )|2

if δ(c, d) = w,

and so using Lemma 3.3 again we have    1 χeiθ (P n A∗w ) p(n) (c, d) = dθ1 dθ2 + o(λn1 ) (3.2) 2 2
(w)+3 |c(eiθ )|2 24π q − − √ for small √  > 0. Let In be the double integral in (3.2). Let ϕ1 = nθ1 and ϕ2 = nθ2 . By Lemma 3.4 we have   1 q6 √ = g(ϕ)n−3 1 + O(n−1 ) , where g(ϕ) = ϕ21 ϕ22 (ϕ1 + ϕ2 )2 (q − 1)6 |c(eiϕ/ n )|2 and so   q6 In = n−4 1 + O(n−1 ) (q − 1)6



√

n

√ − n



√

n

√ − n

g(ϕ)χeiϕ/√n (P n A∗w ) dϕ1 dϕ2 .

By Lemma 3.5 we have

  √ 1 χeiϕ/√n (P n A∗w ) = Cw λ1 (ϕ/ n)n 1 + O(n− 2 ) + o(λn1 ).

Writing h(ϕ) = ϕ21 + ϕ1 ϕ2 + ϕ22 , Lemma 3.6 gives  n √ λ1 (ϕ/ n)n = λn1 1 − βh(ϕ)n−1 + O(n−3/2 )    n   = λn1 e−βh(ϕ)/n + O n−3/2 = λn1 e−βh(ϕ) 1 + O n−1/2 . Therefore

  −1/2  q6 n −4 In = Cw λ n 1 + O n (q − 1)6 1



√ n

√ − n



√

n

√ − n

g(ϕ)e−βh(ϕ) dϕ1 dϕ2 .

The integral tends to  ∞ ∞  ∞ ∞ 1 8π g(ϕ)e−βh(ϕ) dϕ1 dϕ2 = 4 g(ϕ)e−h(ϕ) dϕ1 dϕ2 = √ , β −∞ −∞ 9 3β 4 −∞ −∞ and the result follows from (3.2).


Random Walks on Buildings

31

√ 3(q−1)+ q2 +34q+1 Remark 3.8. The spectral radius formula λ1 = agrees with 6q computations made by Saloff-Coste and Woess in [36, Example 6]. Remark 3.9. Let G = SL3 (F) where F is a non-archimedean local field. Let I be the standard Iwahori subgroup of G, defined by the following diagram, where o is the ring of integers in F and where θ : F → k is the canonical homomorphism onto the residue field k (for example, F = Fq ((t)), o = Fq [[t]], k = Fq and θ = evt=0 ). G

=

∪
K

∪
=

∪
I

SL3 (F) SL3 (o)

θ

−→ SL3 (k)

∪
=

θ−1 (B(k))

∪
θ

−→

B(k)

where B(k) is the subgroup of upper triangular matrices in SL3 (k). Then G/I is the set of chambers of an A˜2 building (and G/K is the set of type 0 vertices of that building). Since Cw (gI) = (gIwI)/I our local limit theorem gives a local limit theorem for bi-I-invariant probability measures on G.

Part II: Harmonic analysis on affine Hecke algebras In this part we give an outline of some well-known structural theory of affine Hecke algebras. We prove Opdam’s generating function formula for the trace functional. Our argument is slightly different to Opdam’s [28] (we prove the formula by applying the harmonic analysis on the centre of the Hecke algebra). This formula is at the heart of harmonic analysis on affine Hecke algebras. We apply it to prove the Plancherel Theorem for type A˜2 , following the general technique of [29].

4. Affine Weyl groups and alcove walks In this section we fix some standard notation on affine Weyl groups, and briefly discuss the combinatorics of alcove walks. Alcove walks control many aspects of the representation theory of Lie algebras and Hecke algebras. Standard references for this section include [5], [19] and [32]. 4.1. Root systems and affine Weyl groups Let us fix some notation, mainly following [5]. • Let h be an n-dimensional real vector space with inner product ·, ·. • For nonzero α ∈ h let α∨ = 2α/α, α. • Let R be a reduced irreducible root system in h (see [5] for the classification). • Let {α1 , . . . , αn } be a set of simple roots of R. • Let R+ be the set of positive roots. Let ϕ ∈ R be the highest root. • For α ∈ R let Hα = {λ ∈ h | λ, α = 0} be the hyperplane orthogonal to α.

32

J. Parkinson and B. Schapira • • • •

For α ∈ R let sα ∈ GL(h) be the reflection sα (λ) = λ− λ, αα∨ through Hα . ∨ + ∨ ∨ Let Q = Zα∨ 1 +· · ·+Zαn be the coroot lattice, and Q = Z≥0 α1 +· · ·+Z≥0 αn . Let {ω1 , . . . , ωn } be the dual basis to {α1 , . . . , αn } defined by ωi , αj  = δij . Let P = Zω1 +· · ·+Zωn be the coweight lattice, and P + = Z≥0 ω1 +· · ·+Z≥0 ωn be the cone of dominant coweights.

In Figure 1 the lattice Q consists of the centres of the solid hexagons, and the lattice P consist of all vertices in the picture. The Weyl group W0 of R is the subgroup of GL(h) generated by {sα | α ∈ R}. The Weyl group is a finite Coxeter group with distinguished generators s1 , . . . , sn (where si = sαi ) and thus has a length function : W0 → Z≥0 , with (w) being the smallest ≥ 0 such that w = si1 · · · si
. Let w0 be the (unique) longest element of W0 . The inversion set of w ∈ W0 is R(w) = {α ∈ R+ | w−1 α ∈ −R+},



and

(w) = |R(w)|.

The open connected components of h\ α∈R Hα are Weyl sectors. These are open simplicial cones, and W0 acts simply transitively on the set of Weyl sectors. The fundamental Weyl sector is S0 = {λ ∈ h | λ, αi  > 0 for i = 1, . . . , n}, and P + = P ∩ S0 , where S0 is the closure of S0 in h. The roots α ∈ R can be regarded as elements of h∗ by setting α(λ) = λ, α for λ ∈ h. Let δ : h → R be the (non-linear) constant function with δ(λ) = 1 for all λ ∈ h. The affine root system is Raff = R + Zδ. The affine hyperplane for the affine root α + jδ is Hα+jδ = {λ ∈ h | λ, α + jδ = 0} = {λ ∈ h | λ, α = −j} = H−α−jδ . The affine Weyl group is the subgroup W of Aff(h) generated by the reflections sα+kδ with α+kδ ∈ Raff , where the reflection sα+kδ : h → h is given by the formula sα+kδ (λ) = λ − (λ, α + k)α∨ for λ ∈ h. Let α0 = −ϕ + δ (with ϕ the highest root of R). The affine Weyl group is a Coxeter group with distinguished generators s0 , s1 , . . . , sn , where s0 = sα0 . For µ ∈ h, let tµ : h → h be the translation tµ (λ) = λ + µ for all λ ∈ h. Then sα+kδ = t−kα∨ sα and W is the semidirect product W = Q  W0 .  The open connected components of h\ β∈Raff Hβ are chambers (or alcoves). The fundamental chamber is c0 = {λ ∈ h | λ, αi  > 0 for all i = 0, . . . , n} ⊂ S0 . The affine Weyl group acts simply transitively on the set of chambers, and therefore W is in bijection with the set of chambers. Identify 1 with c0 . ˜ = P W0 acts transitively (but in general The extended affine Weyl group W ˜ is not a Coxeter not simply transitively) on the set of chambers. In general W ˜ → Z≥0 group, but it is “nearly” a Coxeter group: There is a length function : W ˜ defined by (w) = |{Hα+jδ | Hα+jδ separates c0 from wc0 }|, and for w ∈ W ⊆ W ˜ | (w) = 0}. Then this agrees with the Coxeter length function. Let Γ = {w ∈ W ˜ ˜ W = W  Γ, and Γ is isomorphic to the finite abelian group P/Q. Therefore W

Random Walks on Buildings

33

˜ can be thought acts simply transitively on the set of chambers in h × Γ, and so W of as |Γ| copies of W . ˜ = P  W0 we define the weight wt(w) ∈ P and the final direction If w ∈ W θ(w) ∈ W0 by the equation w = twt(w) θ(w).

(4.1)

The Bruhat partial order on W is defined as follows: v ≤ w if and only if v is a ‘subexpression’ of a reduced expression w = si1 · · · si
for w. Here subexpression means an expression obtained by deleting one or more factors from the expression w = si1 · · · si
. If v ≤ w then v is a subexpression of every reduced expression ˜ by setting v ≤ w if and only if w = w
γ for w. The Bruhat order extends to W


and v = v γ with w , v ∈ W and γ ∈ Γ and v
≤ w
. 4.2. Alcove walks Each affine hyperplane Hα+kδ determines two closed half-spaces of h. Define an orientation on the affine hyperplane Hα+kδ by declaring the positive side to be the half-space which contains a subsector of the fundamental sector S0 . Explicitly, if α ∈ R+ and k ∈ Z then the negative and positive sides of Hα+kδ are − Hα+kδ = {x ∈ h | x, α + kδ ≤ 0} = {x ∈ h | x, α ≤ −k}, + Hα+kδ = {x ∈ h | x, α + kδ ≥ 0} = {x ∈ h | x, α ≥ −k}.

See the picture in Example 1.3; note that this orientation is translation invariant. ˜ , with γ ∈ Γ. A positively Let w  = si1 · · · si
γ be an expression for w ∈ W ˜, folded alcove walk of type w  is a sequence of steps from alcove to alcove in W ˜ starting at 1 ∈ W , and made up of the symbols . − .... + − .. + xsi xsi −. ... + x x ...................... xsi .................... x (4.2) . . . . . .......... . . . . . . ... ... ... (positive i-crossing) (positive i-fold ) (negative i-crossing) where the kth step has i = ik for k = 1, . . . , . To take into account the sheets ˜ , one concludes the alcove walk by “jumping” to the γ sheet of h × Γ. Our of W pictures will always be drawn without this jump by projecting h × Γ → h × {1}. Let p be a positively folded alcove walk. For each i = 0, 1, . . . , n let fi (p) = #(type i-folds in p). ˜ . Define Let w  = si1 · · · si
γ be a reduced expression for w ∈ W P(w)  = {all positively folded alcove walks of type w}. 

(4.3)

˜ be the alcove where p ends. By the definition of the Bruhat order Let end(p) ∈ W it is clear that if p ∈ P(w)  (with w  reduced) then end(p) ≤ w

in Bruhat order.

(4.4)

Define the weight wt(p) ∈ P and final direction θ(p) ∈ W0 by the equation end(p) = twt(p) θ(p).

(4.5)

34

J. Parkinson and B. Schapira

The dominance order on P is given by µ  λ if and only if λ − µ ∈ Q+. It is not difficult to show that if p ∈ P(w)  then wt(w)  wt(p).

(4.6)

This is a consequence of the paths being ‘positively’ folded. Example 4.1. The positively folded alcove walk p . .. .. ..... .. ... ..... .. .. ...... .. ... ..... .. ... . . . .. . . . .. ..... .......... . . . ............................................ ..... ........... . . ...................................... ..... ............ . . . . . . . . . . ... .. .. .... .. . . ............. v.. .. .... .. .. ...... .. .... .... . .. ... . .. .. . .. .. . . . .... ... . . . ......................................... ..... ........ . ..... . ........................................ ..... ............. . . . ...................................... . . . ..... .. . . . . . .. ... ..... ... ............................ ... . . ...... ... .... .... ... ... .. . . . . . . . . . . . . . . . . . .. . . . . .. ..... ........ . . . ....................................................... ........ . . . ......................................... ..... .......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . ... . . .. . . ... ....... ......... ......... ......... ........ ......... . ... ... ... . .. ... ... . .... ..... ..... ....... . .. ....... ... . ... ... . . ............................................. ..... ........... . . . ........................................ ..... ......... . . . . ........................................ .. .. ... ..... .. ... .... .. ... ..... .. ... .... .. ... .. ... . ... . ... .. .. ... . . . .. . .. .. ..... .......... . . . ............................................ ..... .......... . . .......................................... ..... ............. . . . . .. .. .. .. .. .. .. .. .. .. .. . .. ..

............ = 0 ..... .... = 1 ... =2

Figure 4. A positively folded alcove walk in type A˜2 has type w  = s0 s1 s2 s0 s1 s0 s2 s1 s0 s1 s2 s0 (this is reduced). The end chamber of p is end(p) = v = s0 s1 s2 s0 s1 s2 s1 s0 s2 s0 = s0 s1 s2 s0 s2 s1 s0 s2 ≤ w, and f0 (p) = 1, f1 (p) = 1, and f2 (p) = 0. We have wt(p) = 4ω1 − ω2 and wt(w) = 5ω1 − 6ω2 . Note ∨ + that wt(p) − wt(w) = α∨ 1 + 3α2 ∈ Q and so wt(w)  wt(p). Example 4.2. In type A˜2 , the set P(s1 s2 s1 s0 ) consists of the 10 paths in Figure 5 (arranged according to wt(p)).

........... ................ . . ........ ...

....... ...... ............................. ........................ ....... .......

....................................... ...

.. ........ ...... ....................... ....... ........ .... ..... .. .......... .......................... ... . ............ . . . . . . . . . ....... .... ....................... . ........... ....... ....... . . . . . . . . . . . . . . . ... ......... .. ..... .......... .......

Figure 5. Positively folded alcove walks of type w  = s1 s2 s1 s0 The bottom path is w  (the path with no folds). Note that all other paths have end(p) ≤ w and wt(w)   wt(p).

Random Walks on Buildings

35

4.3. Parameter systems ˜ , etc be as above. A parameter system is a set q = {q0 , q1 , . . . , qn } Let R, W0 , W, W such that (i) qi > 1 for each i = 0, 1, . . . , n, and (ii) qi = qj whenever si and sj are conjugate in W . For example, the parameters of a locally finite regular building form a parameter system. A parameter system is reduced if it satisfies (iii) if R is of type A1 then q0 = q1 , and if R is of type Cn then q0 = qn . The ‘reduced’ hypothesis can be removed, but without it some of the subsequent formulae become more complex. Let q be a reduced parameter system. By [5, IV, §1, No.5, Prop 5] qw := qi1 · · · qi


if w = si1 · · · si
∈ W is a reduced expression

˜ does not depend on the choice of reduced expression. Extend this definition to W be setting qwγ = qw whenever w ∈ W and γ ∈ Γ. For α ∈ R, define qα by qα = qi

if α ∈ W0 αi .

Since α ∈ W0 αi ∪ W0 αj implies that sj = wsi w−1 for some w ∈ W0 this definition is unambiguous. 4.4. Extended affine Hecke algebras ˜ be an extended affine Weyl group and let q be a reduced Definition 4.3. Let W ˜ and parameter system. The extended affine Hecke algebra with Weyl group W ˜ parameter system q is the algebra H over C with generators Tw (w ∈ W ) and defining relations Tu Tv = Tuv

if 1 2

Tw Tsi = Twsi + (qi −

−1 qi 2 )Tw

(uv) = (u) + (v)

if (wsi ) = (w) − 1.

We will usually drop the adjective ‘extended’ and call H the affine Hecke algebra. Remark 4.4. We often write Ti in place of Tsi for i = 0, 1, . . . , n. One immediately 1

−1

sees that each Ti is invertible, with inverse Ti−1 = Ti − (qi2 − qi 2 ), and that ˜ , is invertible. Tγ−1 = Tγ −1 for γ ∈ Γ. It follows that each Tw , w ∈ W Remark 4.5. If q0 , q1 , . . . , qn are the parameters of a locally finite regular building then the subalgebra HW of H generated by Tw , w ∈ W , is isomorphic to A , 1/2 with Tw → qw Aw (see Proposition 1.13). This renormalisation leads to neater formulas in the Hecke algebra theory. Also it is more convenient to work in the larger extended Hecke algebra.

5. Structure of affine Hecke algebras This section is classical and well known to experts. Standard references include [23], [25], [27], and [42]. The main results we describe are: • The Bernstein presentation. This realises the semidirect product structure ˜ = P  W0 of the extended affine Weyl group at the Hecke algebra level. W

36

J. Parkinson and B. Schapira • The computation of the centre of H . This is useful because the centre of an algebra plays an important role in its representation theory. • The derivation of the Macdonald formula. This formula is key to the Plancherel formula on the centre of H .

5.1. Bernstein presentation of H ˜ , and choose any expression v = si · · · si γ for v (not necessarily Let v ∈ W 1
reduced). Interpret this expression as an alcove walk with no folds starting at the ˜ . Let 1 , . . . , 
∈ {−1, +1} be the signs of the crossings of this walk. alcove 1 ∈ W The element xv = Ti11 · · · Ti

Tγ does not depend on the particular expression for v chosen (see [15]). ˜ , and choose a reduced expression w Proposition 5.1. Let w ∈ W  = si1 · · · si
γ. Then n   1 −1 Tw = Q(p)xend(p) where Q(p) = (qi2 − qi 2 )fi (p) . i=0

p∈P(w)

1

− 12

Proof. This is an easy induction using the formula Ti = Ti−1 + (qi2 − qi

).




˜ } is a basis of H . The transition matrices Corollary 5.2. The set {xv | v ∈ W ˜ } and {xv | v ∈ W ˜ } are upper triangular converting between the bases {Tw | w ∈ W with respect to the Bruhat order, and have 1s on the main diagonal. For µ ∈ P , define xµ = xtµ . The relations in the following presentation of H are the algebra analogues of the defining relations: si sj si · · · = sj si sj · · ·,      

s2i = 1,

mij terms

tλ tµ = tλ+µ = tµ+λ ,

si tλ = tsi λ si .

mij terms

˜. (i, j = 1, . . . , n and λ, µ ∈ P ) in the extended affine Weyl group W Theorem 5.3 (Bernstein Presentation). For all i, j = 1, . . . , n and all λ, µ ∈ P we have 1

− 12

Ti2 = 1 + (qi2 − qi

)Ti

Ti Tj Ti · · · = Tj Ti Tj · · · λ µ

λ+µ

x x =x

(mij terms on each side)

µ λ

=x x

1

− 12

Ti xµ = xsi µ Ti + (qi2 − qi

)

xµ − xsi µ ∨ 1 − x−αi

(the Bernstein relation).

Proof. These facts can be deduced from the alcove walk setup. See [32].




Remark 5.4. The ‘fraction’ appearing in the Bernstein relation is actually an element of C[P ], because si µ = µ − µ, αi α∨ i , and µ, αi  ∈ Z since µ ∈ P .

Random Walks on Buildings

37

Corollary 5.5. The sets {xµ Tw | µ ∈ P, w ∈ W0 }

and

{Tw xµ | µ ∈ P, w ∈ W0 }

are both bases for H . Proof. Since tµ is in the ‘1-position’ of tµ W0 , and since the orientation on the hyperplanes is translation invariant we have xtµ w = xµ Tw−1 −1

for all µ ∈ P and all w ∈ W0 .

Therefore {xµ Tw−1 −1 | µ ∈ P, w ∈ W0 } is a basis of H , and the result follows from Corollary 5.2 and the Bernstein relation.
It is not difficult to use the Bernstein relation to compute the centre of H . Let C[P ] denote the C-span of the elements xλ , λ ∈ P . Then C[P ] carries a natural W0 -action (with w · xλ = xwλ ), and we write C[P ]W0 = {p ∈ C[P ]W0 | w · p = p for all w ∈ W0 }. Corollary 5.6. The centre of H is Z(H ) = C[P ]W0 . Proof. If z ∈ C[P ]W0 then Theorem 5.3 gives Tw z = zTw and xµ z = zxµ for all w ∈ W0 and µ ∈ P . Therefore z ∈ Z(H ). Conversely suppose that z ∈ Z(H ). Use Corollary 5.5 to write  z= pw (x)Tw where pw (x) ∈ C[P ]. w∈W0

Let w be a maximal element of W0 (in the Bruhat order) subject to the condition that pw (x) = 0. Since xλ zx−λ = z the Bernstein relation gives pw (x) = xλ−wλ pw (x) for all λ ∈ P , and so w = 1. Therefore z ∈ C[P ]. Then for i = 1, . . . , n we have zTi = Ti z = (si z)Ti + z
for some z
∈ C[P ], and so zTi = (si z)Ti by Corollary 5.5. Thus z = si z for each i, so z ∈ C[P ]W0 .
5.2. The Macdonald formula It is natural to seek modifications τw of the elements Tw which satisfy the “simplified Bernstein relation” τw xµ = xwµ τw

for all w ∈ W0 and µ ∈ P .

For each i = 1, . . . , n define the intertwiner τi ∈ H by ∨

1

− 12

τi = (1 − x−αi )Ti − (qi2 − qi

).

By Theorem 5.3 we have τi xµ = xsi µ τi for all µ ∈ P , and a direct computation (using Theorem 5.3) gives ∨

∨

τi2 = qi (1 − qi−1 x−αi )(1 − qi−1 xαi ) ∈ C[P ]. It can be shown that τw = τi1 · · · τi


(5.1)

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J. Parkinson and B. Schapira

is independent of the choice of reduced expression w = si1 · · · si
∈ W0 , and that the τw are linearly independent over C[P ]. Define 10 ∈ H by   1 1 10 = qw2 Tw , where W0 (q) = qw . (5.2) W0 (q) w∈W0

w∈W0

Induction on (w) shows that 1

for all w ∈ W0 ,

Tw 10 = 10 Tw = qw2 10

and so

120 = 10 .

(5.3)

Therefore 1

− 12

∨

10 τi = qi2 10 (1 − qi−1 xαi ) and τi 10 = −qi

∨

∨

x−αi (1 − qi−1 xαi )10 .

(5.4)

Define elements d(x), n(x) ∈ C[P ] by   ∨ ∨ d(x) = (1 − x−α ) and n(x) = (1 − qα−1 x−α ). α∈R+

(5.5)

α∈R+

Theorem 5.7. We have qw0  − 12 d(x)10 = qw cw τw , W0 (q)

where



cw =

∨

(1 − qα−1 x−wα ).

α∈R(w −1 w0 )

w∈W0

q

0 In particular, the coefficient of τe in d(x)10 is W0w(q) n(x).  Proof. We have d(x)10 = w∈W0 aw τw for some polynomials aw ∈ C[P ] (because each d(x)Tw with w ∈ W0 has this property). This expression is unique, because 1/2 the τw are linearly independent over C[P ], and obviously aw0 = qw0 W0 (q)−1 . On the one hand using (5.4) we see that for each i = 1, . . . , n we have  1 1 ∨ ∨ d(x)10 τi = d(x)10 qi2 (1 − qi−1 xαi ) = qi2 aw (1 − qi−1 xwαi )τw ,

w∈W0

and on the other hand direct computation gives   d(x)10 τi = aw τw τi = awsi τw + w:wsi <w

w∈W0



awsi τw τi2 .

w:wsi >w

Since τi2 ∈ C[P ] we deduce that 1

∨

qi2 aw (1 − qi−1 xwαi ) = awsi

whenever (wsi ) = (w) − 1.

Write w0 = wsi1 · · · si
with = (w0 ) − (w). Then 1

∨

aw = qi21 awsi1 (1 − qi−1 x−wαi1 ) = · · · = qw−1 w0 aw0 1 1/2



∨

(1 − qα−1 x−wα ),

α

where the product is over α ∈ {αi1 , si1 αi2 , . . . , si1 · · · si
−1 αi
} = R(w−1 w0 ).




Random Walks on Buildings

39

 Lemma 5.8. Let ρ = 12 α∈R+ α∨ . If w ∈ W0 then    1 ∨ τw 10 = (−1)
(w) qw2 x−ρ+wρ (1 − qβ−1 x−wβ ) 10 . β∈R(w −1 )

Proof. It follows from (5.4) that if w = si1 · · · si
is reduced then  1 ∨ ∨ τw 10 = −qα2 x−α (1 − qα−1 xα ) 10 , α

where the product is over α ∈ {αi1 , si1 αi2 , . . . , si1 · · · si
−1 αi
} = R(w). Since R(w) = −wR(w−1 ) it follows that    1  ∨ ∨ τw 10 = (−1)
(w) qw2 x β∈R(w−1 ) wβ (1 − qβ−1 x−wβ ) 10 , and the result follows since wρ − ρ =

β∈R(w −1 )



β∈R(w −1 )

wβ ∨ .




Theorem 5.9. For all µ ∈ P we have the Macdonald formula   1 − q −1 x−α∨ q w0  α µ µ 10 x 10 = Pµ (x)10 where Pµ (x) = w x . W0 (q) 1 − x−α∨ + w∈W0

α∈R

Proof. By Theorem 5.7 and Lemma 5.8 we have qw0  − 12 d(x)10 xµ 10 = qw cw xwµ τw 10 W0 (q) w∈W0 qw0 −ρ  = x (−1)
(w) (wn(x))xwµ+wρ 10 . W0 (q) w∈W0

By Bourbaki [5, VI, §3, No. 3, Proposition 2] the polynomial  p(x) = (−1)
(w) (wn(x))xwµ+wρ is divisible by

xρ d(x),

w∈W0 ρ

and since w(x d(x)) = (−1)
(w) xρ d(x) we have   1 − q −1 x−wα∨ p(x) α wµ = x , xρ d(x) 1 − x−wα∨ + w∈W0

completing the proof.

α∈R




Remark 5.10. The above computation can be used to prove the Satake isomorphism 10 H 10 ∼ = Z(H ) = C[P ]W0 , because {10 xλ 10 | λ ∈ P + } is a basis for 10 H 10 and {Pλ (x) | λ ∈ P + } is a basis for C[P ]W0 .

40

J. Parkinson and B. Schapira

5.3. Some representation theory The calculation of the centre of H has important implications for the representation theory of H . Proposition 5.11. Let (π, V ) be an irreducible representation of H over C. 1. There is an element t ∈ Hom(P, C× ) such that π(z) = ht (z)I

for all z ∈ C[P ]W0 ,

where ht : C[P ] → C is the evaluation homomorphism given by ht (xλ ) = tλ

for all λ ∈ P , where tλ := t(λ).

2. We have ht (z) = ht (z) for all z ∈ C[P ]W0 if and only if t
∈ W0 t. 3. V is necessarily finite dimensional. Proof. 1. The algebra H has countable dimension. Therefore by Dixmier’s infinitedimensional generalisation of Schur’s Lemma (see [39, §5.3, Lemma 9]), if (π, V ) is an irreducible representation of H then Z(H ) = C[P ]W0 acts on V by scalars. Thus π : H → End(V ) induces an algebra homomorphism h : C[P ]W0 → C by π(z) = h(z)I for all z ∈ C[P ]W0 . Since C[P ] is integral over C[P ]W0 each algebra homomorphism h : C[P ]W0 → C is the restriction of some homomorphism C[P ] → C. Therefore h = ht for some t ∈ Hom(P, C× ). 2. Exercises 12 and 13 in [2, Chapter V] show that the homomorphisms ht and ht agree on C[P ]W0 if and only if t
∈ W0 t. 3. Since C[P ] is integral over C[P ]W0 , and since {Tw xλ | w ∈ W0 , λ ∈ P } is a vector space basis of H , it follows that H is finite dimensional as a C[P ]W0 module, and hence V is finite dimensional (by part 1).
Remark 5.12. It can be shown that if (π, V ) is an irreducible representation of H then dim(V ) ≤ |W0 | (see [20]). Definition 5.13. The element t ∈ Hom(P, C× ) in Proposition 5.11 is the central character of (π, V ). To be more precise, the central character is the orbit W0 t. Note that H = H0 ⊗ C[P ] where H0 is the |W0 |-dimensional subalgebra generated by Tw , w ∈ W0 . This allows us to write down finite-dimensional representations of H by inducing representations of the commutative subalgebra C[P ] to H . For t ∈ Hom(P, C× ) let Cvt be the one-dimensional representation of C[P ] with action xλ · vt = tλ vt . Definition 5.14. Let t ∈ Hom(P, C× ). The principal series representation of H with central character t is (πt , V (t)), where V (t) = IndH C[P ] (Cvt ) = H ⊗C[P ] (Cvt ). We have h · (h
⊗ vt ) = (hh
⊗ vt ) and (xλ ⊗ vt ) = tλ (1 ⊗ vt ). Therefore V (t) has basis {(Tw ⊗ vt ) | w ∈ W0 }, and hence has dimension |W0 |.

Random Walks on Buildings

41

The importance of these representations is given by Kato’s Theorem: Theorem 5.15 (see [20]). We have ∨

1. (πt , V (t)) is irreducible if and only if tα = q ±1 for each α ∈ R. 2. If (π, V ) is an irreducible representation of H with central character t then (π, V ) is a composition factor of (πt , V (t)). See Section 8 for explicit computations of representations in type A˜2 . Another way of building representations is to induce representations from parabolic subalgebras. Again, see Section 8 for some examples. We now give a simple combinatorial formula for the matrix elements of the principal series representation. Recall the definitions of wt(p) ∈ P and θ(p) ∈ W0 from (4.5). It is convenient to generalise the definition of P(w)  from (4.3) to allow ˜ a alcove walks that start at an alcove different from 1. Given u ∈ W0 and w  ∈W reduced expression, let P(w,  u) = {positively alcove walks of type w  starting at the alcove u}. ˜ . Relative to the basis {(T −1 Tw0 ⊗ vt ) | u ∈ W0 }, the Theorem 5.16. Let w ∈ W u matrix elements of the principal series representation (πt , V (t)) are given by  [πt (Tw−1 )]v,u = Q(p)t−w0 (wt(p)) {p∈P(w,u)|θ(p)=v}

where Q(p) is as in Proposition 5.1. ˜ , then by Proposition 5.1 we have Proof. If u ∈ W0 and w ∈ W  −1 (Tw−1 Tu−1 )∗ = Tu−1 Q(p)xwt(p) Tθ(p) −1 Tw = −1 p∈P(w,u)

(the involution ∗ is described in (6.3)). Since (xλ )∗ = Tw0 x−w0 λ Tw−1 we get 0 Tw−1 · (Tu−1 Tw0 ⊗ vt ) = Tw−1 Tu−1 · (Tw0 ⊗ vt )  −1 = Q(p)t−w0 (wt(p)) (Tθ(p) Tw0 ⊗ vt ), p∈P(w,u)

and the result follows.




The positivity of this formula has some very useful applications, for example see the proof of Lemma 3.2.

6. Harmonic analysis for the Hecke algebra In the previous section we recalled some of the well-known structural theory of affine Hecke algebras. In the current section we describe the beginnings of the harmonic analysis on H , following the main line of argument in [28]. The outline

42

J. Parkinson and B. Schapira

is as follows. Define a trace Tr : H → C on H by linearly extending Tr(Tw ) = δw,1 . For fixed t ∈ Hom(P, C× ) define a function Ft : H → C by  Ft (h) = t−µ Tr(xµ h) whenever the series converges. (6.1) µ∈P ∨

We show that the series converges provided each |tαi | < r is sufficiently small. We will see that  1 − q −1 t−α∨ ft (h) n(t) α Ft (h) = , where c(t) = = , (6.2) qw0 c(t)c(t−1 ) d(t) 1 − t−α∨ + α∈R

where d(t) and n(t) are as in (5.5), and where d(t)ft (h) is a linear combination of terms {tλ | λ ∈ P }. Hence ft (h) has a meromorphic continuation (as a function of t). Furthermore ft is related to the character of the principal series representation (πt , V (t)) by f˜t = χt , where f˜t is the symmetrisation of ft . It follows from (6.1) and (6.2) that if dt is normalised Haar measure on the product Tn of n circle groups T then  ft (h) Tr(h) = dt. −1 ) (rT)n qw0 c(t)c(t A more general version of this formula is the main result of [28], and it is at the heart of the harmonic analysis and Plancherel measure for H . 6.1. The C ∗ -algebra Define an involution ∗ on H and a function Tr : H → C by   ∗    cw Tw = cw Tw−1 and Tr cw Tw = c1 . ˜ w∈W

˜ w∈W

(6.3)

˜ w∈W

An induction on (v) using the defining relations in the algebra H shows that Tr(Tu∗ Tv ) = δu,v , and so Tr(h1 h2 ) = Tr(h2 h1 ) It follows that

for all h1 , h2 ∈ H .

(h1 , h2 ) := Tr(h∗1 h2 )

defines a Hermitian inner product on H . Let |h|2 = acts on itself, and the corresponding operator norm is

(6.4)

 (h, h). The algebra H

|h| = sup{|hx|2 : x ∈ H , |x|2 ≤ 1}. Let H denote the completion of H with respect to this norm. It is a noncommutative C ∗ -algebra. Recall from Remark 4.5 that if there is an underlying building then there is an isomorphism ψ : HW → A , where HW is the subalgebra of H generated by 1/2 {Tw | w ∈ W }. The isomorphism is given by ψ(Tw ) = qw Aw for all w ∈ W . It is not immediately clear that the operator norms on A and HW (written as  ·  and | · | respectively) are compatible with ψ, and so we pause to prove the following:

Random Walks on Buildings

43

Proposition 6.1. If there is an underlying building then |h| = ψ(h) for all h ∈ HW . Therefore H W ∼ =A. Proof. Let o ∈ C be a fixed chamber of the underlying building (C, δ). Let 2o (C) be the subspace of 2 (C) consisting of functions which are constant on each set Cw (o). −1 Since Aw δo = qw 1Cw−1 (o) the injective map ω : A → 2o (C), A → Aδo , embeds 2 A into o (C) as a dense subspace (the subspace of finitely supported functions). Therefore η := ω ◦ ψ : HW → 2o (C) embeds HW as a dense subspace of 2o (C), and a straight forward computation shows that |h|2 = η(h)2 for all h ∈ HW . Therefore by the density of η(HW ) in 2o (C) it follows that |h| = ψ(h)o

for all h ∈ HW ,

where  · o is the 2 -operator norm on B( 2o (C)) (note that each A ∈ A maps

2o (C) into itself). It remains to show that Ao = A for all A ∈ A . To see this, note that the homomorphism Φ : A → B( 2o (C)), A → A|
2o (C) is injective, for if Φ(A) = 0 then Aδo = 0, and so A = 0 by Lemma 2.2. But by [10, Theorem I.5.5] an injective homomorphism between C ∗ -algebras is necessarily an isometry, and so A = Ao for all A ∈ A .
We return to the study of the trace functional Tr : H → C. 6.2. A formula for the trace on H 10 It is easy to derive a formula for the trace on H 10 = C[P ]10 using the harmonic analysis on Z(H ). The key idea is that Tr(xµ 10 ) = Tr(xµ 120 ) = Tr(10 xµ 10 ), and then 10 xµ 10 = Pµ (x)10 , where Pµ (x) ∈ Z(H ) is the Macdonald spherical function. First a formula for the trace on Z(H )10 . Theorem 6.2. Let p(x) ∈ C[P ]W0 . Then Tr(p(x)10 ) =

W0 (q) |W0 |qw0

 Tn

p(t) dt. c(t)c(t−1 )

Proof. Since {Pλ (x) | λ ∈ P + } is a basis for C[P ]W0 it suffices to check the formula when p(x) = Pλ (x) for some λ ∈ P + . It is not hard to see that if λ ∈ P + then −1  1 ) −1/2 qw0 W0λ (q 10 xλ 10 = qtλ qw2 Tw , 2 W0 (q) w∈W0 tλ W0

where W0λ = {w ∈ W0 | wλ = λ} (see [27, Lemma 2.7]). Therefore by Theorem 5.9 Tr(Pλ (x)10 ) = Tr(10 xλ 10 ) = δλ,0 . On the other hand since c(t)c(t−1 ) is W0 -invariant we have    W0 (q) Pλ (t) 1  twµ c(wt) tµ dt = dt = dt. −1 ) −1 ) |W0 |qw0 Tn c(t)c(t−1 ) |W0 | Tn c(wt)c(wt Tn c(t w∈W0

∨ Q≥0 α∨ 1 +· · ·+Q≥0 αn

If λ ∈ P then λ ∈ (see [5, VI, §1, No.10]). Therefore if λ = 0 is dominant then the integral is zero by regarding it as an iterated contour integral +

44

J. Parkinson and B. Schapira

of a function that is analytic inside the contours in each variable tω1 , . . . , tωn . The result follows.
Corollary 6.3. If µ ∈ P then



Tr(xµ 10 ) = If µ ∈ / −Q

+

Tn

tµ dt. c(t−1 )

µ

then Tr(x 10 ) = 0.

Proof. If µ ∈ P then by (5.3), (6.4), and Theorem 5.9 we have Tr(xµ 10 ) = Tr(xµ 120 ) = Tr(10 xµ 10 ) = Tr(Pµ (x)10 ), where Pµ (x) ∈ C[P ]W0 . The result now follows from Theorem 6.2 and its proof.
6.3. Opdam’s trace generating function formula Let t ∈ Hom(P, C× ). Define a function Ft : H → C by (6.1). Let us deal im˜ then mediately with the issue of convergence of this series. Recall that if v ∈ W v ∈ tµ W0 for a unique µ ∈ P , and we write wt(v) = µ. ˜ . Then: Lemma 6.4. Let v ∈ W 1/2

1. |Tr(xv )| ≤ 2
(v) qv . 2. If Tr(xv ) = 0 then wt(v) ∈ −Q+ . Proof. 1. We use induction on (v) to prove that  xv = cvw Tw with |cvw | ≤ 2
(v) qv1/2 for all v, w. w∈W

The result follows since Tr(xv ) = cv1 . The case (v) = 0 is trivial, since xγ = Tγ for all γ ∈ Γ. Suppose that (vsi ) = (v) + 1. Then  xvsi = xv Ti = cvw Tw Ti where  = +1 or  = −1. w∈W

If  = +1 then the relations in H give  cvwsi vsi 1 cw = −1 cvwsi + (qi2 − qi 2 )cvw and if  = −1 the relations in H give  1 −1 cvwsi − (qi2 − qi 2 )cvw vsi cw = cvwsi

if (wsi ) > (w) if (wsi ) < (w),

if (wsi ) > (w) if (wsi ) < (w).

Therefore in all cases the induction hypothesis implies that 1

− 12

v 2 i |cvs w | ≤ |cwsi | + (qi − qi

1

1/2 )|cvw | ≤ (1 + qi2 )2
(v) qv1/2 ≤ 2
(vsi ) qvs . i

Random Walks on Buildings

45

2. Recall the path theoretic formula from Proposition 5.1. Using (4.4) and ˜ we have (4.6) this formula shows that for all v ∈ W  Tv = xv + avu xu . {u|u
Inverting this change of basis formula gives  xv = Tv +

cvu Tu .

{u|u
But Tr(xv ) = and hence Tr(xv ) = 0 implies that wt(v)  wt(1) = 0. Therefore by the definition of the dominance order we have wt(v) ∈ −Q+ .
cv1 ,

Corollary 6.5. There exists r > 0 such that for all h ∈ H the series Ft (h) converges ∨ uniformly if each |tαi | < r.


Proof. This is immediate from Lemma 6.4.

Let T be the circle group, and let dt = dt1 · · · dtn be the normalised Haar measure on Tn . Let r > 0 be as in Corollary 6.5, and write Tr = rT. Then  Tr(h) = Ft (h) dt for all h ∈ H . Tn r

This is the starting point for the harmonic analysis on H . Our first task is to compute Ft (h). The following very nice properties are useful. ∨

Proposition 6.6. Let t ∈ Hom(P, C× ) with |tαi | < r for each i = 1, . . . , n. The function Ft : H → C satisfies: 1. Ft is linear. 2. Ft (xλ hxµ ) = tλ+µ Ft (h) for all λ, µ ∈ P and all h ∈ H . 3. Ft (τw ) = δw,1 Ft (1) for all w ∈ W0 . Proof. The first statement is obvious. For the second statement, using the fact that Tr(ab) = Tr(ba) and making a change of variable in the summation gives   Ft (xλ hxµ ) = t−ν Tr(xλ+ν hxµ ) = t−ν Tr(xλ+µ+ν h) = tλ+µ Ft (h). ν∈P

ν∈P

Finally, since τw xλ = xwλ τw for all λ ∈ P and w ∈ W0 , we have tλ Ft (τw ) = Ft (τw xλ ) = Ft (xwλ τw ) = twλ Ft (τw ), and so (tλ − twλ )Ft (τw ) = 0. The condition on t ∈ Hom(P, C× ) implies that if w = 1 then xλ − xwλ = 0 for all λ ∈ P , and the result follows.
∨

Proposition 6.7. Let t ∈ Hom(P, C× ) with |tαi | < r for each i = 1, . . . , n. Then Ft (h) = ft (h)Ft (1)

for all h ∈ H ,

where d(t)ft (h) is a polynomial in {t | λ ∈ P } with complex coefficients. λ

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J. Parkinson and B. Schapira

Proof. If w ∈ W0 then d(x)Tw can be written as a linear combination of the elements {τv | v ∈ W0 } with complex coefficients. Therefore if h ∈ H then d(x)h is in the C-span of {xλ τw | w ∈ W0 , λ ∈ P }. Writing  d(x)h = pw (x)τw with pw (x) ∈ C[P ] w∈W0

and using Proposition 6.6 gives d(t)Ft (h) = Ft (d(x)h) =



pw (t)Ft (τw ) = p1 (t)Ft (1).




w∈W0

Proposition 6.7 shows that for each h ∈ H , the function t → ft (h) has a meromorphic continuation to t ∈ Hom(P, C× ) with possible poles at the points ∨ where tα = 1 for some α ∈ R. Proposition 6.6 immediately implies the following. Corollary 6.8. The function ft : H → C satisfies: 1. ft is linear. 2. ft (xλ hxµ ) = tλ+µ ft (h) for all λ, µ ∈ P and all h ∈ H . 3. ft (xλ τw ) = tλ δw,1 for all λ ∈ P and w ∈ W0 . Let f˜t : H → C be the symmetrisation of ft . That is,  f˜t (h) = fwt (h) for all h ∈ H . w∈W0

The following theorem gives an important connection between f˜t and the character χt of the principal series representation. First a quick lemma. ∨

Lemma 6.9. If tα = 1 for all α ∈ R then V (t) has basis {τw ⊗ vt | w ∈ W0 } and   χt (xλ τw ) = δw,1 tw t . w  ∈W0

Proof. For each w ∈ W0 induction shows that d(x)Tw can be written as a linear combination of elements {xλ τw | w
∈ W0 , λ ∈ P }, and the first claim follows. For all λ ∈ P and w, u ∈ W0 we have −1

(xλ τw ) · (τu ⊗ vt ) = (τw τu xu

w −1 λ

−1

⊗ vt ) = tu

w −1 λ

(τw τv ⊗ vt ).

It follows from (5.1) that for all w, u ∈ W0 we have τw τu ∈ τwu C[P ]. Therefore if w = 1 then diagonal entries of πt (xλ τw ) relative to the basis {τw ⊗ vt | w ∈ W0 } are all zero, and so χt (xλ τw ) = 0. If w = 1 then the diagonal of πt (xλ ) consists of  the terms tw λ with w
∈ W0 , and the result follows.
∨ Theorem 6.10. If tα = 1 for all α ∈ R then f˜t (h) = χt (h) for all h ∈ H .

Proof. Since C[P ]W0 = Z(H ) it is clear that πt (p(x)) = p(t)I for all p ∈ C[P ]W0 . Therefore for all h ∈ H we have χt (p(x)h) = tr(πt (p(x))πt (h)) = tr(p(t)πt (h)) = p(t)χt (h),

Random Walks on Buildings

47

and Corollary 6.8 gives f˜t (p(x)h) = p(t)f˜t (h). Pick p(x) = d(x)d(x−1 ) (this is symmetric). Since d(x)Tw is in the C[P ]-span of {τv | v ∈ W0 } for all w ∈ W0 we see that p(x)h can be written as  p(x)h = pw (x)τw with pw (x) ∈ C[P ]. w∈W0

By Corollary 6.8, Lemma 6.9, and the above observations we see that  p(t)χt (h) = χt (p(x)h) = p1 (wt) = f˜t (p(x)h) = p(t)f˜t (h). w  ∈W0 ∨ So if tα = 1 for all α ∈ R we have f˜t (h) = χt (h) (since p(t) = d(t)d(t−1 ) = 0).




Since Ft (h) = ft (h)Ft (1) the final piece in the puzzle is to compute Ft (1). Theorem 6.11. Let t ∈ Hom(P, C× ) be such that Ft (1) converges. Then Ft (1) =

−1 qw 0 , c(t)c(t−1 )

where

c(t) =

 1 − q −1 t−α∨ n(t) α = . d(t) 1 − t−α∨ + α∈R

∨

Thus for all h ∈ H the series Ft (h) converges if |tα | < qα−1 for each α ∈ R+ . Proof. Let µ ∈ P . By Corollary 6.3 and the definition of Ft we have   ∨ n    t−µ uµ uαi du −µ µ Ft (10 ) = t Tr(x 10 ) = du = . ∨ ∨ −1 α α i i ) u − t c(u−1 ) Tn c(u Tn + + µ∈−Q

i=1

µ∈−Q

Considering this integral as a iterated contour integral, and computing the simple ∨ ∨ residues at uαi = tαi gives Ft (10 ) =

1 . c(t−1 )

(6.5)

By Theorem 5.7 we have d(x)10 = qw0 n(x) +



aw (x)τw

for some polynomials aw (x) ∈ C[P ]

w∈W0 ,w =1

and so by Proposition 6.6 we have d(t)Ft (10 ) = Ft (d(x)10 ) = qw0 Ft (n(x)) = qw0 n(t)Ft (1) and the result follows from (6.5).




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˜2 7. The Plancherel Theorem for A Let us prove the Plancherel Theorem in the A˜2 case. In fact we will prove the Plancherel formula for the algebra HW (this is slightly more convenient; similar computations work for the extended affine Hecke algebra). This is the affine Hecke algebra with generators Tw , w ∈ W , and relations  Twsi if (wsi ) = (w) + 1 Tw Tsi = 1 − 12 2 Twsi + (q − q )Tw if (wsi ) = (w) − 1, where W is the Coxeter group from Figure 1. Alternatively this algebra has basis {xµ Tw | w ∈ W0 , µ ∈ Q}, where W0 is the parabolic subgroup of W generated ∨ by s1 and s2 , and where Q = Zα∨ 1 + Zα2 . The relations that hold amongst the elements of this basis are given by Theorem 5.3. Summarising the results of the last section, for all h ∈ HW we have    ft (h) Tr(h) = Ft (h) dt = ft(h)Ft (1) dt = dt (7.1) −1 ) q c(t)c(t 2 2 2 Tr Tr Tr w0 where r > 0 is sufficiently small, and where ft (h) = φh (t)/d(t) for some linear combination φh (t) of the terms tλ , λ ∈ Q, with f˜t (h) = χt (h). Therefore we are lead to consider integrals of the form  f (t) If = dt where f (t) = φ(t)/d(t) with φ analytic on (C× )2 . −1 ) c(t)c(t 2 Tr Lemma 7.1. Let If be as above, and let c(t) and c1 (u) be as in Theorem 2.3. Then   f (t) q(q − 1)2 gf (u) q 3 (q − 1)3 dt + du + f (q −1 , q −1 ), If = 2 q 2 − 1 T |c1 (u)|2 q3 − 1 T2 |c(t)| 1

1

1

1

where gf (u) = f (q 2 u, q −1 ) + f (q − 2 u−1 , q − 2 u) + f (q −1 , q 2 u). Proof. This is just some residue calculus. Write   If = Iφ (t2 ) dt2 where Iφ (t2 ) = Tr

Tr

φ(t)d(t−1 ) dt1 . n(t)n(t−1 )

1 dz Fix t2 ∈ Tr . Consider the integral Iφ (t2 ) as a contour integral (dt1 = 2πi ) z along Cr (the circular contour with radius r and centre 0 traversed once counterclockwise). We will shift this contour to C1 . In doing so we will pick up residue contributions from the poles of the integrand lying between Cr and C1 . Since −1 −1 −1 n(t) = (1 − q −1 t−1 t2 )(1 − q −1 t−1 1 )(1 − q 1 t2 ) we see that the only pole between Cr and C1 (for fixed t2 ∈ Tr ) comes from the term n(z, t2 )−1 at z = q −1 . Therefore a residue computation gives  q(q − 1) If = Jφ (t1 ) dt1 + 2 Kφ , q −1 T

Random Walks on Buildings where



Jφ (t1 ) =

Tr



φ(t)d(t−1 ) dt2 n(t)n(t−1 )

and

Kφ =

Tr

49

t2 φ(q −1 , t2 ) dt2 . −2 t ) (1 − q −1 t−1 2 2 )(1 − q

For fixed t1 ∈ T, the poles of the integrand of Jφ (t1 ) between the contours Cr and C1 are at z = q −1 and z = q −1 t−1 1 . Residue calculus shows that Jφ (t1 ) equals   φ(t)d(t−1 ) q(q − 1) t1 φ(t1 , q −1 ) φ(t1 , q −1 t−1 1 ) dt + 2 − . −1 ) 2 −2 t ) (1 − q −1 t )(1 − q −2 t−1 ) q − 1 (1 − q −1 t−1 T n(t)n(t 1 1 1 )(1 − q 1 (This computation assumes t1 = 1. At t1 = 1 the pole has order 2, but this is a set of measure zero.) Residue computations give  t2 φ(q −1 , t2 ) q2 Kφ = dt − φ(q −1 , q −1 ). 2 −1 3 −1 t −2 t ) q − 1 (1 − q )(1 − q T 2 2 Putting these computations together and using the formula φ(t) = d(t)f (t) it follows that If equals   (1 − t)(1 − qt−1 )gf
(t) f (t) q(q − 1)2 q 3 (q − 1)3 dt + dt + f (q −1 ,q −1 ). −1 ) q 2 − 1 T (1 − q −1 t−1 )(1 − q −2 t) q3 − 1 T2 c(t)c(t 1

where gf
(t) = f (t, q −1 )+f (t−1 , q −1 t)+f (q −1 , t). After a change of variable t = q 2 u 1 in the second integral it becomes an integral over q − 2 T, and the integrand has no poles between this contour and T. Therefore we can expand the contour to T for free. The result follows.
(1)

We have already encountered the representations πu and π (2) . See the next section for the details. Lemma 7.2. We have f

1

(q 2 u,q−1 )

(h) + f

1

1

(q− 2 u−1 ,q− 2 u)

(h) + f

1

(q−1 ,q 2 u)

(h) = χ(1) u (h)

for all h ∈ HW ,

and f(q−1 ,q−1 ) (h) = χ(2) (h) for all h ∈ HW . Proof. The first statement is similar to Theorem 6.10. Here is an outline (see the 1 1 next section). If u = q − 2 , q 2 then the representation space V (u) = HW ⊗H1 (Cvu ) has basis {1 ⊗ vu , τ2 ⊗ vu , τs1 s2 ⊗ vu }. One computes τ1 ⊗ vu = 0, and it easily (1) follows that the diagonal entries of the matrices πu (τw ) with w = 1 are all zero. (1) Therefore χu (τw ) = 0 for all w = 1. The result easily follows from Corollary 6.8. ∨ To see that f(q−1 ,q−1 ) (h) = χ(2) (h) for all h ∈ HW , note that xα1 = ∨ ∨ ∨ T2−1 T0 T2 T1 and xα2 = T1−1 T0 T1 T2 , and so χ(2) (xα1 ) = χ(2) (xα2 ) = q −1 . On ∨ ∨ the other hand, f(q−1 ,q−1 ) (xkα1 +
α2 τw ) = δw,1 q −k−
, and the result follows.
We can now prove the Plancherel Theorem for A˜2 .

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J. Parkinson and B. Schapira

Proof of Theorem 2.3. By (7.1) and Lemmas 7.1 and 7.2 we have   (1) 1 ft (h) (q − 1)2 χu (h) (q − 1)3 (2) Tr(h) = 3 dt + du + 3 χ (h). 2 2 2 2 q T2 |c(t)| q (q − 1) T |c1 (u)| q −1 The result follows from Theorem 6.10 (by symmetrising the first integral).




8. Some explicit representations (1)

Let us construct the representations πt , πu , and π (2) of HW that appear in the Plancherel Theorem. A good reference for the representation theory of rank 2 affine ˜2 , and G ˜ 2 ) is [33]. Hecke algebras (that is, those of type A˜1 × A˜1 , A˜2 , B Principal series representations. The principal series representation with central character t ∈ Hom(Q, C× ) is (πt , V (t)), where V (t) = HW ⊗C[Q] Cvt . Here Cvt is the one-dimensional representation of C[Q] given by xλ · vt = tλ vt . The representation space V (t) has basis {Tw ⊗ vt | w ∈ W0 }. The following example illustrates how to compute relative to this basis. Write ∨ ∨ ∨ 1 1 q = q 2 − q − 2 . The Bernstein relation gives xα1 T1 = T1 x−α1 + q(1 + xα1 ), and so ∨ xα1 acts on the basis element T1 ⊗ vt by  ∨ ∨ ∨  xα1 · (T1 ⊗ vt ) = T1 x−α1 + q(1 + xα1 ) ⊗ vt ∨

∨

= t−α1 (T1 ⊗ vt ) + q(1 + tα1 )(1 ⊗ vt ). Computing the matrices πt (T1 ) and πt (T2 ), is straight forward, and we recover −1/2 the matrices from Section 2.3 (remember that Aw ↔ qw Tw ). One can compute ϕ∨ πt (T0 ) using x = T0 T1 T2 T1 where ϕ = α1 + α2 , but it is quicker to use  −1 ∨ t−w ϕ (Tsϕ w ⊗ vt ) if ϕ ∈ R(w) T0 · (Tw ⊗ vt ) = −w−1 ϕ∨ t (Tsϕ w ⊗ vt ) + q(Tw ⊗ vt ) if ϕ ∈ / R(w) which follows from [25, (3.3.6)]. Therefore the matrix πt (T0 ) is given by ∨

T0 · (1 ⊗ vt ) = q(1 ⊗ vt ) + t−ϕ (Ts1 s2 s1 ⊗ vt ) ∨

T0 · (T1 ⊗ vt ) = q(T1 ⊗ vt ) + t−α2 (Ts1 s2 ⊗ vt ) ∨

T0 · (T2 ⊗ vt ) = q(T2 ⊗ vt ) + t−α1 (Ts2 s1 ⊗ vt ) ∨

T0 · (Ts1 s2 ⊗ vt ) = tα2 (T1 ⊗ vt ) ∨

T0 · (Ts2 s1 ⊗ vt ) = tα1 (T2 ⊗ vt ) ∨

T0 · (Ts1 s2 s1 ⊗ vt ) = tϕ (1 ⊗ vt ). Induced representations. Let H1 be the (infinite-dimensional) subalgebra of HW generated by T1 and C[Q]. Let u ∈ C× , and let Cvu be a 1-dimensional representation of H1 with 1

T1 · vu = −q − 2 vu ,

∨

xα1 · vu = q −1 vu ,

and

∨

1

xα2 · vu = q 2 u vu .

Random Walks on Buildings

51

(1)

Let (πu , V (u)) be the induced representation of HW with representation space W given by V (u) = IndH H1 (Cvu ) = HW ⊗H1 Cvu . The representation space V (u) has basis {1 ⊗ vu , T2 ⊗ vu , Ts1 s2 ⊗ vu }, and straightforward computations give 1

T1 · (1 ⊗ vu ) = −q − 2 1 ⊗ vu T1 · (T2 ⊗ vu ) = Ts1 s2 ⊗ vu T1 · (Ts1 s2 ⊗ vu ) = T2 ⊗ vu + q Ts1 s2 ⊗ vu

T2 · (1 ⊗ vu ) = T2 ⊗ vu T2 · (T2 ⊗ vu ) = 1 ⊗ vu + q T2 ⊗ vu 1

T2 · (Ts1 s2 ⊗ vu ) = −q − 2 Ts1 s2 ⊗ vu (1)

giving the matrices stated in Section 2.3. One can compute πu (T0 ) using the ∨ formula xϕ = T0 T1 T2 T1 , or by using [25, (3.3.6)]. The result is given in Section 2.3. The 1-dimensional representation. The representation π (2) with representation ∨ 1 space C has π (2) (Ti ) = −q − 2 for i = 0, 1 and 2, and since xα1 = T2−1 T0 T2 T1 and ∨ ∨ ∨ xα2 = T1−1 T0 T1 T2 we have π (2) (xα1 ) = π (2) (xα2 ) = q −1 . Acknowledgment The first author thanks Donald Cartwright for helpful discussions on related topics over many years. It is a pleasure to present this paper at a conference in honour of his birthday. Indeed both Donald Cartwright and Jean-Phillipe Anker suggested this problem to us, and we thank them both very warmly. The first author also thanks Arun Ram for teaching him about affine Hecke algebras. We also thank E. Opdam for helpful conversations regarding his work on the Plancherel measure of affine Hecke algebras. Finally, thank you to Wolfgang Woess for organising the workshop Boundaries in Graz, Austria, June-July 2009. Part of the research for this paper was undertaken in Graz where the first author was supported under the FWF (Austrian Science Fund) project number P19115-N18.

References [1] P. Abramenko and K. Brown, Buildings: Theory and Applications, Graduate Texts in Mathematics, 248. Springer, New York, (2008). [2] M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, AddisonWesley Series in Mathematics, Addison-Wesley, (1969). [3] H. Baumg¨ artel, Analytic perturbation theory for matrices and operators, Operator Theory: Advances and Applications, 15. Birkh¨ auser Verlag, Basel, (1985). [4] P. Bougerol, Th´eor` eme central limite local sur certains groupes de Lie, Ann. Sci. ´ Ecole Norm. Sup. (4) 14 (1981), no. 4, 403–432. [5] N. Bourbaki, Lie Groups and Lie Algebras, Chapters 4–6, Elements of Mathematics, Springer-Verlag, Berlin Heidelberg New York, (2002). [6] K. Brown and P. Diaconis, Random walks and hyperplane arrangements Ann. Probab. 26 (1998), no. 4, 1813–1854. [7] L.J. Billera and K. Brown and P. Diaconis, Random walks and plane arrangements in three dimensions, Amer. Math. Monthly 106 (1999), no. 6, 502–524.

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˜n , Monatsh. [8] D.I. Cartwright, Spherical harmonic analysis on buildings of type A Math. 133 (2001), no. 2, 93–109. ˜d , [9] D.I. Cartwright and W. Woess, Isotropic random walks in a building of type A Math. Z. 247 (2004), no. 1, 101–135. [10] K. Davidson, C ∗ -algebras by example, Fields Institute Monographs, 6, American Mathematical Society, Providence, RI, (1996). [11] P. Diaconis, From shuffling cards to walking around the building: An introduction to modern Markov chain theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math. 1998, Extra Vol. I, 187–204. [12] P. Diaconis and A. Ram, Analysis of systematic scan Metropolis algorithms using Iwahori-Hecke algebra techniques, Michigan Math. J. 48 (2000), 157–190. [13] J. Dixmier, C ∗ -algebras, Translated from the French by F. Jellett. North-Holland Mathematical Library, Vol. 15. North-Holland Publishing Co., Amsterdam-New York-Oxford, (1977). [14] W. Fulton, Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bulletin of the American Mathematical Society, Vol. 37, No. 3, 209–249 (2000). [15] U. G¨ ortz, Alcove walks and nearby cycles on affine flag manifolds, J. Alg. Comb., 26, (2007), 415–430. [16] Y. Guivarc’h, Loi des grands nombres et rayon spectral d’une marche al´ eatoire sur un groupe de Lie, Ast´erisque, vol. 74, (1980), 47–98. [17] Y. Guivarc’h and M. Keane and P. Roynette, Marches al´eatoires sur les groupes de Lie, LNM Vol. 624, Springer-Verlag, (1977). [18] Harish-Chandra Collected papers, Vol IV, 1970–1983. Edited by V.S. Varadarajan. Springer-Verlag, New York, (1984). [19] J.E. Humphreys, Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29. Cambridge University Press, Cambridge, (1990). [20] S.-I. Kato, Irreducibility of principal series representations for Hecke algebras of affine type, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 28 (1981), no. 3, 929–943. [21] T. Kato, Perturbation theory for linear operators, Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, (1976). [22] M. Lindlbauer and M. Voit, Limit theorems for isotropic random walks on triangle buildings, J. Aust. Math. Soc. 73 (2002), no. 3, 301–333. [23] G. Lusztig, Affine Hecke algebras and their graded versions, Journal of the American Mathematical Society, Vol. 2, No. 3, (1989). [24] I.G. Macdonald, Spherical functions on a group of p-adic type, Publications of the Ramanujan Institute, No. 2. Ramanujan Institute, Centre for Advanced Study in Mathematics, University of Madras, Madras, (1971). [25] I.G. Macdonald, Affine Hecke algebras and orthogonal polynomials, Cambridge Tracts in Mathematics, 157. Cambridge University Press, Cambridge, (2003). [26] H. Matsumoto, Analyse harmonique dans les syst`emes de Tits bornologiques de type affine, Lecture Notes in Mathematics, Vol. 590. Springer-Verlag, Berlin-New York, (1977).

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[27] K. Nelson and A. Ram, Kostka-Foulkes Polynomials and Macdonald Spherical Functions, Surveys in Combinatorics, 2003 (Bangor), London Math. Soc. Lecture Note Ser., Cambridge University Press, 307, (2003), 325–370. [28] E. Opdam, A generating function for the trace of the Iwahori-Hecke algebra, Studies in memory of Issai Schur (Chevaleret/Rehovot, 2000), 301–323, Progr. Math., 210, Birkh¨ auser Boston, Boston, MA, (2003). [29] E. Opdam, On the spectral decomposition of affine Hecke algebras, J. Inst. Math. Jussieu 3 (2004), no. 4, 531–648. [30] J. Parkinson, Buildings and Hecke algebras, J. Algebra 297 (2006), no. 1, 1–49. [31] J. Parkinson, Isotropic random walks on affine buildings, Ann. Inst. Fourier (Grenoble) 57 (2007), no. 2, 379–419. [32] A. Ram, Alcove walks, Hecke algebras, spherical functions, crystals and column strict tableaux, Pure Appl. Math. Q. 2 (2006), no. 4, part 2, 963–1013. [33] A. Ram, Representations of rank two affine Hecke algebras, in Advances in Algebra and Geometry, University of Hyderabad conference 200, Ed. C. Musili, Hindustan Book Agency, 2003, 57–91 [34] M. Ronan, A construction of buildings with no rank 3 residues of spherical type, Buildings and the Geometry of Diagrams (Como 1984), Lecture Notes in Math., 1181, 242–284, (1986). [35] M. Ronan, Lectures on buildings, Revised edition, University of Chicago press, (2009). [36] L. Saloff-Coste and W. Woess, Transition operators, groups, norms, and spectral radii, Pacific Journal of Mathematics, Vol. 180, No. 2, (1997). [37] S. Sawyer, Isotropic random walks in a tree, Z. Wahrsch. Verw. Gebiete 42 (1978), no. 4, 279–292. [38] F. Tolli, A local limit theorem on certain p-adic groups and buildings, Monatsh. Math. 133 (2001), no. 2, 163–173. [39] V.S. Varadarajan, An introduction to harmonic analysis on semisimple Lie groups, Cambridge Studies in Advanced Mathematics, 16. Cambridge University Press, Cambridge, (1989). [40] A.D. Virtser, Central limit theorem for semi-simple Lie groups, Theory Probab. Appl. 15 (1970), 667–687. [41] W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, (2000). [42] N.H. Xi, Representations of affine Hecke algebras, Lecture Notes in Mathematics, 1587. Springer-Verlag, Berlin, (1994). James Parkinson School of Mathematics and Statistics University of Sydney, Australia Bruno Schapira D´epartment de Math´ematiques d’Orsay Universit´e Paris-Sud, France

Progress in Probability, Vol. 64, 55–64 c 2011 Springer Basel AG


On Continuity of Range, Entropy and Drift for Random Walks on Groups Anna Erschler Abstract. We study continuity of various characteristics of random walks on groups with respect to strong convergence of measures. Mathematics Subject Classification (2000). Primary 60B15; Secondary 60G50. Keywords. Range of random walk, entropy, drift.

1. Introduction Let G be a finitely generated group and µ a probability measure on G. The range of a random walk is defined as   R(n, µ) R(µ) = E lim , n→∞ n where R(n, µ) is the number of distinct points visited by the random walk until the moment n. By subadditivity this limit always exists. We recall that a random walk is said to be recurrent, if with probability 1 it visits the origin at least once after the moment 1. It is clear that this is equivalent to the condition that the random walk visits the origin infinitely many times with probability one. The range of the random walk is zero if and only if the random walk is recurrent [2].  We recall that the entropy of a probability measure µ is defined by H(µ) = − G µ(g) log µ(g) < ∞. The entropy of the random walk is defined by H(µ∗n ) . n→∞ n

h(µ) = lim

The entropy criterion states that for any measure µ of finite entropy the Poisson boundary of (G, µ) is trivial if and only if h(µ) = 0 (Kaimanovich, Vershik [9, Theorem 1.1] and Derriennic [2]).

56

A. Erschler

If the measure µ has finite first moment with respect to a word metric lS , that is if lS (g)µ(g) < ∞, one can define the drift of the random walk  ∗n g∈G l(g)µ (g) lS (µ) = lim . n→∞ n It is known that for finite first moment symmetric measures h(µ) = 0 if and only if lS (µ) = 0 (see Varopoulos [12] for finitely supported measures and Karlsson Ledrappier [10] for measures with finite first moment). We say that a sequence of probability measures µi on G converges strongly to µ, if there exists a finite set K ⊂ G such that the supports of µi lie inside K for all sufficiently large i and µi (g) converges to µ(g) for all g ∈ K. If we do not assume that the supports of the measures are finite, as it is done in the definition of the strong convergence, then there are various classes of groups where entropy of random walks is known to be not continuous with respect to point-wise or other stronger types of convergence of the defining measures. Kaimanovich ([8], Theorem 3.1 and remarks after that theorem) shows that an infinite symmetric group admits a finite entropy symmetric measure with nontrivial boundary (and, therefore, with positive entropy of the random walk). All finitely supported approximations of this defining measure define a random walk on a finite group, and thus have zero entropy of the random walk. A similar behavior occurs on some other infinite locally finite groups (see Theorem 4.1 [8] for the case of wreath products of (Z/2Z)∞ with finite groups of cardinality at least two). Another well-known example is G = Z2  A, where A is a finite group. The group G admits finite first moment symmetric measures with non-trivial boundary. Indeed, it suffices to consider a symmetric transient finite first moment probability measure ν1 on Z2 , and put µ = 1/2(ν1 + ν2 ), where ν2 is a symmetric measure with the support equal to A. By [9] the boundary of the random walk (G, µ) is non-trivial, while finitely supported symmetric approximations of this measure have a trivial boundary, and thus zero entropy of the random walk. In this paper we study continuity of entropy and other characteristic of random walks with respect to strong convergence of (finitely supported) measures. We show that entropy of random walks is continuous for random walks on wreath products of Z with finite groups. We show that for some groups range, drift and entropy of random walks can all be discontinuous with respect to strong convergence of the defining measures. In the case of range, drift and entropy we give examples of groups where a sequence of random walks with zero characteristics converges to a measure with positive characteristics. A special case of interest is the case of symmetric measures. For discontinuity of drift we construct examples also inside the class of symmetric measures. As the entropy of the random walks for the examples we construct is continuous, they show also that h/vl can be discontinuous with respect to strong convergence of symmetric measures (see [13] for more question about h/vl). As for the range, we show that a result from [1] implies the continuity of range for strong convergence of symmetric measures. It

On Continuity of Entropy

57

is unknown whether this type of discontinuity for symmetric measures can occur for entropy. The measures in the examples, admitting discontinuity, that we construct in this paper are not non-degenerate: the group generated by the support of the limit measure is smaller than the group under consideration. One can ask whether one can find a similar phenomenon among non-degenerate measures.

2. Range and entropy First we recall the following simple fact. Lemma 1. Let µi converge strongly to µ. Then (i)

R(µ) ≥ lim sup R(µi ).

(ii)

h(µ) ≥ lim sup h(µi ).

(iii) For any finite generating set S lS (µ) ≥ lim sup lS (µi ). ∗n Proof. Note that for each n the sequence µ∗n i converges strongly to µ , as i tends to infinity. This implies that for each fixed n it holds R(n, µi ) → R(n, µ) and H(n, µi ) → H(n, µ), where H(n, ν) denotes the entropy of the nth convolution of ν and L(n, µi ) → L(n, µ), where L(n, ν) is the average distance to the origin after n steps. Since R(n), L(n) and H(n) are each sub-additive, this implies the statement of the lemma.


2.1. Continuity of range for symmetric measures Proposition 2 (continuity of range, the limit measure is symmetric). Let µi converge strongly to µ. Assume that µ is symmetric. Then R(µ) = lim R(µi ). Proof. We know that R(µ) ≥ lim sup R(µi ). For the proof of the proposition it is sufficient to show that for any  > 0 there exists N such that R(µ) ≤ (1 + )R(µi ) for any i ≥ N . Observe that for any positive  and any sufficiently large i it holds µi > (1 − )µ. Applying [1] we get that for all such i the expected numbers of visits of the origin satisfy the following inequality ∞  n=0

µ∗n i (e) ≤

∞ 1  ∗n µ (e). 1 −  n=0

Observe that for any G and any µ the range is defined by the expected number of visits to the origin: ∞  1 = µ∗n (e). R(µ) n=0

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In the formula above we allow R(µ) to be equal to 0, in this case we use the notation 1/0 = ∞. Indeed, let F be the probability to return at least once to the origin. We have R(µ) = 1 − F, (see, e.g., Lemma 1 in [3]: a well-known argument in the Abelian group case [11] works for any group). On the other hand it is clear that ∞ 

1 . 1 − F n=0 ∞ In particular, R = 0 if and only if F = 1 if and only if n=0 µ∗n (e) = ∞ if and only if the random walk is transient. Therefore we get that for any positive  and any sufficiently large i 1 1 1 ≤ , Ri 1−R (that is, if R = 0, then for sufficiently large i the range Ri = 0 and satisfies the inequality above). It implies that for sufficiently large i 1 R≤ Ri . 1− This shows that R ≤ lim inf Ri , and completes the proof of the lemma.
µ∗n (e) = 1 + F + F 2 + F 3 + · · · =

2.2. Discontinuity of range and entropy Lemma 3 (Discontinuity of range). Let G be the infinite dihedral group. For any  > 0 there exists a sequence of probability measures µi , which strongly converges to a probability measure µ, such that R(µ) = 1 −  and R(µi ) = 0 for all i. Proof. G is infinite dihedral group, generated by a and b such that a2 = b2 = e. µp is the measure such that µp (ab) = p, µp (ba) = 1 − p. The support of µp generates an infinite cyclic group, and its range is |1 − 2p|. Let µp,i = (1 − 1/i)µp + 1/iµ∗ , where µ∗ is a measure, supported on a, b such that µ∗ (a) = µ∗ (b) = 1/2. Then µp,i converges strongly to µp . First observe that the drift of µp,i is zero for all i. Indeed, if this drift would be positive, then by a general result of [10] we would see that the dihedral group admits a non-zero homomorphism to R, which is impossible, since this group is generated by two elements of order 2. In fact, in this particular case of the dihedral group one can see this claim directly, without using [10], (see Lemma 4.5 [6]). Observe that then l(Xni )/n → 0 with probability one, where Xni is a trajectory of the random walk, defined by the measure µp,i . Since dihedral group has linear growth, this implies that the range of the random walk (G, µp,i ) is zero. Observe also that the range of the limit measure µp is |1 − 2p|, which is non-zero for any p = 1/2.


On Continuity of Entropy

59

Recall  that the wreath product of the groups A and B is a semidirect product  of A and A B, where A acts on A B by shifts: if a ∈ A, f : A → B, f ∈ A B, then f a (x) = f (a−1 x), x ∈ A. Let A  B denote the wreath product. The group A is a quotient of the wreath product AB, and hence any random walk on A  B projects naturally to A. Lemma 4 (Discontinuity of entropy). Consider the wreath product H = G  A, where G is infinite dihedral group and A is finite, of cardinality at least 2. There exists a sequence of probability measures µi on H, which strongly converge to a probability measure µ, such that H(µ) > 0 and for any i it holds H(µi ) = 0. Proof. Consider the infinite cyclic subgroup Z inside G, generated by the product of generators of G. Let µ be any finitely supported non-degenerate measure on ZA, such that its projection on Z has positive mean. For example, we can take µ such that its support is the union of 1 and −1 in Z and A, such that µ(1) > µ(−1). It is well known in this case that (G, µ) has non-trivial boundary (since the projection on Z is transient [9]), and hence the entropy of the random walk is positive. In fact, one can also see directly in this example that the entropy is positive [5]. Now take any non-degenerate generating set S of H. Let µS be the probability measure, equidistributed on S. Put µi = (1−1/i)µ+1/iµS . Obviously, the sequence µi converges strongly to µ. Consider the projection of the random walk (H, µi ) to the dihedral group G. This projected random walk is a non-degenerate random walk on G. It has zero drift and zero range, as we have already mentioned in the proof of the previous lemma. Let SH ⊂ H be a finite subset of H, containing the support of µi and let C be the maximum of the length (with respect to some fixed word metric on G) of the elements in the support of h, h ∈ SH . Observe that if the trajectory of the random walk (H, µi ) visits at the moment n the element Xn = (gn , fn ), where gn ∈ G and fn : G → A, then any element in the support of fn lies at distance at most C from a point of G, visited by the projection of the random walk in the time interval between 0 and n. Therefore we know that with positive probability the support of fn belongs to a sublinear set. This implies that, for all i, the nth step distribution of the random walk (H, µi ) is supported on a sub-exponential set with positive probability, and thus for all i the entropy of the random walk (H, µi ) is zero.
Remark. One can easily check that the sequence of measures µi , converging to µ, constructed in the proof above has the following property: ρ(µi ) = lim sup(µni (e))1/n = 1,

where as

n→∞

ρ(µ) = lim sup(µni (e))1/n < 1, n→∞

and, moreover, lim sup(sup µni (x))1/n < 1. n→∞

x

It is clear for a sequence of symmetric measures (on any group) such phenomenon can not occur.

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3. Random walks on Z
A In the previous section we have discussed the examples of discontinuity of entropy. It seems plausible that for large classes of groups entropy of random walks is continuous. Below we prove that this is the case for random walk on wreath products of Z with finite groups. We hope to return to this question for other classes of groups elsewhere. Theorem 5. Let G = Z  A, A is a finite group. Consider a sequence of probability measures µi that converges strongly to µ. Then h(µi ) converges to h(µ). Proof. Lemma 6. Let G be a finitely generated group and µi is a sequence of probability measures converging strongly to µ. Suppose that there exists A(n), not depending on i, such that A = A(1) + A(2) + A(4) + A(8) + · · · < ∞ and such that for all i entropy of the convolutions of µi satisfies Hµi (2n) ≥ 2Hµi (n) − A(n)2n. Then h(µi ) → h(µ) ¯ µ (n) → h(µi ). ¯ µi (n) = Hµi (n)/n. By definition of entropy we have H Proof. Let H i ¯ µ (2n) ≥ H ¯ µ (n) − A(n). This By the assumption of the lemma we see that H i i implies that for all n and all k ¯ µ (2k+j ) ≥ H ¯ µ (2j ) − A(2j ) − A(2j+1 ) − · · · − A(2j+k−1 )) H i i ¯ µ (2j )−(A(2j )+A(2j+1 )+A(2j+2 )+ Therefore, for all i and all j it holds h(µi ) ≥ H i ∗2j ∗2j ¯ µ (2j ) converges to H ¯ µ (2j ), we see that . . . ) Since µi converges to µ and H i h(µ) ≤ lim inf h(µi ). In view of Lemma 1 this implies that h(µi ) → h(µ).
Lemma 7. For each U > 0 there exists C > 0 such that the following holds. Take any probability measure ν on Z such that the support of ν lies inside [−U, U ] and such that s = xν(x) ≥ 0. Consider a trajectory of the random walk, defined by ν: X1 , X2 , . . . Let mi and Mi be the minimal and, respectively, the maximal point visited until the moment i. Then i) with probability at least 1 − C/n it holds m(n) ≥ −Cn2/3 . ii) With probability at least 1 − C/n it holds M (n) ≤ Xn + Cn2/3 . Proof. There exists a constant C1 , depending on R only, such that the n step transition probability satisfies pi (0, x) ≤ exp(−C1 (|x| − si)2 /i). In particular, for all x > 0 it holds pi (0, −x) ≤ exp(−C1 x2 /i). This implies that for all integers y > 0 probability that for all i ≤ n it holds P [Xi ≤ −y] ≤

∞  x=y

exp(−C1 x2 /i) ≤ exp(−C2 y 2 /i),

On Continuity of Entropy

61

where C2 is a positive constant depending on C1 . Therefore n  exp(−C2 y 2 /i) ≤ n exp(−C2 y 2 /n), P [m(n) ≤ −y] ≤ i=1

and this implies the first statement of the lemma. The second statement follows from the first one, as one observes looking onto the correspondence X1 , X2 , . . . , Xn → Xn − Xn−1 , Xn − Xn−2 , . . . , Xn − X1 , Xn , which one-two-one measure preserving correspondence on the space of trajectories of length n.
Given a group G, we denote in the sequel by φ the mapping from G × G to G that sends (g, g
) to gg
. Lemma 8. Let U = U1 × U2 ⊂ G × G be a subset of cardinality at most V , µU = µU1 × µU2 be a probability measure on U (where µU1 and µU2 are probability measures on U1 and U2 respectively). Suppose that there exists U0 ⊂ G × G such that µU (U0 ) > 1 −  and such that for any g ∈ G the cardinality of the preimage under φ in U0 is at most C. Then H(φ(µU )) ≥ H(µU1 ) + H(µU2 ) − log C −  log V = H(µU ) − log C −  log V Proof. Since µU = µU1 × µU2 , we know that H(µ) = H(µU1 ) + H(µU2 ). Let µ0 be the restriction of µU to U0 . (The measure µ0 is not a probability measure!) Nevertheless, we can consider its entropy and claim that H(φ(µ0 )) ≥ H(µ0 ) − log(C). On the other hand entropy of a positive measure of total mass , defined on a set of cardinality V is not greater than  log V . Therefore, H(µ) ≤ H(µ0 ) +  log V . Therefore, H(φ(µU )) ≥ H(φ(µ0 )) ≥ H(µ0 ) − log(C) ≥ H(µ) −  log V − log(C).




For g = (a, f ) ∈ Z  A we denote by I(g) the minimal interval in Z containing a and the support of f . Lemma 9. Given a finite group A and R > 0 there exists C > 0 such that the following holds. Take any probability measure µ on G = Z  A such that the support of µ lies in the ball of radius R in the standard generators of G. Let X1 , X2 , . . . , ¯ n = X −1 X2n , and Yn be the X2n be a trajectory for the random walk (G, µ). Let X n projection of Xn to Z. i) With probability at least 1 − C/n the length of the intersection I ∗ of I(Xn ) ¯ n ) is at most Cn2/3 . and Yn + I(X (n) ii) There exists a subset W0 in the space of the trajectories of length 2n of (n) probability at least 1 − C/n such that the following holds. Let U0 be the (n) projection of W0 to G × G, under the mapping that sends X1 , . . . , X2n to ¯ n . The number of preimages in U (n) under φ of any point g ∈ G is at Xn , X 0 2/3 most 2nR#ACn .

62

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Proof. i) follows from Lemma 7, since for any i ≥ 1 any element in the support of the configuration of Xi is a point at distance at most R from a point in Z, visited by the trajectory of the projection to Z in the time interval between 0 and i.
ii) follows from i). Indeed, let X1 , . . . , X2n and X1
, . . . , X2n be two trajec
tories of length 2n such that the projection of Xn and Xn to Z coincide (and are
equal to Yn ). Suppose that X2n = X2n . Observe that the configuration of Xn and
∗ ¯ n and X ¯ n coincide in all Xn coincide in all points outside I . And that similarly X ∗ points outside I − Yn . We know also that the projection of Xn to Z is equal to
the projection of Xn
to Z, and since X2n = X2n this implies that the projections
¯ ¯ of Xn and Xn to Z are equal.
Lemma 10. Let A be a finite group. For all R > 0 there exists C0 > 0 such that the following holds. Consider a probability measure µ on G = Z  A and suppose that the support of µ belongs to the ball of radius R. Then for all n it holds Hµ (2n) ≥ 2Hµ (n) − C0 n2/3 Proof. Combining the second claim of Lemma 9 with Lemma 8 we obtain the proof of this lemma.
Now we return to the proof of the theorem. Since µi converges strongly to µ, we know that the there exists R such that the supports of µi belong to the ball of radius R for all sufficiently large i. Therefore by Lemma 10 we know that for all sufficiently large i and all n Hµi (2n) ≥ 2Hµi (n) − C0 n2/3 , where C0 is some positive constant not depending on i. Let A(n) = C0 /(2n1/3 ). Observe that A(1) + A(2) + A(4) + · · · < ∞. Hence we can apply Lemma 6 and conclude that h(µi ) converges to h(µ).


4. Symmetric examples of discontinuity of drift Proposition 11 (Discontinuity of drift, symmetric measures). There exists a group G, generated by a finite set S, and a sequence of finitely supported symmetric probability measures µi on G, strongly converging to a finitely supported, symmetric probability µ such that the drifts of these measure with respect to the word metric lS satisfy lS (µ) > lim sup lS (µi ). Proof. Consider a free non-Abelian group Fm , m ≥ 2. Consider the involution of G = Fm ×Fm , that permutes the generators of the first copy of Fm with generators of the second copy of Fm . Let G be the extension of Fm × Fm with respect to this involution. The involution is denoted by c. The group G is a wreath product of Z/2Z with Fm . Let ν1 and ν2 be the probability measures, that are equidistributed on the 2m standard generators and their inverses of each copy of Fm under consideration.

On Continuity of Entropy

63

The free generating sets of two copies of Fm we denote by Sa = {a1 , a2 , . . . , am }, and Sb = {b1 , b2 , . . . , bm } respectively. Let µp = pν1 + (1 − p)ν2 . Let ν0 be the Dirac probability measure supported on c. µp,i = (1 − 1/i)µp + 1/iν0 . It is clear that µp,i converges strongly to µp . ±1 Take S which consists of all possible products of the form a±1 i bj (1 ≤ i, j ≤ m) and of the element c. Consider the word metric on G with respect to this generating set S. For this word metric and for any p = 1/2 lS (µp,i )  lS (µp ). Indeed, observe that the word length with respect to the generating set under consideration satisfies lS ((g, h)) = max{lSa (g), lSb (h)}, for all g, h ∈ Fm . Therefore, with probability one, a trajectory Xn = (Zn , Tn ) of the random walk (G, µp ) satisfies lSa (Zn )/n → pl0 , lSb (Tn )/n → (1 − p)l0 , where l0 = (m − 1)/m is the drift of the simple random walk on Fm , corresponding to the free generating set of Fm . Hence m−1 lS (µp ) = (max p, 1 − p)l0 = (max p, 1 − p) . m Now observe that any element of G either belongs to Fm × Fm , or it is a product of c with an element in this direct product. In the latter case we have lS ((g, h)c) − max lSa (g), max lSb (h) ≤ 1. Consider a trajectory Xn of the random walk (G, µp,i ). We write Xn in the form (Zn , Tn )δ, where Zn and Tn are from the first and respectively second copy of Fm inside G, and δ = c or δ = eG . Observe that both Zn and Tn can be described at the points, visited by two trajectories of the simple random walk on Fm , at times moment t1,n and t2,n respectively. Here t1,n /n and t2,n /n both tend to (1 − 1/i)/2. This implies that for all i 1 − 1/i 1 − 1/i m − 1 1 − 1/i 1 − 1/i m − 1 lSa (Zn )/n → l0 = , lSb (Tn )/n → l0 = . 2 2 m 2 2 m Therefore, the limit of l(µp,i ) as i tends to infinity is equal to l0 /2 = (m − 1)/(2m), and this completes the proof of the proposition.


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References [1] P. Baldi, N. Lohou´e, J. Peyri`ere, Sur la classification des groupes recurrents, C. R. Acad. Sci., Paris, S´er. A 285, 1103–1104 (1977). [2] Y. Derriennic, Quelques applications du th´ eor`eme ergodique sous-additif, Ast´erisque 74, (1980), 183–201. [3] A. Dyubina, Characteristics of random walks on wreath products of groups, Journal Math. Science, vol. 107, no. 5, (2001). [4] A. Erschler, On drift and entropy growth for random walks on groups, Ann. Prob. 31 (2003) 1193–1204 [5] A. Erschler, Liouville property for groups and manifolds, Inv. Mathematicae, 155, 2004, 55–80. [6] L. Gilch, Acceleration of Lamplighter Random Walks, Markov Process. Related Fields 14 (2008), no. 4, 465–486. [7] Y. Guivarch, Sur la loi des grands nombres et le rayon spectral d’une marche al´ eatoire, Ast´erisque 74, 47–98, Soc. Math. France, 1980. [8] V.A. Kaimanovich, Examples of non-commutative groups with non-trivial exit boundaries, Differential geometry, Lie groups and mechanics, V. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 123 (1983), 167–184. [9] V.A. Kaimanovich, A.M. Vershik, Random walks on discrete groups: boundary and entropy, Ann. Prob. 11 (1983) 457–490 [10] A. Karlsson, F. Ledrappier, Linear drift and Poisson boundary for random walks, Pure Appl. Math. Q. 3 (2007) 1027–1036 [11] F.Spitzer, Principles of random walks, Van Nostrand, Princeton, (1964) [12] N.Th. Varopoulos, Long range estimates for Markov chains, Bull. Sci. Math., II. S´er. 109, 225–252 (1985). [13] A.M. Vershik, Dynamic theory of growth in groups: Entropy, boundaries, examples, Russ. Math. Surv. 55, No.4, 667–733 (2000); translation from Usp. Mat. Nauk 55, No.4, 59–128 (2000). [14] W. Woess, Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, 138, Cambridge University Press, 2000. Anna Erschler Laboratoire de Math´ematique d’Orsay Equipe de Topologie/Dynamique Universit´e Paris-Sud Bˆ atiment 425 F-91405 Orsay Cedex, France e-mail: [email protected]

Progress in Probability, Vol. 64, 65–74 c 2011 Springer Basel AG


Polynomial Growth, Recurrence and Ergodicity for Random Walks on Locally Compact Groups and Homogeneous Spaces Yves Guivarc’h and C.R.E. Raja Abstract. Let G be a locally compact group, E a homogeneous space of G. We discuss the relations between recurrence of a random walk on G or E, ergodicity of the corresponding transformations and polynomial growth of G or E. We consider the special case of linear groups over local fields. Mathematics Subject Classification (2000). 22E30, 31C12, 37A17, 37A50. Keywords. Random walk; ergodicity homogeneous spaces; polynomial growth.

1. Introduction Let G be a locally compact separable group and λG be a left Haar measure on G. We denote by µ a given probability measure on G, by Gµ the closed subgroup generated by its support. We study recurrence properties of random walks defined by µ, either on G, or on homogeneous spaces of G. We also discuss ergodicity of associated homeomorphisms on path spaces. This paper can be considered as an introduction to the more complete article [17] where detailed proofs are given. In particular we sketch proofs of the following results of [17].  Theorem 1. Assume G is a closed subgroup of GL(di , Fi ) where I is finite and i∈I

Fi is a local field. Then G carries a recurrent random walk whose support generates G if and only if G has at most quadratic growth. Theorem 2. Let G be a semisimple Lie group of real rank 1, Γ a discrete cocompact subgroup, Γ
a normal subgroup of Γ such that Γ/Γ
= Z. Let µ be a symmetric probability measure on G such that supp µ is compact and generates a non amenable subgroup of G. We endow E = G/Γ
with the Haar measure and we consider the Markov measure µ⊗Z ⊗ m on GZ × E. Then the corresponding Markov shift on GZ × E is ergodic with respect to µ⊗Z ⊗ m.

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Y. Guivarc’h and C.R.E. Raja

We observe that Theorem 1 is based on the following simple fact which seems not to have been noted before: Proposition 1. Assume G is locally compact and carries a recurrent random walk whose support generates G. Then every closed subgroup of G is unimodular. For basic information and results we refer to the surveys [7], [14] and to the books [6], [12], [29], [30]. We thank F. Ledrappier P. Bougerol, Y. Derriennic and T. Steger for useful comments on ergodicity of bilateral Markov shifts with infinite invariant measure.

2. Polynomial growth Definition 1. We say that G has polynomial growth of degree at most d ≥ 0 if for any compact neighborhood W of e, there exists CW > 0 such that for every n ∈ N, λG (W n ) ≤ CW nd . A typical example of polynomial growth is the following: N is a nilpotent compactly generated group, K is a compact group of automorphisms of N and G is a semi-direct product of K and N . Then d can be calculated in terms of the descending series of N (See[10]). We give now a few structural facts. It is well known that polynomial growth of G implies amenability and unimodularity for G as well as for its closed subgroups. A fundamental result of [9] says that if G is finitely generated with polynomial growth then G is virtually nilpotent. If G is locally compact and compactly generated with polynomial growth then G has a compact normal subgroup K such that G/K is a real Lie group (connected or not) [21]. The connected Lie groups with polynomial growth are the groups of rigid type [10]. Rigid type for a connected Lie group G means that for every g ∈ G, the automorphism Adg of the Lie algebra of G has only eigenvalues of modulus one.

3. Transience and recurrence 3.1. Definitions ! = GZ ) endowed with the shift We consider the product space Ω = GN (resp. Ω N ! θ and the product measure P = µ (resp. P = µZ ) and we denote by Xk (ω) ! If E is a locally compact G-space and the coordinates of ω ∈ Ω (resp. ω ∈ Ω). x ∈ E, the random walk on E of law µ, starting from x, is the sequence of random variables Sn (ω)x (n ∈ N ∪ {0} or Z) defined by: S0 (ω)x = x Sn (ω)x = Xn (ω) · · · X1 (ω)x Sn (ω)x =

−1 Xn+1 (ω) · · · X0−1 (ω)x

(n > 0) (n < 0)

Polynomial Growth, Recurrence and Ergodicity

67

If E = G, the asymptotic behavior of Sn (ω)x is essentially independent of x ∈ G. This is not the case in general if E = G and we will discuss the situation in terms of a Radon measure λ quasi invariant under the µ-action on E. In particular if λ is " = P⊗ λ. a µ-invariant measure (µ∗ λ = λ) we will consider on Ω× E the measure λ " P ⊗ λ), with θ(ω, " x) = (θω, X1 (ω)x) will play a basic The skew product (Ω × E, θ, " " role since λ is θ-invariant. If E = G, we will take λ = λG . We will need also to ! x) = (θω, X1 (ω)x) which ! × E : θ(ω, consider the extended bilateral shift θ! on Ω leads to consider the bilateral random walk Sn (ω) on E (n ∈ Z). The Markov operator P on Cb (E), the space of continuous bounded functions on E, which is defined by  P ϕ(x) =

ϕ(gx)dµ(g),

allows to express various quantities of probabilistic interest. For example if A ⊂ E, the expected number of visits of the random walk to A is   ∞ ∞  1A (Sn (ω)x)dP(ω) = P k 1A (x). 0

If E = G, we have

0 ∞ 

P k 1A (e) =

0

∞ 

µk (A)

0

k

where µ is the kth convolution power of µ. We will now mainly restrict to the case E = G, and we will come back to the general case in the last section. Definition 2. We will say that µ or Sn (ω) is recurrent on G if for every neighborhood W of e we have P-a.e: Sn (ω) ∈ W infinitely often. If P-a.e., Sn (ω) escapes to infinity we will say that µ or Sn (ω) is transient. It is easy to see that Sn (ω) is either transient or recurrent. On can show, using Hopf maximal ergodic lemma, that a necessary and sufficient condition for transience of µ is the existence of W , a relatively compact neighborhood of e, such that ∞  µn (W ) < ∞. 0

We will assume µ adapted, i.e., Gµ = G, and also we will exclude the case G = Z and µ = δa (a ∈ Z). The following important concept will be used below. Definition 3. Let f be a Borel #function on G. We say that f is left µ-harmonic (resp. # left µ-superharmonic) if f (hg)dµ(h) = f (g), for any g ∈ G (resp., f ≥ 0 and f (hg)dµ(h) ≤ f (g) for any g ∈ G). We observe that we could have considered Sn
(ω) = X1 (ω) · · · Xn (ω), instead of Sn (ω). But the laws of Sn (ω) and Sn
(ω) are equal to µn . Then the above implies that recurrence (resp. transience) of Sn (ω) is equivalent to recurrence (resp. transience) of Sn
(ω). Hence we can speak of µ being recurrent (resp. transient).

68

Y. Guivarc’h and C.R.E. Raja If µ is transient and u ∈ Cc+ (G), the nonnegative function ∞ ∞   P k u(x) = (µk ∗ δx )(u) 0

0

is continuous and left-superharmonic. Then we have Proposition 2. Assume µ is as above. Then we have the equivalence a) Sn (ω) is recurrent b) Any continuous left µ-superharmonic function is constant ! ⊗ λG . c) θ! is ergodic with respect to P " A natural connection between θ-invariant functions and µ-harmonic functions modulo λG is as follows. If F ∈ L∞ (Ω × G, P ⊗ λG ) satisfies F oθ" = F then the function f ∈ L∞ (G) defined by  f (g) = F (ω, g)dP(ω) # satisfies f (hg)dµ(h) = f (g), i.e., f is µ-harmonic mod λG . As is well known, this correspondence is bijective. Ergodicity of θ" amounts to constancy of bounded µ-harmonic functions. Ergodicity of θ! is a stronger property. If G = Z3 it is easy to ! construct non trivial θ-invariant functions. However bounded µ-harmonic functions are constant if µ is adapted. Such a construction is as follows. For any x ∈ Z3 , we denote ! A+ x = {ω ∈ Ω ; ∃n > 0, x + Sn (ω) = 0} − ! ; ∃n ≤ 0, x + Sn (ω) = 0} Ax = {ω ∈ Ω − Ax = A+ ∪ A x x , A = ∪ 3 Ax × {x}. x∈Z

! × Z3 is the set of paths of the random walk which pass through Then A ⊂ Ω ! 0 at some time n ∈ Z; clearly A is θ-invariant. On the other hand, since Sn (ω) is + − ! ! transient, if x = 0: 0 < P(Ax ) < 1 , 0 < P(Ax ) < 1 (see [27]). Since Sn (ω) (n > 0) and Sn (ω) (n ≤ 0) are independent, the complements of ! + ! − ! + − Ax , A− x are also independent, hence 0 < (1−P(Ax ))(1−P(Ax )) = 1− P(Ax ∪Ax ) < 3 ! ! ×Z . 1. It follows that A is a nontrivial θ-invariant set of Ω Definition 4. We say that the group G is recurrent if there exists a probability measure µ on G which is adapted and recurrent. One can show (see below) that if G is recurrent then G and its closed subgroups are unimodular. Also in this case amenability of G is valid and follows from the constancy of bounded harmonic functions. 3.2. Examples a) It is well known that Z2 is recurrent while Z3 is transient (see [27]). The free group Fd with d generators (d ≥ 2) is transient since it is non amenable (see [8]). b) Let G2 be the group of motions of the Euclidean plane. We identify G2 with the group of maps of the form gz = az + b where z ∈ C and |a| = 1, b ∈ C. Let

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α∈ / Q and a, b ∈ G2 defined by az = e2iπα z + 1, bz = e−2iπα z + 1, µ = 12 (δa + δb ). Then one can show that µ is recurrent, hence G2 is recurrent. The proof relies on the estimation of the polymer sums Sn
(ω)0 = 1 +

n−1 

e2iπsk (ω)

1

where

sk (ω) =

k 

εk (ω)

1

# and εk (ω) = ±1 with probability 1/2; one gets |Sn
(ω)0|2 dP(ω) ∼ Cn with C > 0 and this can be shown to imply recurrence (see [12]). c) Let G = SL(2, R), Γ ⊂ G a cocompact discrete subgroup of G, Γ
a normal subgroup of Γ such that Γ/Γ
= Z or Z2 , E = G/Γ
, µ = δa with a = diag (e, e−1 ). We consider the Haar measure m on E and the action of a on E. The corresponding dynamical system is the time-one geodesic flow on E and can be ! P⊗m). ! ! identified with (Ω×E, θ, One can show (see [13], [24], [28]) that this system is ergodic. Here recurrence of the a-action is an easy first step based on Birkhoff ergodic theorem.

4. Growth and recurrence The following quadratic growth conjecture was stated in [11]: G is recurrent if and only if G has polynomial growth of degree at most 2. One can show using [21], that if G has polynomial growth of degree at most 2, then G is recurrent (see [23]). Also for the converse one can assume that G is compactly generated. In [19] it was already conjectured that finitely generated recurrent groups have non exponential growth. The general quadratic growth conjecture is based on this idea but is clearly stronger. Also its possible proof can be seen to involve necessarily a group theoretic argument. One can expect the information of [21] to be sufficient. Then in the discrete case the quadratic growth conjecture has been settled in [29] using [9]: recurrent finitely generated groups are virtually Z or Z2 . On the other hand, as shown in [1], recurrent connected Lie groups are closed subgroups of G2 , up to normal compact subgroups. Then, using the analysis developed in [1], [12] and [29], one can solve the quadratic growth conjecture for Lie groups, connected or not. Also, for Lie groups over p-adic fields, the conjecture is solved in [23]. But this leaves unsolved the conjecture if G is totally disconnected. As a first step, it is natural to consider the case where G is a closed subgroup of GL(V ) with V = Fd and F is a non-archimedean local field. We observe that, if F has positive characteristic, G is not a Lie group in general. In this case, due to the ultrametric triangular inequality, one can expect a much more simpler analysis and result than in [12] and [29]. This is indeed the case as shown below. Furthermore, in the general case of a recurrent compactly generated and totally disconnected group G, one can expect the following: G has an open compact normal subgroup K such that G/K is recurrent and finitely generated. Then, the corresponding result would follow from [29].

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5. Linear groups over local fields One has the following. Theorem 1 Assume G is a closed subgroup of a finite product of



GL(di , Fi )

i∈I

where any Fi is a local field. Then G is recurrent if and only if G has at most quadratic growth. One will sketch the main ideas of the proof of necessity of quadratic growth in the case I = {1} and F1 = F a non-archimedean local field. Also one will assume G compactly generated recurrent and show the existence of a compact normal subgroup K such that G/K is virtually Z or Z2 . One will need to consider the contraction subgroup Cg of g ∈ G, i.e., Cg = {h ∈ G; lim g n hg −n = e} n→∞

If G is a closed linear group it is easy to show that Cg is also closed. Furthermore if Cg = {e} is closed, the closed subgroup of G generated by g and Cg is not unimodular. Then the two main steps are given by the following propositions. Proposition 1 Assume G is locally compact and recurrent. Then every closed subgroup of G is unimodular. It follows from the above remark that, if furthermore G ⊂ GL(V ) is closed, then Cg = Cg−1 = {e} Proposition 3. Assume G is a compactly generated closed subgroup of GL(d, F) where F is a non archimedean local field. Then, if Cg = {e} for any g ∈ G, G has a compact open normal subgroup. In the proof of this result an important role is played by the following lemmas. Lemma 1. (See [18]) Assume G is totally disconnected and g ∈ G satisfies Cg = Cg−1 = {e}. Then there exists a compact open subgroup Kg such that gKg g −1 = Kg . This lemma is a consequence of the study of the so-called tidy subgroups [2]. Lemma 2. (See [3], [17]) Assume G ⊂ GL(d, F) is compactly generated, F is nonarchimedean, and any g ∈ G generates a bounded subgroup. Then G is bounded. One may observe that, on the real field, even if G is finitely generated, the corresponding statement is false, which gives a negative answer to a question of S. Ulam (see [3]). But, on ultrametric fields, S. Ulam’s question has a positive answer. On the other hand, if G is assumed to be closed, the conclusion of Lemma 2 is valid for any local field (see [3], [17]). Let us give the proof of Proposition 1. Let µ be adapted and recurrent on G. Then, if we denote by P the Markov operator on E = G/H associated to ∞  µ, we have for any u ∈ Cc+ (E), u = 0 : P k u = +∞ on E. Hence, using 0

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[20], one sees that there exists a P -invariant Radon measure λ on E. For any ϕ ∈ Cc+ (E), we #consider the nonnegative function f on G given by f (g) = gλ(ϕ). Clearly f (g) = f (gh)dµ(h), i.e., f is µ-harmonic on G. Since µ is recurrent and adapted, Proposition 1 implies that f is constant, i.e., gλ(ϕ) = λ(ϕ). Since g and ϕ are arbitrary λ is G-invariant. As is well known from general theory of invariant measures on homogeneous spaces this implies ∆G (h) = ∆H (h) for any h ∈ H, where ∆G (resp. ∆H ) is the modular function of G (resp. H). If we take for H the closed subgroup generated by g ∈ G, we get ∆G (g) = ∆H (g) = 1; hence G is unimodular. From above it follows ∆H (h) = 1 for any h ∈ H, hence H is unimodular.

6. Homogeneous spaces Various examples of recurrent and ergodic behaviors take place (see [5], [17], [12]). Here we only develop one example and formulate a general question. 6.1. A framework Let E be a locally compact G-space, P be the Markov operator on Cb (E) defined by:  P ϕ(x) = ϕ(gx)dµ(g), we consider a P -invariant Radon measure λ on E, i.e., P λ = λ. Then we can " x) = (θω, X1 (ω)x) ! consider the extended shifts on Ω×E and Ω×E defined by θ(ω, ! ! and θ(ω, x) = (θω, X1 (ω)x), we endow Ω×E and Ω×E with their natural structure of Polish spaces so that θ" is continuous and θ! is a homeomorphism we write " = P ⊗ λ on Ω × E is θ-invariant. " Ω− = G−N∪{0} . Then the Markov measure λ ! which is ! One can show that there exists on Ω × E a unique Radon measure λ ! " on Ω × E. θ-invariant and has projection λ In the general case of a dynamical system with σ-finite invariant measure, the ! can be construction goes back to V.A. Rokhlin ([25]). In our special situation, λ defined as follows. Let P ∗ be the adjoint operator of P in L2 (E, λ). Since P λ = λ, ∨ P ∗ is also a Markov operator and we denote by Pλ the corresponding Markov measure on Ω− ×E. If we denote by θ − the shift on Ω− , we can define an extended ∨ shift on Ω− × E by θ"− (ω− , x) = (θ− ω− , g0−1 x). Then Pλ is θ"− -invariant and has #∨ ∨ ∨ projection λ on E, hence we can disintegrate Pλ as Pλ = Py ⊗ δy dλ(y), along the #∨ != P fibers Ω− × {y}. Then one can verify that the measure λ y ⊗ P ⊗ δy dλ(y) is ! θ-invariant. Definition 5. Let (E, P, λ) be as above. Then one says that (E, P, λ) has property R if for any relatively compact open set U ⊂ E, and P ⊗ λ-a.e. (ω, x) ∈ Ω × U , we have Sn (ω)x ∈ U infinitely often. If E = G, λ = λG and P is as above then property R for (E, P, λ) is equivalent to recurrence of µ on G. Then one has the

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Proposition 4. Assume E is not the G-space Z with µ = δg acting by translation on Z. Then one has the equivalence a) Property R is valid for (E, P, λ) and the equation P f = f , f ∈ L∞ (λ) implies f is constant. b) Property R is valid for (E, P, λ) and λ is extremal in the cone of P -invariant Radon measures. ! ! × E with respect to λ. c) The homeomorphism θ! is ergodic on Ω 6.2. An example Assume G is semisimple of real rank 1, Γ is a discrete cocompact subgroup and Γ
is a normal subgroup of Γ which satisfies Γ/Γ
= Z. We endow E = G/Γ
with the Haar measure λ = m and observe that G/Γ
is an abelian cover of the compact ! × E and the homogeneous space G/Γ. We consider the homeomorphism θ! of Ω ! ! measure P ⊗ m = λ. Proposition 5. With the above notations we assume µ is symmetric with compact ! ⊗ m. support and Gµ is non amenable. Then θ! is ergodic with respect to P For the proof we use the equivalence of a) and c) in the above proposition. Property R follows easily from the symmetry of µ and the G-invariance of m, using ergodicity of the random walk on G/Γ. For the study of the equation P f = f , f ∈ L∞ (E), one uses induced unitary representations, as follows. One observes that Γ/Γ
= Z acts on G/Γ
and this action commutes with the G-action. Taking a Borel fundamental domain ∆ ⊂ E of this action one can identify G/Γ
with ∆ × Z. One denotes by x → g.x the transformation of G/Γ defined by g ∈ G. Then for x ∈ ∆ ⊂ E one can write: gx = (g.x)z(gx) with z(gx) ∈ Z, g.x ∈ ∆. For any character χ of Γ/Γ
= Z one can define a unitary representation ρχ of G in L2 (G/Γ) by ρχ (g)f (x) = f (g −1 .x)χ(z(g −1 x)) If χ(Γ) = 1, i.e., χ = 0 we get the natural representation ρ0 of G in L2 (G/Γ) and we know that ρ0 restricted to L20 (G/Γ) do not contain weakly the identity representation. Since the one-dimensional representation of Γ defined by χ do not contain weakly identity, the same is valid for the induced representation ρχ from Γ to G (see [22] Prop. 1.11 p. 112). Then one can use the non-amenability of Gµ and a result of [26] (see also [4], [16]) to conclude that if χ = 0 the operator ρχ (µ) defined by  ρχ (µ)f (x) = f (gx)χ[z(gx)]dµ(g) satisfies ||ρχ (µ)|| < 1. Then the same analysis as in [15] can be performed with the natural Fourier decomposition of L2 (E): we get that the condition P f = f , f ∈ L∞ (E) implies f = cte.

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6.3. A question Let G be a connected Lie group or an algebraic group defined over a local field F, H a closed subgroup, λ a P -invariant Radon measure on E = G/H, x ∈ supp λ. Assume that (E, P, λ) satisfies property R. Then, is it true that, for any n ∈ N : λ(W n x) ≤ CW n2 , where W is a compact neighborhood of e and CW > 0? Is it possible to describe geometrically the systems (E, P, λ) if λ is finite? Is it true that the basic building blocks for such an E are either boundaries of (G, µ) or spaces of the form G/Γ where Γ is a lattice in G?

References [1] P. Baldi. Caract´erisation des groupes de Lie connexes r´ecurrents, Ann. Inst. H. Poincar´e Sect. B (N.S.) 17 (1981), 281–308. [2] U. Baumgartner and Willis. Contraction groups and scales of automorphisms of totally disconnected locally compact groups, Israel J. Math. 142 (2004), 221–248. [3] J.-P. Conze and Y. Guivarc’h. Remarques sur la distalit´e dans les espaces vectoriels, C.R. Acad. Sci. Paris S´er. A 278 (1974), 1083–1086. [4] Y. Derriennic and Y. Guivarc’h. Th´eor`eme de renouvellement pour les groupes non moyennables, C.R.A.S. Paris 277 (1973) A 613-A 615. [5] A. Eskin and G.A. Margulis. Recurrence properties of random walks on finite volume homogeneous manifolds, Random walks and geometry, 431–444, Walter de Gruyter, Berlin 2004. [6] S.R. Foguel. The ergodic theory of Markov processes. Van Nostrand Mathematical Studies, No. 21 Van Nostrand Reinhold Co., New York-Toronto, Ont.-London 1969. [7] A. Furman. Random walks on groups and random transformations. Handbook of dynamical systems, Vol. 1A, 931–1014, North-Holland, Amsterdam, 2002. [8] H. Furstenberg. Random walks and discrete subgroups of Lie groups. Advances in Probability and Related topics (Ed. P. Ney) vol. 1, p. 1–63, M. Dekker (1971). [9] M. Gromov. Groups of polynomial growth and expanding maps, I.H.E.S. Publ. Math. No. (1981), 53–73. [10] Y. Guivarc’h. Croissance polynomiale et p´eriodes des fonctions harmoniques, Bull. Soc. Math. France, 101 (1973), 333–379. [11] Y. Guivarc’h and M. Keane. Marches al´eatoires transitoires et structure des groupes de Lie. Symposia Mathematica, Vol. XXI (Convegno sulle Misure su Gruppi e su Spazi Vettoriali, Convegno sui Gruppi e Anelli Ordinati, INDAM, Rome, 1975), 197–217. Academic Press, London 1977. [12] Y. Guivarc’h, M. Keane and B. Roynette. Marches al´eatoires sur les groupes de Lie, Lecture Notes in Mathematics 624, Springer-Verlag, Berlin-New York, 1977. [13] Y. Guivarc’h. Propri´et´es ergodiques, en mesure infinie, de certains syst`emes dynamiques fibr´es, Ergodic Theory Dynam. Systems 9 (1989), 433–453. [14] Y. Guivarc’h. Marches al´eatoires sur les groupes, Development of Mathematics 1950– 2000, 577–608, Birkh¨ auser, Basel, 2000. [15] Y. Guivarc’h and A.N. Starkov. Orbits of linear group actions, random walks on homogeneous spaces and toral automorphisms. Ergodic Theory Dynam. Systems 24 (2004), 767–802.

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[16] Y. Guivarc’h. Limit Theorems for Random walks and Products of Random Matrices. In Probability Measures on Groups Eds.: S.G. Dani and P. Graczyk, pp. 256–332, 2006 T.I.F.R. Narosa Publ. House, New Delhi, India. [17] Y. Guivarc’h and C.R.E. Raja. Recurrence and Ergodicity of random walks on locally compact groups and on homogeneous spaces. Preprint 2009. [18] W. Jaworksi and C.R.E. Raja. The Choquet-Deny theorem and distal properties of totally disconnected locally compact groups of polynomial growth, New York J. math. 13 (2007), 159–174. [19] H. Kesten. The Martin boundary of recurrent random walks on countable groups 1967 Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Vol. II; Contributions to Probability Theory, Part 2, pp. 51–74. Univ. California Press, Berkeley, Calif. [20] M. Lin. Conservative Markov processes on a topological space, Israel J. Math. 8 (1970), 165–186. [21] V. Losert. On the structure of groups with polynomial growth. Math. Z. 195 (1987), 109–117. [22] G. Margulis. Discrete subgroups of semisimple Lie groups, Ergebnisse der Mathematik und ihrer Grenzgebiete. Springer-Verlag, Berlin, 1991. [23] C.R.E. Raja. On growth, recurrence and the Choquet-Deny Theorem for p-adic Lie groups, Math. Z. 251 (2005), 827–847. [24] M. Rees. Checking ergodicity of some geodesic flows with infinite Gibbs measure, Ergodic Theory Dynam. Systems 1 (1981), 107–133. [25] V.A. Rokhlin Selected topics from the metric theory of dynamical systems Uspehi Mat. Nauk. 4 (1949), 57–125 A.M.S. Transl. Ser. 249 (1966). [26] Y. Shalom. Explicit Kazhdan constants for representations of semisimple and arithmetic groups, Annales de l’Institut Fourier, 50 (1979), 171–202. [27] F. Spitzer. Principles of Random walk, Van Nostrand, 1964. [28] D. Sullivan. The density at infinity of a discrete group of hyperbolic motions, I.H.E.S. Publ. Math. No. 50 (1979), 171–202. [29] Th. Varopoulos, L. Saloff-Coste and T. Coulhon. Analysis and geometry on groups, Cambridge Tracts in Mathematics, 100. Cambridge University Press, Cambridge, 1992. [30] W. Woess. Random walks on infinite graphs and groups, Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. Yves Guivarc’h IRMAR, Universit´e de Rennes I Campus de Beaulieu F-35042 Rennes Cedex, France e-mail: [email protected] C.R.E. Raja Stat. Math. Unit, Indian Statistical Institute 8th Mile Mysore Road Bangalore 560059, India e-mail: [email protected]

Progress in Probability, Vol. 64, 75–89 c 2011 Springer Basel AG


Ergodic Theorems for Homogeneous Dilations Michael Bj¨orklund Abstract. In this paper we prove a general ergodic theorem for ergodic and measure-preserving actions of Rd on standard Borel spaces. In particular, we cover R.L. Jones’ ergodic theorem on spheres. Our main theorem is concerned with almost everywhere convergence of ergodic averages with respect to homogeneous dilations of certain Rajchman measures on Rd . Applications include averages over smooth submanifolds and polynomial curves. Mathematics Subject Classification (2000). Primary 37A30; Secondary 42B25. Keywords. Pointwise ergodic theorems, Fourier dimension of a measure, maximal inequalities.

1. Introduction The first multidimensional pointwise ergodic theorem is due to N. Wiener [19], who proved that if (X, B, µ) is a standard Borel space and T is an ergodic measurepreserving action of Rd on X, then for all f ∈ L1 (X), the limit   1 lim f (Tλt x) dt = f dµ, λ→∞ |B| B X exists almost everywhere on X, where B denotes the unit ball in Rd and |B| the volume of B. It is not hard to extend this theorem to the more general setting where the normalized characteristic function of a ball is replaced by an absolutely continuous probability measure on Rd . In this paper we prove Wiener’s ergodic theorem with respect to not necessarily absolutely continuous probability measures on Rd ; more precisely, we are interested in the class Cp of probability measures ν for which   lim f (Tλt x) dν(t) = f dµ λ→∞

Rd

X p

exists almost everywhere on X for all f in L (X), where the range of p is allowed to depend on ν. It is obviously necessary that ν is continuous, i.e., does not give positive mass to individual points, for this to true. We will say that Wiener’s

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ergodic theorem holds for ν if the limit above exists almost everywhere on X. It was proved by R.L. Jones [9] that the induced Lebesgue measure on S d−1 in Rd d for d ≥ 3 belongs to the class Cp for p > d−1 . This was later extended to d = 2 by M. Lacey [11]. We will extend their result to a much larger class of measures. We stress however that the techniques in this paper are not new, and many of the results were probably already known to the experts. We hope that the paper will serve as a survey of some classical ideas in the harmonic analysis approach to ergodic theory. Recall that the Fourier dimension of a probability measure ν on Rd is defined as the supremum over all 0 ≤ a ≤ d such that |ˆ ν (ξ)| ≤ C|ξ|−a/2

as ξ → ∞.

For instance, if S is a smooth hypersurface in Rd with non-vanishing Gaussian curvature and ν is the induced Lebesgue measure on S, then the Fourier dimension of ν is at least n − 1 [15], which motivates the terminology. Note that there are many non-smooth sets (e.g., random Cantor sets [2]) in Rd which support probability measures with high Fourier dimension. However, by Frostman’s lemma (see, e.g., [12]), the Fourier dimension is always bounded from above by the Hausdorff dimension of the support of ν. In this paper we prove Wiener’s ergodic theorem for probability measures ν on Rd with sufficiently large Fourier dimensions. Theorem 1.1. Let (X, B, µ) be a standard Borel probability measure space, and suppose T is an ergodic Borel measurable action of Rd on X which preserves µ. If ν is a compactly supported probability measure on Rd with Fourier dimension a > 1, then   lim f (Tλt x) dν(t) = f dµ λ→∞

Rd

X

almost everywhere on X, for all f in Lp (X) for p > pa , where pa =

1+a . a

The question of mean convergence is much simpler and an immediate consequence of the spectral theorem for unitary operators. Recall that a probability measure ν on Rd is a Rajchman measure if the Fourier transform of ν decays to zero at infinity. Note that a Rajchman measure is always continuous, but not necessarily absolutely continuous. Indeed, there are Rajchman measures supported on the set of Liouville numbers in R [3], which have zero Hausdorff dimension. The following proposition is well known and we only include it for completeness. Proposition 1.2. Let U be a unitary representation of Rd on a separable Hilbert space H. Let ρ be a Rajchman measure on Rd , and define for x ∈ H, the operator  Aλ x = Uλt x dρ(t), λ > 0. Rd

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77

Let P denote the projection onto the space of invariant vectors of U in H. Then ||Aλ x − P x||H → 0 for all x in H. Given positive real numbers a1 , . . . , ad , we define the associated homogeneous dilation by λ on Rd : λ.t = (λa1 t1 , . . . , λad td ),

t ∈ Rd .

We will use the following two simple facts about homogeneous dilations; if t, ξ ∈ Rd , then λ.(ξ + t) = λ.ξ + λ.t

and ξ, λ.t = λ.ξ, t

for all λ > 0,

where ·, · is the standard inner product on Rd . We use the dot-notation to separate between the standard dilations (a1 = · · · = ad = 1) on Rd and homogeneous dilations. An important ingredient in the proof of Theorem 1.1 is the following version of Calderon’s maximal theorem [7]. We include a proof for completeness. Proposition 1.3. Suppose ν is a Rajchman measure on Rn for which the maximal operator     M φ(x) = sup  φ(λ.t) dν(t), φ ∈ S(Rd ) λ>0

Rd

is of weak type (p, p) for some 1 ≤ p < ∞. For every standard Borel probability space (X, B, µ) and ergodic measure-preserving action T by Rd on X,   lim f (Tλ.t x) dν(t) = f dµ λ→∞

Rd

X p

almost everywhere on X for every f in L (X). By A.P. Calderon’s transfer principle [7], the proof of this theorem is reduced to finding a dense subspace Lν in Lp (X) on which Wiener’s ergodic theorem for ν holds almost everywhere. It turns out (see Lemma 3.3) that we can choose this subspace, independently of the Rajchman measure ν, to consist of the constants and all functions of the form  f (x) = f0 (Ts x)φ(s) ds, Rd

where f0 ∈ L∞ (X) and φ ∈ L10 (Rd ). This fact is established by using fairly standard smoothing arguments and some results on vague convergence of signed measures and is the main new ingredient in this paper. Using the fundamental works by E.M. Stein on homogeneous dilations and maximal inequalities for dilated polynomial curves, we can establish the following theorem.

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Theorem 1.4. For every standard Borel probability space (X, B, µ) and ergodic measure-preserving action T by Rd on X,  1  lim f (Tλ.q(w) x) dw = f dµ λ→∞

0

X p

almost everywhere on X for every f in L (X), where p > 1 and q(w) = (q1 w, . . . , qd wd ),

w ∈ [0, 1],

for some non-zero real numbers q1 , . . . , qd and λ.t = (λt, . . . , λd t),

t ∈ Rd , λ > 0.

Extensions of this result to more general curves is possible; see, e.g., [17].

2. Measure theory and Fourier analysis 2.1. Basic measure theory Let G be a Hausdorff locally compact second countable group, and let BG denote the σ-algebra of Borel subsets of G. Let (X, B) be a measurable space such that there exists a Polish topology on X for which B is the induced σ-algebra. If µ is a σ-finite measure on B, we refer to (X, B, µ) as a standard Borel space. We say that G admits a Borel measurable action on X if there is a map π : G × X → X satisfying π(gg
, x) = π(g, g
x) for each g in G and x in X such that π is a measurable map from (G × X × BG × B) to (X, B). We will write π(g, x) = gx for short. The action is measure-preserving if µ(gE) = µ(E) for each g in G and E in B. All actions in this paper are assumed to be Borel measurable. For functions on X, measurability will refer to the completion of the σ-algebra B. By a result due to V.S. Varadarajan [18], there is a compact metric G-space Y with a jointly continuous G-action and G-invariant Borel subset Y0 ⊂ Y together with a G-equivariant measurable and bijective map ψ : X → Y0 . Let µ
denote the push-forward of the measure µ on X. This construction yields a Borel measurable measure-preserving action of G on the Borel subset Y0 . Suppose λ → νt is a weakly continuous from R+ to the convex set of probability measures M 1 (G) on G. Define, for a continuous function f on Y and λ in R+ , the linear operator  Aλ f (x) = f (g −1 x) dνλ (g). G

In this paper we will be concerned with the associated maximal function, i.e.,     M f (x) = sup Aλ f (x). λ>0

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79

Note that since λ → νλ is weakly continuous, M f is a measurable function on Y0 , since we can realize it as     M f (x) = sup  f (g −1 x) dνλ (g). λ>0 λ∈Q

G

It is easy to see that if a measurable function f is a pointwise monotone increasing limit of a sequence fn of nonnegative functions for which the maximal functions M fn are measurable, then M f is measurable. By rather technical, but standard, approximation arguments (see, e.g., [13]), M f is measurable for every measurable function f on X. Note however that this argument only ensures that M f is measurable with respect to the completion of the σ-algebra B with respect to the probability measure µ. We now restrict our attention to the group G = Rd for d ≥ 1. If µ is absolutely continuous with respect to the Haar measure on Rd , weak-Lp bounds for the sublinear operator M are implied, via Calder´ on’s transfer argument, by the corresponding weak Lp -bounds for the Hardy–Littlewood maximal function (see, e.g., [7]). Analogous transfer arguments for maximal functions works equally well for singular measures and will be described in Subsection 3.3. 2.2. Fourier analysis In this subsection we recall some basic notions from classical Fourier analysis. If ν is a complex measure on Rd , we define the Fourier transform of ν by  νˆ(ξ) = e−i ξ,t dt, ξ ∈ Rd . Rd

It is not hard to see that νˆ is a uniformly continuous function on Rd . We let A1 (Rd ) denote the class of absolutely continuous complex measures ν on Rd for which νˆ is integrable with respect to the Lebesgue measure, and by A10 (Rd ), the subspace of A1 (Rd ) with ν(Rd ) = 0. If ν is in A1 (Rd ), we can reproduce the density ρ of ν by  ρ(t) = ei ξ,t νˆ(ξ) dm(ξ), Rd

where m denotes the Plancherel measure on Rd . We let S(Rd ) denote the class of Schwartz functions on Rd and S0 (Rd ) the subspace of S(Rd ) with zero integral. Note that S(Rd ) is strictly included in A1 (Rd ). The Fourier dimension of a probability measure ν is defined as the supremum over all real numbers 0 ≤ a ≤ d such that |ξ|a/2 ||ˆ ν (ξ)| ≤ C,

∀ ξ ∈ Rd ,

for some finite constant C. If the support of ν is compact, the Fourier dimension can always majorized by the Hausdorff dimension of the support.

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3. Ergodic theorems 3.1. Mean ergodic theorems Before we turn to the pointwise ergodic theorems, we give a proof of Proposition 1.2. Recall that a probability measure ν on Rd is Rajchman if the Fourier transform of ν decays to zero at infinity. If a homogeneous dilation has been chosen on Rd , we let νλ denote the measure   φ(t) dνλ (t) = φ(λ.t) dν(t), φ ∈ Cc (Rd ), Rd

Rd

where Cc (Rd ) is the space of compactly supported continuous functions on Rd . The proof of Proposition 1.2 consist of simple calculations with spectral measures. We give the proof for completeness. Proof of Proposition 1.2. We only have to prove that Aλ x → 0 for all x for which P x = 0. Note that   2 ||Aλ x||H = Uλ.t x, Uλ.s x dρ(t)dρ(s) d d R  R  = e−i ξ,λ.(t−s) dνx (ξ)dρ(t)dρ(s) d d d R R R 2 = |ρ(λ.ξ)| ˆ dνx (ξ) Rd

→ νx ({0}) as λ → ∞, where νx is the spectral measure of U located at x. Since x is in the complement of invariant vectors we have νx ({0}) = 0, which finishes the proof.
Remark 3.1. The theorem also holds for isometric representations of more general uniformly convex Banach spaces from standard approximation arguments. Note that the analogous theorem for representations of Zd is completely false. This is essentially due to the discreteness of Zd . Let U be a unitary representation of Zd on a separable Hilbert space H, and define for x ∈ H  An x = ρk Unk x. k∈Zd

Proposition 3.2. For all non-zero x ∈ H, and for any probability measure ρ on Zd , lim inf n→∞

||An x||H ≥ ||ρ||
2 (Zd ) > 0. ||x||H

Proof. We define the sets Ek = {θ ∈ Td | k, θ = 0},

k ∈ Zd .

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81

Note that E0 = Td , and by Parseval’s relation, N −1 N −1 1  1    ||An x||2H = lim ρk ρl Unk x, Unl x N→∞ N N →∞ N d d n=0 n=0

lim

k∈Z l∈Z

1 N →∞ N

= lim

= lim

N →∞

= lim =

 

 ρk ρl

n=0 k∈Zd l∈Zd

N −1   n=0



N →∞



1 N

N −1 

Td

e−2πi θ,n(k−l) dνx (θ)

|ˆ ρ(nθ)|2 dνx (θ)

(ρ ∗ ρˇ)k

k∈Zd

Td

N −1 1  νˆx (−nk) N n=0

(ρ ∗ ρˇ)k νx (Ek ) ≥ ||ρ||2
2 (Zd ) ||x||2H > 0,

k∈Zd

for all N ≥ 1. Thus, ||An x||H does not converge to 0 for any non-zero x in H.




3.2. Pointwise ergodic theorems We follow the general approach to pointwise ergodic theorems: We first establish Wiener’s ergodic theorem for a dense subspace of Lp (X), and then prove a maximal inequality, which implies that the subspace of functions for which the Wiener ergodic theorems is true is closed in Lp (X). Since the necessary maximal inequalities follow from A.P. Calderon’s transfer lemma, which we present below, the key result in this paper is the following lemma. # Lemma 3.3. The linear span L of all functions on the form f (x)= Rd f0 (Ts x)φ(s)ds, where f0 ∈ L∞ (X) and φ ∈ L10 (Rd ), is dense in L10 (X) and if ν is a Rajchman measure on Rd , then  Aλ f (x) = f (Tλ.t x) dν(t) → 0 Rd

almost everywhere on X for all f ∈ L. Proof. Suppose there is a function h ∈ L∞ 0 (X) such that     h(x) f0 (Ts x)φ(s)ds dµ(x) = φ(s) f0 (Ts x)h(x)dµ(x) ds X Rd Rd X  = f0 (x) h(T−s x)φ(s)ds dµ(x) = 0, X ∞

for all f0 ∈ L (X) and φ ∈ S0 (R ). Thus,  h(T−s x) φ(s) ds = 0,

Rd

d

a.e. [µ].

Rd

Thus, h is invariant under the action of T and by ergodicity, h must be almost everywhere zero.

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We now turn to almost everywhere convergence for f in the subspace L. We can without loss of generality assume that f is of the form  f (x) = f0 (Ts x)(φ ∗ ψ)(s) ds, X

where φ ∈ L1 (Rd ) and ψ ∈ S01 (Rd ), since every element in L1 (Rd ) can be approximated arbitrarily well in L10 (Rd ) by such functions. We note that    f (Tλ.t x) dν(t) = f0 (Tλ.t+s x)ψ(s)φ(t − s) dsdν(t) d d Rd  R R = f0 (Ts x) ψ(s − λ.t − r)φ(r) drdν(t) ds Rd Rd    = f0 (Ts x)φ(r + s) ds ψ(−λ.t − r) dν(t) dr Rd

Rd

Rd

We define, for a fixed x in X, the functions  g(r) = f0 (Ts x)φ(r + s) ds Rd

and

 hλ (r) =

Rd

ψ(−λ.t − r) dν(t).

We observe that for a conull subset of X, g is a bounded and uniformly continuous function on Rn and the set {hλ }λ>0 is contained in some ball of finite radius in L10 (Rd ) for all λ > 0. We think of hλ as Borel measures on Rn with uniformly bounded total variations. It now suffices to prove that the sequence hλ tends to zero in the weak topology, i.e.,  g dhλ → 0, Rd

for all bounded and continuous functions on Rn . Since the variation norm of hλ is uniformly bounded in λ, this is implied by the uniform convergence of the Fourier transforms of hλ to 0 and the tightness of the sequence hλ (see, e.g., [1]), Note that ˆ λ (ξ) = νˆλ (ξ)ψ(ξ), h and the uniform convergence of this sequence to 0 is immediate from the fact that ˆ ν is a Rajchman measure and ψ(0) = 0.
Remark 3.4. Note that Lemma 3.3 also implies that L ∩ Lp0 (X) is dense in Lp0 (X) for all 1 ≤ p < ∞.

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3.3. Maximal inequalities Let λ → νλ be a weakly continuous map from R+ into M 1 (Rd ). Suppose (X, B, µ) is a standard Borel probability measure space, and T a Borel measurable action of Rd on X, which preserves µ. We define  Aλ f (x) = f (Tt x) dνλ (t), λ > 0. X

and M f (x) = sup |Aλ f (x)|. λ>0

By the arguments in Section 2.1, M f is measurable with respect to the completion of the σ-algebra B. We also define Dλ φ = φ ∗ νλ ,

φ ∈ S,

and the corresponding maximal operator Sφ(x) = sup |Dλ φ(x)|. λ>0

Both of these operators have been extensively studied in classical harmonic analysis; see, e.g., [14] and [16]. We recall the following lemma by A.P. Calderon [7] which allows us to transfer bounds on S to bounds on M . We include a proof for completeness. Lemma 3.5 (Calderon’s Transfer Lemma). Suppose λ → νλ is a weakly continuous map from R+ into M 1 (Rd ). If (X, B, µ) is a Borel standard probability space and T a Borel measurable action of Rd on X, which preserves the measure µ, then if S is of strong or weak type (p, p) for some 1 ≤ p ≤ ∞, then so is M . Proof. We only give the proof in the case when S if of strong type (p, p). Suppose f is in Lp (X) and define F (s, x) = f (Ts x) for s ∈ Rd and x ∈ X. Note that F (s + t, x) = F (s, Tt x) for all s, t in Rd and for almost every x ∈ X. Now set  G(s, x) = sup | F (s, Tt x) dνλ (t)|, λ>0

Rd

and observe that G(s + r, x) = G(s, Tr x) for all s, r in Rd . For R > 0, we define  F (s, x) if |s| < R FR (s, x) = 0 otherwise, and GR (s, ·) = SFR (s, ·). Since S is a positive sublinear operator, G(s, ·) = SF (s, ·) = S(FR+ε (s, ·) + F (s, ·) − FR+ε (s, ·)) ≤ SFR+ε (s, ·) + S(F (s, ·) − FR+ε (s, ·)),

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for all ε > 0. The last term is zero, since the function F − FR+ε vanishes in the region |s| < R + ε. Let BR denote the Euclidean ball in Rd of radius R. Note that if S is strong type (p, p), then  ||M f ||pp = G(0, x)p dm(x) X   1 = G(s, x)p dµ(x)dm(s) m(BR ) BR X   1 ≤ GR+ε (s, x)p dm(s)dµ(x) m(BR ) BR X   1 = SFR+ε (s, x)p dm(s)dµ(x) m(BR ) X BR   Cp ≤ |FR+ε (s, x)|p dm(s)dµ(x) m(BR ) X BR m(BR+ε ) = Cp ||f ||pp . m(BR ) If we let R → ∞ and use the polynomial volume growth of balls in Rd , we conclude that ||M f ||p ≤ C||f ||p ,


for all f in Lp (X).

The following theorem is now an easy and straightforward consequence of Lemmata 3.3 and 3.5. Theorem 3.6. Suppose ν is a probability measure on Rn for which the maximal operator     M φ(x) = sup  φ(λ.t) dν(t), φ ∈ S(Rd ) λ>0

Rd

is of weak type (p, p) for some 1 ≤ p < ∞. For every standard Borel probability space (X, B, µ) and ergodic measure-preserving action T by Rd on X,   lim f (Tλ.t x) dν(t) = f dµ λ→∞

Rd

X p

almost everywhere on X for every f in L (X). Proof. We can without loss of generality restrict our attention to f in the subspace Lp0 (X). The almost sure convergence above holds for f in the dense subspace L of Lp0 (X) defined in lemma 3.3. Thus, it suffices to prove that the subspace C = {f ∈ Lp (X) | lim Aλ f (x) λ→∞

exists a.e.}

Ergodic Theorems for Homogeneous Dilations

85

is closed in Lp0 (X). Suppose fk is a sequence in C which converges to f in Lp0 (X), and note that if λ and η are positive, then for all k, |Aλ f (x) − Aη f (x)| = |Aλ fk (x) − Aη fk (x)| + |Aη (f − fk )(x)| + |Aλ (f − fk )(x)|. The first term clearly goes to zero almost everywhere, since fk is in C. Thus, since M is of weak type (p, p) we have µ({x ∈ X | lim sup |Aλ f (x) − Aη f (x)| > α}) ≤ µ({x ∈ X | 2M (f − fk ) > α}) λ,η→∞

≤C

 p 2 ||f − fk ||pp , α


which can be made arbitrarily small for k large. 3.4. Rubio de Francia’s maximal inequality Let ν be a probability measure on Rd and define the maximal function     Sφ(x) = sup  φ(λt) dν(t). λ>0

Rd

These functions were studied by J.L. Rubio de Francia in [14], where the following striking theorem was established. Theorem 3.7. Suppose ν is a compactly supported measure in Rd with Fourier exponent a > 1. Then S is strong type (p, p) for all p > pa , where pa =

1+a . a

By Theorem 3.6, this result immediately implies the following theorem. Theorem 3.8. Let (X, B, µ) be a standard Borel probability measure space, and suppose T is a Borel measurable action of Rd on X which preserves µ. If ν is a compactly supported probability measure on Rd with Fourier exponent a > 1, then   lim f (Tλt x) dν(t) = f dµ λ→∞

Rd

X

almost everywhere on X, for all f in Lp (X) for p > pa , where pa =

1+a . a

Weaker versions of Theorem 3.7 has been extended to a more general class of homogeneous dilations in [8].

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4. Applications We will now present some straightforward applications of Theorems 3.8 and 3.6. 4.1. Ergodic theorems for dilations of submanifolds The following theorem can be found in Chapter 8 in [15] Theorem 4.1. Suppose S is a smooth hypersurface in Rd , whose Gaussian curvature is non-zero everywhere, and let dν = ψdσ, where the support of ψ ∈ C0∞ (Rd ) is assumed to intersect S in a compact set. Then the Fourier exponent of ν is at least d − 1. By Theorem 3.6 we have the following theorem. Corollary 4.2. Let S and ν be as in Theorem 4.1 above, and suppose d ≥ 3. For every standard Borel probability space (X, B, µ) and ergodic measure-preserving action T by Rd on X,   lim f (Tλt x) dν(t) = f dµ λ→∞

Rd

X

almost everywhere on X for every f in Lp (X), where d p> . d−1 Remark 4.3. J. Bourgain [5] proved that if d = 2 and ν denotes the arc-length measure on the boundary of a smooth centrally symmetric convex body in R2 , then maximal operator in Subsection 3.4 is of strong type (p, p) for p > 2. Hence, the analogue of Corollary 4.2 holds in this case. The special case of circles was proved by M. Lacey in [11]. 4.2. Ergodic theorems for polynomial curves Let λ > 0 and q1 , . . . , qd non-zero real numbers. We define the polynomial curve q : [0, 1] → Rd by q(w) = (q1 w, . . . , qd wd ), w ∈ [0, 1] . Let m denote the Lebesgue measure on [0, 1] and set ν = q∗ m, where q∗ denotes the push-forward induced by q. Note that the Fourier dimension of ν is generally of order 2/d and thus Theorem 1.1 does not apply. We introduce the homogeneous dilation on Rd defined by λ.t = (λt1 , . . . , λd td ),

t ∈ Rd .

For φ ∈ S we define the maximal operator  1    M φ(t) = sup  φ(s − λ.q(w)) dw. λ>0

0

The following theorem is due to E.M. Stein and S. Wainger [17], who also studied maximal inequalities for more general curves. Theorem 4.4. The maximal operator M is of strong type (p, p) for 1 < p < ∞.

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87

Corollary 4.5. For every standard Borel probability space (X, B, µ) and ergodic measure-preserving action T by Rd on X,  1  lim f (Tλ.q(w) x) dw = f dµ λ→∞

0

X

almost everywhere on X for every f in Lp (X), where d p> . d−1 4.3. Ergodic theorems for Salem sets The Hausdorff dimension of a subset E of Rd can be defined as the supremum of the a in [0, d] such that, for some probability measure ν which is supported on E,  |ˆ ν (ξ)|2 |ξ|a−d dξ < ∞. |ξ|≥1

This reformulation of the standard definition is justified by Frostman’s lemma; see, e.g., [12]. Note that this does not imply that ν is a Rajchman measure. However, if C |ˆ ν (ξ)| ≤ a/2 , |ξ| then the integral above is finite, and we conclude that the Fourier dimension of ν is always majorized by the Hausdorff dimension. If E supports a probability measure with Fourier dimension equal to the Hausdorff dimension of E, we say that E is Salem set. With no known exception, these sets are constructed either from probabilistic arguments or number theoretic arguments. It is known (see, e.g., [10]) that the image of any closed subset E
of R+ under a d-dimensional Brownian motion B is almost surely a Salem set with Fourier dimension equal to twice the Hausdorff dimension of E
. Thus, if E
is any compact subset of R+ of Hausdorff dimension b > 1/2 and d ≥ 2 then, almost surely, there is a probability measure ν on the (almost surely totally disconnected) on B(E
) such that Theorem 1.1 holds 2b for all p ≥ 2b−1 . The measure ν depends of course on the Brownian motion, but can almost surely be used as an average measure for a generalized Wiener ergodic theorem. This is reminiscent to J. Bourgain’s [6] theorem on random subsets of Z for which the pointwise ergodic theorem holds. 4.4. Regularity of return times We shall consider the set of return times of uniquely ergodic and jointly continuous actions (Tt ) of Rd on a compact metric space X. Let µ denote the unique invariant measure, and define, for a given Borel set A ⊆ Y and x ∈ X, the set of return times: RA (x) = {t ∈ Rd | Tt ∈ A}. If A is Jordan measurable with respect to µ, i.e., if µ(∂A) = 0, then the uniform ergodic theorem implies that  1 RA (x) = µ(A), lim m[0,1] λ→∞ λ

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for all x ∈ X, where m[0,1] denotes the normalized Lebesgue measure on the closed interval [0, 1]. This example motivates the following definition: A set B ⊆ Rd is Rajchman regular if for any Rajchman probability measure ν on Rd with compact support, the limit 1 d(B) = lim ν( B) λ→∞ λ exists and is independent of ν. After modifying the proof of Lemma 3.3 to the effect that the density of the subspace L ⊆ C(X) (with fo ∈ C(X)) can be established, we can prove the following proposition. Proposition 4.6. Let X be a compact metrizable space with a uniquely ergodic and jointly continuous action of Rd , and suppose that A ⊆ X is Jordan measurable with positive measure. Then for all x ∈ X, the set RA (x) is Rajchman regular. Moreover, the equality d(RA (x)) = µ(A) holds.

References [1] Baez-Duarte L. Central Limit Theorems for Complex Measures. J. Theor. Prob. Theory. (1993) no. 1. 33–56. [2] Bluhm, C.E. Fourier asymptotics of statistically self-similar measures. J. Fourier Anal. Appl. 5 (1999), no. 4, 355–362. [3] Bluhm, C.E. Liouville numbers, Rajchman measures, and small Cantor sets. Proc. Amer. Math. Soc. 128 (2000), no. 9, 2637–2640. [4] Bochner, S. Harmonic analysis and the theory of probability. University of California Press, Berkeley and Los Angeles, 1955. viii+176 pp. [5] Bourgain, J. Averages in the plane over convex curves and maximal operators. J. Analyse Math. 47 (1986), 69–85. [6] Bourgain, J. On the maximal ergodic theorem for certain subsets of the integers. Israel J. Math. 61 (1988), no. 1, 39–72. [7] Calderon, A.P. Ergodic theory and translation-invariant operators. Proc. Nat. Acad. Sci. U.S.A. 59 1968 349–353. [8] Dappa, H. and Trebels, W. On maximal functions generated by Fourier multipliers. Ark. Mat. 23 (1985), no. 2, 241–259. [9] Jones, R.L. Ergodic averages on spheres. J. Anal. Math. 61 (1993), pp. 29–45. [10] Kahane, J.P. Brownian motion and classical analysis. Bull. London Math. Soc. 8 (1976), no. 2, 145–155. [11] Lacey, M.T. Ergodic averages on circles. J. Anal. Math. 67 (1995), 199–206. [12] Mattila, P. Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. xii+343 pp. ISBN: 0-521-46576-1; 0-521-65595-1 [13] Nevo, A. Pointwise ergodic theorems for radial averages on simple Lie groups. I. Duke Math. J. 76 (1994), no. 1, 113–140. [14] Rubio de Francia, J.L. Maximal functions and Fourier transforms. Duke Math. J. 53 (1986), no. 2, 395–404.

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[15] Stein, E.M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993. xiv+695 pp. ISBN: 0-691-03216-5 [16] Stein, E.M. Maximal functions. I. Spherical means. Proc. Nat. Acad. Sci. U.S.A. 73 (1976), no. 7, 2174–2175. [17] Stein, E.M. and Wainger, S. Problems in harmonic analysis related to curvature. Bull. Amer. Math. Soc. 84 (1978), no. 6, 1239–1295. [18] Varadarajan, V.S. Groups of automorphisms of Borel spaces, Trans. Amer. Math. Soc. 109 (1963), 191–220. [19] Wiener, N. The ergodic theorem. Duke Math. J. 5 (1939), no. 1, 1–18. Michael Bj¨ orklund Einstein Institute of Mathematics Edmond J. Safra Campus Hebrew University, Givat Ram, 91904 Jerusalem, Israel e-mail: [email protected]

Progress in Probability, Vol. 64, 91–110 c 2011 Springer Basel AG


Boundaries from Inhomogeneous Bernoulli Trials Alexander Gnedin Abstract. The boundary problem is considered for inhomogeneous increasing random walks on the square lattice Z2+ with weighted edges. Explicit solutions are given for some instances related to the classical and generalized number triangles. Mathematics Subject Classification (2000). Primary 60J10, 60J50. Keywords. Coin-tossing processes, weighted Pascal graph, boundary, combinatorial triangles.

1. Introduction The homogeneous Bernoulli processes all share a property which may be called lookback similarity : if the number of heads h in any first n trials is given, then independently of the future outcomes the random history of the process can be described in some unified way. Specifically, the conditional history of homogeneous Bernoulli trials has the same distribution as a uniformly random permutation of h heads and t = n − h tails. This property is equivalent to exchangeability, meaning the invariance of distribution under finite permutations of coordinates. A central structural result regarding this class of processes is de Finetti’s theorem, which asserts that infinite sequences of exchangeable trials can be characterized as mixtures of the homogeneous Bernoulli processes. More general coin-tossing processes, in which probability of a head in the next trial may depend on the history through the counts of heads and tails observed so far, are divided into other classes of lookback similar processes, each class with its own random mechanism of arranging h heads and t tails in succession, consistently for every value of n = h + t. Understanding the structure of such classes is important in a variety of contexts, including statistical mechanics, urn models, species sampling, random walks on graphs, dimension theory of algebras, ergodic theory and others. The principal steps of the analysis involve identification of the extremal boundary, which may be characterized as the set of ergodic processes in a

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given class, as well as construction and decomposition of distinguished nonergodic processes, like, e.g., P´olya’s urn processes in the exchangeable case. A major technical issue which must be resolved to set up the scene is the way of specifying conditional probabilities on histories. The backward transition probabilities for the Markov chain counting heads-and-tails are rarely available directly. Sometimes the starting point is a particular reference distribution P ∗ , which determines a class of distributions P by requiring that P had the same conditional probabilities on histories as P ∗ [7, 19], however, in many combinatorial and algebraic contexts there might be no obvious choice for P ∗ with a desirable property of non-degeneracy. For instance, if the distributions on histories are determined by some symmetry condition, like equidistribution of paths in a Bratteli diagram, it might require some effort to construct a nontrivial process. The approach adopted in this paper amounts to defining conditional distributions on histories by means of a multiplicative weight function. The underlying structure is the lattice Z2+ with weighted edges, which we call a weighted Pascal graph. For instance, in the case of unit weights the structure is the Pascal triangle, in which the path-counts (combinatorial dimensions) are given by the binomial coefficients. The coin-tossing process is encoded into a increasing lattice path with unit horizontal and vertical jumps, and the boundary problem is connected to the asymptotic weighted path-enumeration. For various choices of the weight function, the model covers arbitrary coin-tossing processes, for which the joint counts of heads and tails is the sufficient statistic of the history to predict the outcomes of future trials. We will consider some classes of lookback-similar processes, for which the boundary can be determined without direct path-counting. Our strategy boils down to three steps. First of all, we refine a monotonicity idea [16] to show that the ergodic processes are naturally parameterized by the probability of a head in the first trial, so the boundary is always a subset of the unit interval. Then we manipulate with admissible transformations of weights to construct a parametric family of coin-tossing processes. The last step is to either verify that the constructed processes are ergodic, or to obtain all extremes by their decomposition in ergodic components. The generalized Pascal triangles appear sometimes in connection with statistics of combinatorial structures like partition and composition posets. Analysis of random walks on the ‘triangles’ has been proved useful to construct processes with values in these more complex objects [10, 11, 12, 14, 16].

2. Weighted Pascal graphs We shall be dealing with Markov chains S = (Sn , n ≥ 0) on the square lattice Z2+ , with edges directed away from the origin. A standard path starts at (0, 0) and each time increments by either (1, 0) or (0, 1). We think of the increment Sn − Sn−1 = (1, 0) as a head, and Sn − Sn−1 = (0, 1) as a tail in the nth cointossing trial. The components will be denoted Sn = (Hn , Tn ), so Hn + Tn = n.

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93

Let w be a (strictly) positive function on the set of edges. We call the lattice Z2+ with weighted edges a weighted Pascal graph. The weights of the edges (h, t) → (h+1, t) and (h, t) → (h, t+1), which connect (h, t) to its followers, will be denoted w1 (h, t) and w0 (h, t), respectively. Define the weight of a path connecting two given grid points to be the product of weights of edges along the path. The sum of weights of all standard paths with terminal point (h, t) is denoted d(h, t) and called dimension, this quantity is analogous to the partition sum in statistical mechanics. The dimensions satisfy the forward recursion d(h, t) = w1 (h − 1, t)d(h − 1, t) + w0 (h, t − 1)d(h, t − 1), d(0, 0) := 1 (where the terms with some negative arguments vanish). More generally, we define the extended dimension d(h, t; h
, t
) as the sum of weights of paths from (h, t) to (h
, t
). A class P = P(w) of lookback similar distributions for S is defined by the following Conditioning property: for all (h, t) ∈ Z2+ , if the Markov chain S starting at S0 = (0, 0) visits (h, t) ∈ Z2+ with positive probability, then given Sh+t = (h, t) the conditional probability of each standard path with endpoint (h, t) is equal to the weight of the path divided by d(h, t). The Markov property of S need not be assumed, rather it follows from the conditioning property. Note also that if the conditioning property holds for one particular (h
, t
) then it holds for every (h, t) lying on a standard path with endpoint (h
, t
). The class P is a convex, weakly compact set. A finite-dimensional counterpart of P is the class Pn of probability distributions for Markov chains with n steps subject to a restricted conditioning property which holds for h+t = n (hence for h+t ≤ n). These distributions are consistent as n varies, therefore P has the structure of projective limit of the Pn ’s. Since every Pn is n-dimensional simplex, the projective limit P is a Choquet simplex (see, e.g., Proposition 10.21 in [17]), which means that every P ∈ P has a unique representation as a convex mixture of the extreme elements of P. The set of extreme points of P is the extremal boundary denoted extP. The extremes are characterized as ergodic measures P ∈ P, for which every tail event of the process S has P -probability zero or one. Note that the tail sigmaalgebra for S coincides with the exchangeable sigma-algebra, comprised of the events invariant under permutations of the sequence of increments of S. The boundary problem asks one to describe as explicitly as possible the set of extremes extP. Each P ∈ P is uniquely determined by the probabilities of finite standard paths. These probabilities are conveniently encoded into a probability function on Z2+ P (Sh+t = (h, t)) , φ(h, t) := d(h, t) so that the probability of a standard path terminating at (h, t) is equal to the weight of the path multiplied by φ(h, t). The transition probabilities from (h, t) to

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(h + 1, t) and (h, t + 1) are written then as p1 (h, t) :=

w1 (h, t)φ(h + 1, t) w0 (h, t)φ(h, t + 1) , p0 (h, t) := , φ(h, t) φ(h, t)

respectively. The probability functions are characterized as nonnegative solutions to a backward recursion φ(h, t) = w0 (h, t)φ(h, t + 1) + w1 (h, t)φ(h + 1, t),

(2.1)

with the normalization φ(0, 0) = 1. Note that n 

φ(h, n − h)d(h, n − h) = 1,

(2.2)

h=0

since the terms make up the distribution of Sn . The boundary problem has many faces of which we mention a few. Moment problems. For the Pascal triangle, with w ≡ 1, the recursion (2.1) goes back to Hausdorff [18]. Positivity of φ in this case means that the iterated differences of the diagonal sequence φ( ·, 0) are nonnegative, i.e., φ( ·, 0) is completely monotone. The celebrated Hausdorff’s theorem states that such a sequence is uniquely representable as a sequence of moments of a probability measure on [0, 1], which means that extP is naturally identified with [0, 1]. Each π ∈ [0, 1] corresponds to a homogeneous Bernoulli process with probability π for heads. The representability of each P ∈ P as a unique mixture of the extremes is equivalent to de Finetti’s theorem. A similar connection exists in the general case too. To that end, observe that the bivariate array φ is obtainable by weighted differencing of the diagonal sequence φ( · , 0), for instance φ(n, 1) = (φ(n, 0)−w1 (n, 0)φ(n+1, 0))/w2 (n, 0). Thus solving (2.1) means finding all sequences φ( · , 0) for which the weighted differences are non-negative. We will show in the next section that extP is homeomorphic to a subset E ⊂ [0, 1] via π(P ) = P (H1 = 1) = φ(1, 0)w(1, 0), therefore the extremal decomposition  φ(n, 0) = E

φπ (n, 0)µ(dπ),

can be seen as a generalized problem of moments on E, with the kernel φπ (n, 0), in place of π n from Hausdorff’s problem of moments. Quasi-invariance. Finite permutations of N act on infinite paths in Z2+ by rearranging the sequence of increments. In these terms, the distributions P ∈ P can be characterized as measures quasi-invariant under permutations. The characteristic cocycle of the action is uniquely determined by the condition that if a path fragment (h, t) → (h + 1, t) → (h + 1, t + 1) is switched by transposition to (h, t) → (h, t + 1) → (h + 1, t + 1), then probability of the path is multiplied by w0 (h, t)w1 (h, t + 1)w1 (h, t)−1 w0 (h + 1, t)−1 . In the case w ≡ 1 the quasi-invariance means invariance, and we are back to the exchangeability.

Boundaries from Inhomogeneous Bernoulli Trials

95

Martin kernel. By a well-known recipe [21], each extreme φ is representable as a limit point of the Martin kernel φ(h, t) = lim

dim(h, t; h
, t
) , dim(h
, t
)

(2.3)

where h
+ t
→ ∞, and (h
, t
) vary in some way to ensure convergence of the ratios for all (h, t). This relates the boundary problem to the asymptotic weighted path enumeration. h-transform. Call P ∗ ∈ P fully supported, if it gives positive probability to every finite path, or, equivalently, the corresponding probability function φ∗ is strictly positive. Every P ∈ P with probability function φ is uniquely obtainable from P ∗ by a change of measure φ = ψφ∗ where the Radon-Nykodim derivative ψ is a P ∗ -harmonic function, that is satisfying the recursion ψ(h, t) = p∗ (h, t)ψ(h + 1, t) + q ∗ (h, t)ψ(h, t + 1), and p∗ , q ∗ are the transition probabilities of S under P ∗ . Note that the change of measure is just Doob’s h-transform, because the transition probabilities under P are connected to that under P ∗ via p1 (h, t) = p∗1 (h + 1, t)ψ(h + 1, t)/ψ(h, t), p0 (h, t) = p∗0 (h + 1, t)ψ(h + 1, t)/ψ(h, t).

(2.4)

We say that P ∈ P is finitely supported if it is not fully supported. For finitely supported P ∈ extP the probability function is (strictly) positive on one of the sets Im := {(h, t) : h ≤ m}, Jm := {(h, t) : t ≤ m} for some m ≥ 0. The process S under finitely supported measure eventually achieves either a finite number of heads or a finite number of tails. The case m = 0 corresponds to two trivial measures, denoted Q0,∞ and Q∞,0 , each supported by a single infinite path Hn ≡ 0 and Tn ≡ 0, respectively. Combinatorial structures. If the weights are integers, we may consider the graph with multiple edges and fine paths, which distinguish among the edges connecting the neighbouring grid points. The setting with multiple edges appears in the ergodic theory in connection with the ‘adic transform’ of the path space [8, 9]. When the distinction of edges with the same endpoints is made, we understand P as the class of probability measures on fine paths, with a more delicate conditioning property that for every (h, t) all standard fine paths terminating at (h, t) have the same probability φ(h, t). Pascal graphs with multiple edges arise by consistent coarsening the set of nodes in a tree. Suppose T is an infinite rooted tree with the set of vertices Tn (n ≥ 0) at distance n from the root. Suppose each Tn is partitioned in n + 1 nonempty blocks labeled (0, n), (1, n−1), . . . , (n, 0), so that for h+t = n each vertex in block (h, t) ⊂ Tn is connected by the same number w1 (h, t) of edges to every vertex in block (h + 1, t), and by the same number w0 (h, t) of edges to every vertex

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in block (h, t + 1). Merging all nodes in each block (h, t) in a single node we obtain a weighted Pascal graph, whose standard paths, in turn, ramify according to T . Example. The Stirling-I triangle has w0 (h, t) = h + t + 1, w1 (h, t) = 1. The dimensions dim(h, t) are unsigned Stirling numbers of the first kind. The fine paths of length n are in bijection with permutations of n + 1 integers. Write permutation σn of [n] := {1, . . . , n} in the one-row notation, like, e.g., 6 4 5 1 2 3. The boldfaced elements are (lower) records (a record is a number smaller than all numbers to the left of it, if any). A permutation σn+1 of [n + 1] extending σn is obtained by inserting integer n + 1 in one of n + 1 possible positions, e.g., 7 645123, 67 45123, . . . , 6451237 . The extension organizes permutations of integers 1, 2, . . . in a tree, in which each σn has n + 1 followers. Now suppose permutations of [n] are classified by the number of records. Each σn with h records (1 ≤ h ≤ n) is followed by a sole permutation of [n + 1] with h + 1 records, obtained by inserting n + 1 in the leftmost position of σn , and n permutations with h records. Assigning label (h, t) to the class of permutations of [h + t + 1] with h + 1 records, the classification of permutations by the number of records is then captured by the Stirling-I triangle. The same multiplicities appear when permutations are classified by the number of cycles. To this end, arrange elements in each cycle in the clockwise cyclic order. Then extending σn amounts to either inserting n + 1 in some cycle clockwise next to any of the integers 1, . . . , n, or starting a new singleton cycle with n + 1.

3. Probability of the first head As we have seen, each measure P ∈ P is uniquely determined by φ( · , 0), that is by the sequence of probabilities P (Hn = n) = dim(n, 0)φ(n, 0), n ≥ 0. We aim now to show that to uniquely parameterize P ∈ extP it is enough to take the probability of the first head π = π(P ) := P (H1 = 1). Every P ∈ P conditioned on Sh+t = (h, t) coincides, as a measure on the set of standard paths terminating at (h, t), with the same elementary measure Qh,t determined by the Martin kernel in (2.3) restricted to standard paths through (h, t). A path (h
n , t
n ), n ≥ 0, is called regular if the elementary measures Qhn ,tn converge weakly along the path, in which case the limit necessarily belongs to P. We may also say that the limit measure is induced by the path. Denote ext◦ P the set of measures induced by regular paths. By some general theory (see [1], p. 161) it is known that for every P ∈ P, the set of regular paths has P -probability 1, and that P ∈ extP if and only if the set of regular paths that induce P has P -probability 1. In particular, ext◦ P ⊃ extP. This larger set ext◦ P is the entrance Martin boundary for the system of backward transition probabilities determined by the conditioning property.

Boundaries from Inhomogeneous Bernoulli Trials

97

For P, P
∈ P we say that P is stochastically larger than P
if P (Hn ≥ h) ≥ P (Hn ≥ h) for all (h, n − h) ∈ Z2+ . This defines a partial order on P, and we will show that it restricts as a total order on ext◦ P.


Lemma 3.1. Let (hn , n − hn ) and (h
n , n − h
n ), n ≥ 0, be two regular paths which induce measures P and P
, respectively. Then the measures are comparable, and P is stochastically larger than P
iff hn ≥ h
n for all large enough n. In the latter case the measures are distinct iff P (H1 = 1) > P
(H1 = 1). Proof. Choose 0 ≤ h
n < hn ≤ n and consider the elementary measures Qhn ,n−hn and Qhn ,n−hn as the laws of two Markov chains S and S
(respectively) with n moves. We define a coupling of independent S and S
by running the chains in the backward time. Start S, S
simultaneously at states (hn , n − hn ) and (h
n , n − h
n ), respectively, and let them be running independently until the first time τ when they meet at the same state. From time τ on let S
coincide with S: this does not affect the marginal distributions of S
since both chains have the same (backward) transition probabilities. Because the number of heads each time decrements by 0 or 1, by this coupling S always has at least as many heads as S
. It follows that P (Hm = m) ≥ P
(Hm = m) for m ≤ n. Now suppose the elementary measures converge along two paths (hn , n − hn) and (h
n , n − h
n ) to distinct P and P
, respectively. In general, two paths either have infinitely many intersections or there is a definite inequality between hn and h
n for large enough n. The case of infinitely many intersections is excluded since P = P
. Suppose eventually hn > h
n , then by the coupling argument we conclude that P is strictly stochastically larger than P
. Then the probability distribution P (Hn = ·) is strictly stochastically larger than P
(Hn = ·) for a subsequence hence for all n > 0, which for n = 1 means that P (H1 = 1) > P
(H1 = 1).
Proposition 3.2. The set ext◦ P is weakly compact and P → π(P ) is a homeomorphism of ext◦ P onto a subset of [0, 1]. Proof. Assume Pj ∈ ext◦ P converge weakly to some P ∈ P. By Lemma 3.1 it is enough to consider the case when the sequence Pj is monotonic in the stochastic order, say increasing. For each Pj fix a path inducing it. Then it is always possible to choose a path which has the last intersection with the path inducing Pj at some time nj , where nj → ∞. It is readily seen that the path induces P , hence P ∈ ext◦ P. Thus the sequential boundary is compact. It remains to note that P → P (H1 = 1) is a continuous strictly increasing function on ext◦ P.
Henceforth the boundary can be identified with E := {π(P ) : P ∈ extP}. The set E may be fairly arbitrary, but in any case it contains the endpoints 0 and 1 that correspond to the trivial measures. If there exists a finitely supported measure with probability function strictly positive on Im and zero otherwise, we denote this measure Qm,∞ . If Qm,∞ exists, it is extreme and induced by the path (m, n − m), n → ∞. Likewise, a measure with probability function strictly positive on Jm and zero otherwise is denoted

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Q∞,m , m ≥ 0. In some cases the boundary is comprised of only finitely supported measures: Lemma 3.3. Suppose the set of values of m for which the finitely supported measure Qm,∞ exists is an infinite increasing sequence. If along this sequence π(Qm,∞ ) → 1 then these finitely supported measures and the trivial measure Q∞,0 exhaust the boundary. Proof. Let m0 = 0 ≤ m1 ≤ · · · be this sequence, characterized, by Proposition 3.2, by the first-head probabilities πj = π(Qmj ,∞ ) ↑ 1. Suppose there exists yet another P ∈ ext◦ P. Choose any j with πj > π(P ), and let (hn , n − hn ) be any regular path inducing P . By Lemma 3.1 eventually hn < mj , but then P is supported (nonstrictly) by Imj , which is a contradiction since all the Qmj ,∞ ’s are the only finitely supported measures.
Moreover, under assumptions of Lemma 3.3, ext◦ P = extP, and every path eventually satisfying mj ≤ hn < mj+1 is regular and induces Qmj ,∞ , as one shows along the same lines. In some cases the boundary is homeomorphic to [0, 1]: Lemma 3.4. [16] Suppose there is a sequence of positive constants (cn ) with cn → ∞, and for each s ∈ [0, ∞] there is a Ps ∈ P which satisfies Ps (Hn /cn → s) = 1. Suppose s → Ps is a continuous injection from [0, ∞] to P, with 0 and ∞ corresponding to the trivial measures. Then a path (hn , n − hn ) is regular iff hn /cn → s for some s ∈ [0, ∞], in which case the path induces Ps . Moreover, ext◦ P = extP = {Ps , s ∈ [0, ∞]}. In the situation of the last lemma the parameter s has the meaning of the asymptotic frequency of heads on the cn -scale. If cn = n can be chosen, like in the case of exchangeable trials, then we should take for the range of s a finite interval (and not [0, ∞]).

4. Transformations of weights We shall approach the question of constructing some lookback similar processes through admissible transformations of w, which change dimensions but do not affect the conditioning property. The idea is to apply an analogue of the h-transform (2.4) to the weights. Proposition 4.1. Two weight functions w and w ˜ yield the same conditional distributions on histories (hence P(w) = P(w)) ˜ iff there exists a positive function f on Z2+ such that for every edge s → s
w(s ˜ → s
) = w(s → s
)f (s
)/f (s).

(4.1)

Proof. For the ‘if’ part, the product of weights w(s, ˜ s
) along a path depends only on the endpoints of the path, hence the conditioning property is the same as for w. The ‘only if’ part follows by induction on the length of the path.


Boundaries from Inhomogeneous Bernoulli Trials

99

Call w ˜ balanced if w ˜0 (h, t) + w ˜1 (h, t) = σ(h + t) for some function σ, which may be then determined recursively from the corresponding to w ˜ dimension function d˜ as n n−1   ˜ n − h) = d(h, σ(i). h=0

i=0

If w ˜ is balanced then there is a fully supported P ∈ P with transition probabilities w ˜0 (h, t) w ˜1 (h, t) p0 (h, t) = , p1 (h, t) = . (4.2) σ(h + t) σ(h + t) Conversely, for fully supported P ∈ P the transition probabilities p0 , p1 themselves define an equivalent weight function. We see that finding a fully supported P is equivalent to transforming w to a balanced weight function w. ˜ On the other hand, if substitution (4.1) is used with f ≥ 0 strictly positive exactly on Im , then every P satisfying the conditioning property with respect to w ˜ for (h, t) ∈ Im also belongs to P(w) and is (nonstrictly) supported by Im . Transforming w to a balanced w ˜ strictly supported by Im yields Qm,∞ . The same construction applies to Jm . A family of admissible transformations has the form w ˜0 (h, t) = g0 (t)w0 (h, t)/g(h + t), w ˜1 (h, t) = g1 (h)w1 (h, t)/g(h + t),

(4.3)

where g > 0, while the functions g0 , g1 must be either both strictly positive, or one of them strictly positive and another nonnegative and supported by {0, . . . , m} for some m. If it turns that w ˜ is balanced with σ ≡ 1 then the process defined by (4.2) has the probability function $t−1 $h−1 i=0 g0 (i) j=0 g1 (j) φ(h, t) = , (4.4) $h+t−1 g(k) k=0 which may or may not be fully supported. Example: the Pascal triangle revisited. The weight is w ≡ 1 and the dimension  function is d(h, t) = h+t . The conditioning property identifies P as the family of h processes of exchangeable trials. We will not change P if instead we choose w1 > 0 to be an arbitrary function of h and w2 > 0 arbitrary function of t, however the balance condition forces g0 , g1 and g to be linear functions. Choosing π ∈ [0, 1] and g0 ≡ 1 − π, g1 ≡ π, g ≡ 1, the probability function (4.4) becomes φ(h, t) = π h (1 − π)t , for which Pπ is homogeneous Bernoulli. By the law of large numbers Pθ (Hn /n → π) = 1, the family π → Pπ is continuous, and the trivial measures appear for π ∈ {0, 1}, respectively, so by Lemma 3.4 the Ππ ’s are extreme and all extremes are found. Thus E = [0, 1] and we recover Hausdorff-de Finetti’s theorem. Note that the Martin kernel (2.3) is 
%
dim(h, n − h; h
, n
− h
) n −n n = .



dim(h , n − h ) h −h h


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From the criterion of path regularity we see that the ratios of binomial coefficients converge as n
→ ∞ (for all n and 0 ≤ h ≤ n) if and only if h
/n
→ π for some π ∈ [0, 1]. Although the latter fact can be derived directly from the formula for binomial coefficients, it is interesting to note that the conclusion can be made without direct evaluations. For a, b > 0, another transformation of weights with g1 (h) = a + h, g0 (t) = b + t, g(h + t) = a + b + h + t results in a balanced weight function, and (4.4) becomes (a)h (b)t φ(h, t) = (a + b)h+t (where (x)m is Pochhammer’s factorial). The corresponding P is the familiar P´ olya urn process with starting configuration (a, b). The representation of P as mixture of homogeneous Bernoulli processes follows by noting that the limit law of Hn /n under P is the beta distribution with density Γ(a + b) a−1 π (1 − π)b−1 , π ∈ [0, 1]. Γ(a)Γ(b)

5. Generalized Stirling triangles The transformation of weights (4.1) always allows to standardize the weights so that w1 ≡ 1, without changing P for a given weighted Pascal graph. This leaves us with the parameterization of graphs by the function w0 . Still, the number of parameters is too large to be analyzed in full generality. Kerov [20] suggested to study a smaller class of generalized Stirling triangles, which have weights of the form w0 (h, t) = ah+t + bh , w1 (h, t) = 1, (5.1) where an , bh satisfy ah+t + bh > 0 for (h, t) ∈ Z2+ . In this section we survey some previous work [11, 16, 26] and give some new results extending [20]. Introduce polynomials in variable θ An (θ) = (θ + a0 ) · · · (θ + an−1 ), Bn (θ) = (θ − b0 ) · · · (θ − bn−1 ). Transforming w as w ˜1 (h, t) = θ − bh , w ˜0 (h, t) = ah+t + bh , we obtain balanced w ˜ with σ(n) = θ + an , and (4.4) yields a family of measures Pθ with probability functions φθ (h, t) =

Bh (θ) Ah+t (θ)

(5.2)

satisfying (2.1). From (2.2) follows that dimensions are the transition coefficients between two polynomial bases, n  An (θ) = d(h, n − h)Bh (θ), h=0

Boundaries from Inhomogeneous Bernoulli Trials

101

which justifies their name as generalized Stirling numbers [4]. In the case of the Stirling-I numbers, the transition is from powers to factorial powers of θ, and for Stirling-II the other way round. Using (5.2) yields transition probabilities for Pθ ∈ P, θ − bh ah+t + bh p1 (h, t) = , p0 (h, t) = , θ + ah+t θ + ah+t provided that care of positivity is taken. For any bounded range of (h, t) we can indeed achieve that w, ˜ p0 , p1 are all positive by choosing large enough θ. If suph bh < ∞ then a fully supported measure Pθ is defined by choosing θ > suph bh . Proposition 5.1. For every m ≥ 0 satisfying bm > bh for h < m, there is a finitely supported measure Qm,∞ ∈ P with the probability of the first head bm − b0 . (5.3) bm + a0 If suph bh = ∞ then for a sequence of such bmj ↑ ∞ the probabilities (5.3) together with their accumulation point 1 comprise the discrete boundary E ⊂ [0, 1] of the generalized Stirling triangle. π=

Proof. Suppose bm ≤ bm−1 . We first show that the measure Qm,∞ does not exist. Recall that under this measure Markov chain S must eventually proceed along the path (0, 0) → (1, 0) → · · · → (m, 0) → (m, 1) → (m, 2) → · · · , which must have a positive probability and induce the measure. Observe that the weight of the standard path $n−1

(0, 0) → (1, 0) → · · · → (m, 0) → (m, 1) → · · · → (m, n − m)

(5.4)

is j=m (aj +bm ). On the other hand, the dimension is estimated as d(m, n−m) ≥ $n−1 (n − m) j=m (aj + bm ), which follows from bm ≤ bm−1 by estimating the total weight of n − m + 1 standard paths of the kind (0, 0) → (1, 0) → · · · → (m − 1, 0) → (m − 1, 1) → · · · → (m − 1, t) → (m, t) → (m, t + 1) → · · · → (m, n − m). Since the probability of (5.4) under Qm,n−m is equal to the relative weight of the path, this probability goes to zero as n → ∞, thus the path does not induce Qm,∞ , hence the measure does not exist. If bm is strictly larger than b0 , . . . , bm−1 , a similar weighted path-counting shows that the path (5.4) induces Qm,∞ , which coincides with Pθ for θ = bm . We see that Qmj ,∞ ’s are the only measures supported by some Im , and since (5.3) approaches 1 as mj → ∞, the result now follows from Lemma 3.3.
Example. The classical Stirling-II triangle has an ≡ 0 and bh = h + 1. For m ≥ 0 the extreme measure Qm,∞ can be described as follows. The first head appears after geometrically distributed time with success probability m/(m + 1), then the second head appears after independent geometrically distributed time with

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probability of a head (m − 1)/(m + 1), and so on to the last series of trials with probability of a head 1/(m + 1), finally followed by only tails. This is the familiar coupon-collectors’ process in which collector starts with one coupon and keeps on sampling from coupons with m + 1 equal frequencies until all m + 1 distinct types of coupons are discovered. If suph bh < ∞ the situation is more complex. We shall consider three special families of weighted Pascal graphs. 5.1. The linear weights Suppose in (5.1) we take an = n + 1, bh = −α(h + 1) with parameter α < 1, so w0 (h, t) = h + t + 1 − α(h + 1), w1 (h, t) = 1.

(5.5)

This weight function is related to Gibbs exchangeable partitions of N [11, 16]: the connection is that Hn+1 coincides with the number of blocks of the partition restricted to {1, . . . , n}. For suitable range of θ a probability function is defined by φθ (h, t) =

(θ + α)(θ + 2α) · · · (θ + hα) , (θ + 1)h+t

(5.6)

The corresponding transition probabilities are p1 (h, t) =

θ + α(h + 1) h + t + 1 − α(h + 1) , p0 (h, t) = . h+t+1+θ h+t+1+θ

To ensure positivity we must either require that 0 ≤ α < 1 and θ ≥ 0, or that α < 0 and −θ/α ∈ Z+ . As n varies, the number of heads Hn+1 increases as the number of blocks in the Ewens-Pitman partition of {1, . . . , n} [25]. The case α = 0. This is the Stirling-I triangle, intrinsically related to random permutations and other logarithmic combinatorial structures [3]. For 0 < θ < ∞, S under Pθ is the process of Bernoulli trials with probability of a head at trial n ≥ 0 being θ/(θ + n + 1). By the strong law of large numbers the measures (5.6) satisfy Pθ (Hn / log n → θ) = 1, thus by Lemma 3.3 the measures are extreme and exhaust the boundary, which θ may be parameterized by π = θ+1 ∈ [0, 1]. The case 0 < α < 1. This case is related to the excursion theory of recurrent continuous-time Markov processes, like Brownian motion in the case α = 1/2 [25]. The Pθ ’s are not extreme, since Hn /nα has a nontrivial limit distribution. The boundary is continuous, E = [0, 1], as in Lemma 3.4, and the nontrivial extreme measures are obtained by conditioning any Pθ with −α < θ < ∞ on Hn /nα → s for 0 < s < ∞. See [16] for details.

Boundaries from Inhomogeneous Bernoulli Trials

103

The case α < 0. This case is covered by Proposition 5.1. The possible values are θ = −αm, and the boundary is comprised of Qm,∞ , m ≥ 0, and Q∞,0 . In the special case α = −1 there is a transformation of weights with g0 (t) = t + γ, g1 (h) = (h + 1)2 − γ(h + 1) + ζ leading to the balanced weight function w ˜1 (h, t) = (h + 1)2 − γ(h + 1) + ζ, w ˜0 (h, t) = (2h + t + 2)(t + γ),

(5.7)

where γ ≥ 0, and ζ has a range suitable to guarantee positivity of the weights. This yields a nonextreme process with a nondegenerate distribution of the terminal number of heads limn→∞ Hn , as described in [11]. Similarly to the Ewens-Pitman partitions [16, 25], one can construct an exchangeable partition-valued process, for which p1 (h, t) is the probability of a new block at time h+ t+ 1, see [11] for details. It is natural to wonder if the weights (5.5) admit any other probability functions of the form (4.4). The answer is negative: these are either (5.6), or the special family in the case α = −1. The characterization follows from the next lemma. Lemma 5.2. If for nonnegative integers h, t and n = h + t the relation (n + 1 − α(h + 1))g0 (t) + g1 (h) = g(n) holds for α = −1 then g0 (t) = c, for c > 0. If the relation holds for α = −1 then g0 (t) = c1 t + c2 for nonnegative c1 , c2 , with at least one of the constants being nonzero. Proof. Let ∆f (n) = f (n + 1) − f (n). Differencing the relation first in n, then in h yields −(n + 2 − α(h + 1))∆2 g0 (t − 1) − (1 + α)∆g0 (t − 1) = 0. Varying h while keeping t = n − h fixed we obtain ∆2 g0 = 0 and then also (1 + α)∆g0 = 0. From this g0 (t) = c1 t + c2 , where c1 = 0 is only possible if α = −1.
5.2. Generalized Stirling-I Suppose bh ≡ 0 and an > 0, so w0 (h, t) = ah+t , w1 (h, t) = 1. Under Pθ the process S is a process of inhomogeneous Bernoulli trials, sometimes called spacetime random walk. The probability function of Pθ is φθ (h, t) =

θh . (θ + a0 ) · · · (θ + ah+t−1 )

The structure of the boundary was sketched by Pitman [26] (abstract of a conference talk). Here we add some details to [26], in particular we confirm that the criterion for E = [0, 1] is any of the equivalent conditions   min(an , 1) an = ∞ ⇐⇒ = ∞. (5.8) 2 (1 + an ) 1 + an n n For instance, for an = nβ the boundary is [0, 1] if |β| ≤ 1, and the boundary is discrete if |β| > 1.

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The weights can be transformed to the form w ˜0 (h, t) = v0 (h + t), w ˜1 (h, t) = v1 (h+t) provided v0 , v1 satisfy v0 (n)/v1 (n) = an . The criterion (5.8) assumes then the form stressing symmetry between heads and tails:  v0 (n)v1 (n) = ∞. 2 (v 0 (n) + v1 (n)) n It is not difficult to see that when (5.8) fails, none of the measures Pθ with 0 < θ < ∞ is extreme. Indeed, the variance of Hn remains bounded as n → ∞, and the centered Hn ’s converge weakly to a nondegenerate distribution, so the tail sigma-algebra of S is nontrivial. Thus Pθ ∈ extP can only hold if the series diverges. In the latter case our Lemma 3.4 has a restricted applicability, because for general (an ) there might be no common scaling cn → ∞ suitable for the full range of θ ∈ [0, ∞]. It is convenient to re-denote the transition probabilities under Pθ as p1 (n) =

θ an , p0 (n) = , θ + an θ + an

where n = h + t and θ ∈ [0, ∞]. 5.2.1. The case of continuous boundary. Proposition 5.3. If (5.8) holds then extP = {Pθ , θ ∈ [0, ∞]}. Thus the extremal boundary is E = [0, 1], which is the range of π = a0θ+θ . Proof. Under Pθ the tail sigma-algebra of S is trivial. Indeed, by a general criterion for the tail sigma-algebra generated by the  sequence of sums of independent random variables it is enough to check that n min(p1 (n), p0 (n)) = ∞; see [22], Theorem 1 and Corollary on p. 169, or see [23], Theorem 3.1(ii). But the latter follows from (5.8). As θ varies, π runs over the full range [0, 1], thus by Proposition 3.2 all extremes are found.
See [5] for extensions to more general space-time lattice walks, and [2] for conditions of triviality of the exchangeable sigma-algebra for random sequences with more than two values. 5.2.2. The case of discrete boundary. We take θ = 1 and assume now that the probabilities of heads/tails under P1 satisfy n p1 (n)p0 (n) < ∞. We consider P ∗ := P1 as the reference measure, to be decomposed in ergodic components. Recall that the elementary symmetric function of degree k in formal variables x1 , x2 , . . . is the infinite series  ek (x1 , x2 , . . . ) = xi1 · · · xik , i1 <···
Substituting xm = 0 for m > n yields the elementary symmetric polynomial ek (x1 , . . . , xn ) in n variables.

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Introduce the odds ratios rn := p1 (n)/p0 (n). The conditional probability given Sh+t = (h, t) of h heads at times {n1 , . . . , nh } ⊂ {1, . . . , h + t} is rn1 · · · rnh . eh (r1 , . . . , rh+t ) Let Ln = {m ≤ n : p1 (m) ≤ p0 (m)}, Mn = {m ≤ n : p1 (m) > p0 (m)}, L = ∪n Ln , M = ∪n Mn . Since    p1 (n)p0 (n) < ∞ ⇐⇒ p1 (n) < ∞ and p0 (n) < ∞, n

n∈L

n∈M

the Borel-Cantelli lemma implies that P ∗ -almost surely S has finitely many (1, 0)increments at times n ∈ L and finitely many (0, 1)-increments at times n ∈ M . The latter means that Sn = (Hn , Tn ) is essentially converging, i.e., (Hn , Tn ) − (#Mn , #Ln ) → (Z, −Z) P ∗ -a.s., for some integer-valued random variable Z. Let R be the range of Z; this is either Z, or a semi-infinite integer interval if M or L is finite. The variable Z is tail-measurable, thus conditioning P ∗ on the value of Z we obtain a countable family of probabilities {Pz∗ , z ∈ Z} ⊂ P. Every Pz∗ is extreme, since it is supported by a single class of equivalent paths which eventually coincide with the path (hn , tn ) = (#Mn − z, tn = #Ln + z) (where n is large enough). Proposition 5.4. If (5.8) does not hold then extP = {Pz∗ , z ∈ R} ∪ {Q0,∞, Q∞,0 }. Proof. We wish to prove that the list of extremes is full. In the spirit of the monotonicity arguments of Section 3, one concludes that if the sequence |hn −#Mn | is bounded then the elementary measures Qhn ,n−hn may only converge to some Pz∗ . Modifying Lemma 3.3, it remains to show that Pz∗ (H1 = 1) → 1 or 0 as z → +∞ or −∞, respectively. We shall focus on the first relation, the second being analogous.  Consider first the special case n p1 (n) < ∞, when Hn converges P ∗ -almost  surely to some finite random variable H. Then also M is finite, n rn < ∞ and & 'k  ek (r1 , r2 , . . . ) < rn < ∞. n

Since M is finite, conditioning on a large value of Z is the same as conditioning on a large value of H. Thus it is enough to show that r1 eh−1 (r2 , r3 , . . . ) P ∗ (H1 = 1|H = h) = → 1, as h → ∞. eh (r1 , r2 , . . . ) Using the identity eh (r1 , r2 , . . . ) = r1 eh−1 (r2 , r3 , . . . ) + eh (r2 , r3 , . . . ) we are reduced to checking that eh (r2 , r3 , . . . )/eh−1 (r2 , r3 , . . . ) → 0. The latter follows from the term-wise estimates of the series: ri1 · · · rih ≤ sh ri1 · · · rih−1 with sh = maxn>h rn → 0. In the general case, we define H
to be the total number of heads at times n ∈ L ∪ {1}. By independence of the increments of S under P ∗ we get exactly

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as above P ∗ (H1 = 1|H
= h) → 1 as h → ∞. Now P ∗ (H1 = 1|Z = z) → 1 for z → ∞ follows from this and H
≥ Z by conditioning on the number of tails at times n ∈ M \ {1}.
The weights in the decomposition of P ∗ over the extremal boundary extP are the probabilities    −1 P ∗ (Z = z) = ei (r
, ∈ L)ej (rm , m ∈ M) p0 ( ) p1 (m). {(i,j): i−j=z}


∈L

m∈M

Thus the point boundary E ⊂ [0, 1] is discrete, with the unique accumulation  at 1 iff n p1 (n) < ∞, the unique accumulation point at 0 iff p (n) < ∞, n 0  and if both series diverge but n p1 (n)p0 (n) < ∞ both 0 and 1 are (the only) accumulation points.  Remark. In [20] it was conjectured that n 1/an = ∞ implies the continuous boundary. Pitman’s criterion (5.8) disproves the conjecture (the case an = 1/n2 is a counterexample). However, for E = [0, 1] to hold it is enough to assume that n 1/an = ∞ and also that the an ’s are bounded away from 0. 5.3. Generalized Stirling-II The generalized Stirling-II triangle has weight function w1 (h, t) = 1, w0 (h, t) = bh , where bh > 0. The measure Pθ has probability function (θ − b0 ) . . . (θ − bh−1 ) , θh+t where either θ > suph bh , or θ = bm for some m such that bm is the strict maximum of b0 , . . . , bm . We shall assume suph bh < ∞, since the opposite situation is covered by Proposition 5.1. Each such Pbm is the ergodic measure coinciding with Qm,∞ , hence we focus on Pθ ’s with θ > suph bh . The process S under Pθ can be seen as a coupon-collector’s sampling scheme, with every new coupon identified with a ‘head’. The sampling starts at time 0 with coupon labelled 0. Each time n ≥ 0 when coupons 0, . . . , h are in the collection, a new coupon to be labelled h + 1 is drawn with probability 1 − λh for λh = bh /θ. More generally, let P be a probability under which ξ0 , ξ1 , . . . are independent geometric variables with φθ (h, t) =

P (ξh = k) = λk−1 (1 − λh ), k ∈ N. h Define an inhomogeneous ‘renewal process’ which counts heads Hn := max{h ∈ h−1 Z+ : j=0 ξj ≤ n}, and let Tn := n − Hn , Sn = (Hn , Tn ). ∞ Lemma 5.5. The tail sigma-algebra of S is trivial if and only if h=0 λh = ∞. Proof. Let Sn = σ{Sk , k ≥ n} and S = ∩n Sn be the tail sigma-algebra of S. If $ the series n λn converges, the probability P (limn→∞ Tn = 0) = ∞ h=0 (1 − λh ) is strictly between 0 and 1, hence S contains a nontrivial event. Moreover, Tn converge to a finite random variable P -a.s.

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Suppose the series diverges. Since in any case Hn → ∞ P -a.s., we can define finite times of heads τh := min{n : Hn = h} = ξ0 + · · · + ξh−1 for h ≥ 0. Let Gh be the sigma-field generated by the events {τh = n} ∩ A for A ∈ Sn and n ≥ 0, and define G := ∩∞ h=0 Gh . We assert that G coincides with S, up to zero events. Indeed, we have τn ≥ n, whence Gn = Sτn ⊂ Sn and G ⊂ S. In the other direction, {τh < n} ∩ A ∈ Gh provided that A ∈ Sn , whence G ⊃ S obtains as first n → ∞ then h → ∞. Now observe that G coincides with the tail sigma-algebra for the h sequence of sums of j=1 ξj , h ≥ 0. Mineka’s criterion for triviality of the tail sigma-algebra for sums ([22] Theorem 1 and Corollary on p. 169) becomes ∞  ∞ 

min(P (ξh = i), P (ξh = i + 1)) =

h=0 i=0

∞ 

λh = ∞,

h=0

from which G is trivial.




Returning to the generalized Stirling-II triangle we conclude on the structure of the boundary, which might possess both continuous and discrete components.  Proposition 5.6. If h bh < ∞ then each Pθ for θ > sup bh is decomposable, and extP = {Qm,∞ : bm > bj for j < m} ∪ {Q∞,m : m ≥ 0}. In this case E is a discrete  set with the only accumulation point at π = 1 − b0 /(suph bh ). If h bh = ∞ then extP = {Qm,∞ : bm > bj for j < m} ∪ {Pθ : θ > suph bh }. In this case E contains the interval [1 − b0 /(suph bh ), 1]. Proof. If the series converges, then Tn converge weakly under Pθ , with θ > sup bh , to a finite random variable. Conditioning on the terminal value Tn = m yields Qm,∞ . If the series diverges then every Pθ with θ > sup bh is extreme by Lemma 5.5.
Remark. This result disproves the assertion of Theorem 5.2 in [20] in the case  b < ∞. A counterexample is the q-Pascal triangle with 0 < q < 1 (already h h discussed in [20]). q-Pascal triangles. A q-analogue of the Pascal triangle is the generalized Stirling-II   triangle with weights w0 (h, t) = q h , w1 (h, t) = 1, and dimension d(h, t) = h+t t q equal to the q-binomial coefficient (see [20, 13, 14]). For integer q equal to a power of prime number, the lookback similar processes on the q-Pascal triangle encode homogeneous measures on Grassmanians in the infinite-dimensional space over the Galois field with q elements [14, 15]. Suppose q > 1. By Proposition 5.6 the only ergodic measures are Qm,∞ , which are given explicitly by the probability function φ(h, t) = q (m−t)h

t−1 

(1 − q m−j ), m = 1, 2, . . . .

j=0

The boundary is E = {q m , m ≥ 0} ∪ {0}.

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A transformation of weights yields w ˜0 (h, t) = q h+t , w ˜1 (h, t) = 1, which is a generalized Stirling-I triangle, thus inhomogeneous Bernoulli processes with probability for head p1 (n) = θ/(θ + q n ) belong to P. By Proposition 5.8 these measures are not extreme. We refer to [13] for the explicit decomposition over the boundary, for further q-analogues of Bernoulli processes and for q-analogues of P´olya sequences. The case q < 1 is treated by transformation to the weights w ˜0 (h, t) = 1, w ˜1 (h, t) = q −t and transposition of Z2+ about the diagonal, which establish equivalence of the q-Pascal graphs with parameters q and q −1 .

6. Generalized Eulerian triangles We define a generalized Eulerian triangle to be a weighted Pascal graph with w1 (h, t) = t + a, w0 (h, t) = h + b, where a, b > 0. The classical Eulerian triangle is the instance a = b = 1, where dimensions are Eulerian numbers that count descents in permutations. Since the balance condition is fulfilled, there is a natural fully supported measure P ∗ with transition probabilities t+a h+b p1 (h, t) = , p0 (h, t) = . h+t+a+b h+t+a+b On the other hand, changing weights to w1 (h, t) = (θ−h+b)(t+a), w0 (h, t) = (θ + t + a)(h + b) for θ = m + a, m = 0, 1, . . . , we obtain Qm,∞ with transition probabilities p1 (h, t) =

(θ − h − b)(t + a) (θ + t + a)(h + b) , p0 (h, t) = . θ(t + h + a + b) θ(t + h + a + b)

Similarly, the measures Q∞,m are constructed. Proposition 6.1. The measures Qm,∞ , Q∞,m with m ≥ 0, and P ∗ comprise the extremal boundary of the generalized Eulerian triangle. Proof. As m → ∞, the finitely supported measures Qm,∞ , Q∞,m converge to P ∗ . By a version of Proposition 3.3 and compactness, these measures comprise the sequential boundary ext◦ P. It remains to be shown that P ∗ is ergodic, but this follows, because otherwise P ∗ were decomposable in mixture of finitely supported elements of ext◦ P, which is impossible because Hn → ∞ and Tn → ∞ P ∗ -a.s.
The process S under P ∗ is known as the following Friedman’s urn scheme (see [6], Section 2.2). At time h + t there are h + t + a + b balls in an urn, of which h+a are marked ‘heads’ and t+a ‘tails’. A ball is drawn uniformly at random, and returned in the urn together with another ball of the opposite label. If a or b are not integers, this prescription is to be understood as adding 1 to the total weight of ‘heads’ with probability (t + a)/(h + t + a + b). Unlike P´olya’s urn model or the Stirling process in section 5.1, Friedman’s urn exhibits concentration of measure

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in the form P ∗ (Hn /n → 1/2) = 1. The latter fact, combined with the criterion of ergodicity and the observation that Qhn ,tn converge weakly to P ∗ whichever hn → ∞, tn → ∞, confirms that P ∗ is a unique fully supported ergodic measure. See [8, 9, 24] for other proofs of the fact that P ∗ is the unique fully supported ergodic measure for the standard Eulerian triangle with a = b = 1. By the virtue of weight transformation, the existence and uniqueness of a fully supported ergodic measure P ∗ ∈ P, and Proposition 6.1 extend straightforwardly to generalized Eulerian triangles with weights of the form w ˜0 (h, t) = (h + b)g0 (t)/g(h + t), w ˜1 (h, t) = (t + a)g1 (h)/g(h + t), where a, b > 0 and g0 , g1 , g are arbitrary positive functions. For the special case of weights w ˜0 (h, t) = ch + 1, w ˜1 (h, t) = ct + 1, with c ∈ N, the uniqueness of such P ∗ was shown recently in the context of adic dynamical systems by analysis of the dimension function of the graph (see [27], Theorem 2.2). Acknowledgment The author is indebted to a referee for a number of improvements and helpful comments.

References [1] D. Aldous, Exchangeability and related topics. L. Notes Math. 1117, Springer, Berlin, 1985. [2] D. Aldous and J. Pitman, On the zero-one law for exchangeable events. Ann. Prob. (1979), 704–723. [3] R. Arratia, A. Barbour and S. Tavar´e, Logarithmic Combinatorial Structures: A Probabilistic Approach. European Math. Soc., 2003. [4] C. Charalambides, Combinatorial Methods in Discrete Distributions. Wiley, 2004. [5] B.M. Baker and D.E. Handelman, Positive polynomials and time-dependent integervalued random variables. Canadian J. Math. 44 (1992), 3–41. [6] P. Flajolet, P. Dumas and V. Puyhaubert, Some exactly solvable models in urn process theory. DMTCS proc. AG (2006), 58–118. [7] G. Fayolle, R. Iasnogorodski and V. Malyshev, Random walks in the quarter-plane. Springer 1999. [8] S.B. Frick, M. Keane, K. Petersen and I.A. Salama, Ergodicity of the adic transformation on the Euler graph. Math. Proc. Camb. Phil. Soc. 141 (2006), 231–238. [9] S.B. Frick and K. Petersen, Random permutations and unique fully supported ergodicity for the Euler adic transformation. Ann. IHP – Prob. Stat. 2008. [10] A. Gnedin, Coherent random permutations with record statistics. DMTCS proc. AH (2007) 147–158. [11] A. Gnedin, A species sampling model with finitely many types. Elec. Comm. Probab. 2010.

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[12] A. Gnedin and G. Olshanski, Coherent random permutations with descent statistic and the boundary problem for the graph of zigzag diagrams. Intern. Math. Res. Notes Article ID 51968 (2006). [13] A. Gnedin and G. Olshanski, The boundary of the Eulerian number triangle. Moscow Math. J. 6 (2006), 461–475. [14] A. Gnedin and G. Olshanski, A q-analogue of de Finetti’s theorem. Elec. J. Combinatorics (2009) paper R78. [15] A. Gnedin and G. Olshanski, q-Exchangeability via quasi-invariance. Ann. Probab. (2010). [16] A. Gnedin and J. Pitman,Gibbs exchangeable partitions and Stirling triangles. J. Math. Sci. 138 (2006), 5674–5685. [17] K.R. Goodearl, Partially ordered abelian groups with interpolation. Mathematical Surveys and Monographs, number 20, American Mathematical Society, Providence, R.I., 1986. [18] F. Hausdorff, Summationsmethoden und Momentfolgen I. Math. Z. 9 (1921), 74–109. [19] I. Ignatiouk-Robert and C. Loree, Martin boundary of a killed random walk on a quadrant. http://arxiv.org/abs/0903.0070, 2009. [20] S. Kerov. Combinatorial examples in the theory of AF-algebras. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 172 (Differentsialnaya Geom. Gruppy Li i Mekh. Vol. 10) (1989) 55–67, 1989. Translated in J. Soviet Math. (1992) 59(5), 1063–1071. [21] S. Kerov, A. Okounkov and G. Olshanski, The boundary of Young graph with Jack edge multiplicities. Intern. Math. Res. Notices (1998) 173–199. [22] J. Mineka, A criterion for tail events for sums of independent random variables. Prob. Th. Rel. Fields (1973) 25 163–170. [23] S. Orey, Tail events for sums of independent random variables, J. Math. Mech. (1966) 15 937–951. [24] K. Petersen and A. Varchenko, The Euler adic dynamical system and path counts in the Euler graph. http://arxiv.org/abs/0811.1733 (2009). [25] J. Pitman, Combinatorial stochastic processes, Springer L. Notes Math., vol. 1875, 2006. [26] J. Pitman, An extension of de Finetti’s theorem Adv. Appl. Probab. (1978) 10 268–270. [27] G. Strasser, Generalisations of the Euler adic (2010) Preprint. Alexander Gnedin Mathematical Institute Utrecht University PO Box 80 010 NL-3508 TA Utrecht, The Netherlands e-mail: [email protected]

Progress in Probability, Vol. 64, 111–142 c 2011 Springer Basel AG


Resistance Boundaries of Infinite Networks Palle E.T. Jorgensen and Erin P.J. Pearse Abstract. A resistance network is a connected graph (G, c). The conductance function cxy weights the edges, which are then interpreted as conductors of possibly varying strengths. The Dirichlet energy form E produces a Hilbert space structure HE on the space of functions of finite energy. The relationship between the natural Dirichlet form E and the discrete Laplace operator ∆ on a finite network is given by E (u, v) = u, ∆v2 , where the latter is the usual 2 inner product. We describe a reproducing kernel {vx } for E which allows one to extend the discrete Gauss-Green identity to infinite networks:   ∂v E (u, v) = u∆v + u ∂n , G

bd G

where the latter sum is understood in a limiting sense, analogous to a Riemann sum. This formula yields a boundary sum representation for the harmonic functions of finite energy. Techniques from stochastic integration allow one to make the boundary bd G precise as a measure space, and give a boundary integral representation (in a sense analogous to that of Poisson or Martin boundary theory). This is done in terms of a Gel’fand triple S ⊆ HE ⊆ S  and gives a probability measure P and an isometric embedding of HE into L2 (S  , P), and yields a concrete representation of the boundary as a set of linear functionals on S. Mathematics Subject Classification (2000). Primary: 05C50, 05C75, 31C20, 46E22, 47B25, 47B32, 60J10, Secondary: 31C35, 47B39, 82C41. Keywords. Dirichlet form, graph energy, discrete potential theory, graph Laplacian, weighted graph, tree, electrical resistance network, effective resistance, resistance form, Markov process, random walk, transience, Martin boundary, boundary theory, boundary representation, harmonic analysis, Hilbert space, orthogonality, unbounded linear operators, reproducing kernel.

The work of P.E.T. Jorgensen was partially supported by NSF grant DMS-0457581. The work of E.P.J. Pearse was partially supported by the University of Iowa Department of Mathematics NSF VIGRE grant DMS-0602242.

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1. Introduction There are several notions of “boundary” as “points at infinity” associated to infinite graphs. Some of these come directly from graph theory, like the notion of graph ends [9, 55] or ideal boundary [47, 74]. Others come from by way of the associated reversible Markov process, the random walk associated to the graph, like topological notion of Martin boundary [56, 58] or its measure-theoretic refinement, the Poisson boundary [29, 31]. There are also less well-known ideas, like the discrete Royden boundary [33] and discrete Kuramochi boundary [48]. Interrelations amongst these concepts are detailed in several excellent collections of notes, including [70, 72] and [63]. This material has its roots in minimal surface theory, probability theory, ergodic theory, and group theory, and the central ideas are often analogues of a corresponding notion for continuous domains (manifolds, Lie groups, etc.). This paper gives a brief account of a new type of boundary developed in [20, 21, 23] and [18] which we call the resistance boundary; it is denoted bd G. It bears many similarities to the Martin and Poisson boundaries, but pertains to a different class of functions: the functions of finite energy. Let G be a resistance network (i.e., a connected simple weighted graph) with vertex set G0 and edges determined by a symmetric conductance function c which weights the edges: cxy = cyx ≥ 0, and cxy > 0 iff there is an edge from x to y, which is denoted x ∼ y. The energy of a function u : G → C is then defined to be 1  E(u) := cxy |u(x) − u(y)|2 . (1.1) 2 0 x,y∈G

For the most part, it suffices to work with R-valued functions (see Remark 2.14 in particular). However, we will need C-valued functions for some applications of spectral theory in §5. Under suitable hypotheses, if h is a bounded harmonic function on X, then Poisson boundary theory provides a measure space (∂X, µ) with respect to which one has an integral representation of h in terms of a kernel k : X × ∂X → C:  ˜ h(x) = k(x, ξ)h(ξ) dµ(ξ), for x ∈ X (1.2) ∂X

˜ is the extension of h to ∂X, in some sense. This paper provides a synopsis where h of how one can obtain a similar representation for the harmonic functions of finite energy. However, instead of using ergodic theory or (topological) compactifications, we take an entirely different approach: operator theory and functional analysis. After embedding the resistance network into a certain Hilbert space, we construct a space of distributions (i.e., generalized functions) on that Hilbert space. We then show that this space of distributions contains the boundary of the original network, in the sense that it supports integral representations of harmonic functions on the network. We work with the energy space, a Hilbert space whose

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113

inner product is given by the sesquilinear form associated to E by polarizing (1.1):    1  u, vE = E(u, v) := cxy u(x) − u(y) v(x) − v(y) . (1.3) 2 0 x,y∈G

We construct a reproducing kernel for this Hilbert space, and then use it to obtain a Gel’fand triple SE ⊆ HE ⊆ SE
.

(1.4)

Here, SE is a E-dense subspace of HE which is also equipped with a strictly finer “test function topology” (defined in terms of the domain of the Laplacian), and the space SE
is the dual space of SE with respect to this finer topology; the specifics are discussed further just below. For now, however, let us eschew technical details and just say that SE
is strictly larger than HE , and it is in SE
that the boundary bd G lies. This framework allows us to invoke Minlos’ theorem and Wiener’s isometric embedding theorem, powerful tools from the theory of stochastic integration. Boundary theory usually involves an enlargement of the original space, either by topological means (e.g., by compactification or completion, in the case of Martin boundary) or by measure-theoretical means (e.g., by taking the measurable hull of an equivalence relation, as in Poisson boundary). For the resistance boundary bd G, we enlarge HE (the Hilbert space representation of the resistance network) by embedding it into SE
via the inclusion map. Definition 1.1. The Laplacian on a resistance network (G, c) is the linear difference operator ∆ which acts on a function v : G0 → C by  (∆v)(x) := cxy (v(x) − v(y)). (1.5) y∼x

A function v : G0 → C is harmonic iff ∆v(x) = 0 for each x ∈ G0 . The domain of ∆ is specified in Definition 2.17. Note that we adopt the (physicists’) sign convention in (1.5) (so that the spectrum is nonnegative) and thus our Laplacian is the negative of the one commonly found in the PDE literature; e.g., [35, 65]. The study of resistance boundaries begins with the following well-known identity for finite networks. Proposition 1.2. Let G be a finite network. For functions u, v on G0 ,  E(u, v) = u(x)∆v(x).

(1.6)

x∈G0

The right-hand side of (1.6) is often denoted by u, ∆v2 . Theorem 3.3 gives a broad extension of Proposition 1.2 to a certain domain M (see Definition 2.17). Extensions of this type have been studied before (see [34, 43]), but only with

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regard to determining conditions that ensure E(u, v) = u, ∆v2 . By contrast, we are more interested in the situation for which it is replaced by   u, vE = u∆v + u ∂∂v . (1.7) G0

bd G

Theorem 3.3 gives conditions under which (1.7) holds; the notation bd G and ∂∂v are explained precisely in Definition 3.1 and Definition 3.2. In particular, (1.7) holds for any u ∈ HE when v lies in a certain dense subspace of HE which we denote by M. The space M was introduced in [20] for this purpose and also to serve as a dense domain for the possibly unbounded Laplace operator, which will be useful later for the construction of SE . We call (1.7) the discrete Gauss-Green identity by analogy with    ∇u∇v dV = − u∆v dV + u ∂∂v dS. Ω

Ω

∂Ω

The space HE consists of potentials (functions on the vertices of G, modulo constants; see Definition 2.5) and enjoys an orthogonal decomposition into the subspace Fin of finitely supported functions and the subspace Harm of harmonic functions; this is given precisely in Definitions 2.9–2.11 and Theorem 2.12. It turns out that HE has a reproducing kernel {vx }x∈G0 : for any u ∈ HE , one has vx , uE = u(x) − u(o),

∀x ∈ G0 ,

where o ∈ G0 is a fixed reference point. Since the reproducing kernel behaves well with respect to (orthogonal) projections P , we also have reproducing kernels {fx }x∈G0 for Fin and {hx }x∈G0 for Harm, where fx := PFin vx ,

and hx := PHarm vx .

In Theorem 5.1, we apply (1.7) to the reproducing kernels {hx }x∈G0 for Harm, and find that for all h ∈ Harm,  x h(x) − h(o) = h ∂h (1.8) ∂ . bd G

This direct analogue of (1.2) first appeared in [20, Cor. 3.14]. Formula (1.8) gives a boundary sum representation of harmonic functions, but the boundary sum in (1.8) is understood only as a limit of sums taken over boundaries of finite subnetworks. Comparison of (1.8) and (1.2) makes one optimistic that bd G can be realized as a x measure space which supports a measure corresponding to ∂h , thus replacing the ∂ sum in (1.8) with a integral. In Corollary 5.19, we extend (1.8) to such an integral representation for which (1.8) is analogous to a Riemann sum. The primary difference between our boundary theory and that of Poisson and Martin is rooted in our focus on HE : both of these classical theories concern harmonic functions with growth/decay restrictions. By contrast, provided they neither grow too wildly nor oscillate too wildly, elements of HE may be unbounded and may fail to remain nonnegative. From [1], it is known that functions which are E-limits of finitely supported functions must vanish at ∞ (except for a set

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of measure 0 with respect to the usual path-space measure); however see [18, Ex. 13.10] for an unbounded harmonic function of finite energy. Note, however, that functions of finite energy can always be approximated in HE by bounded functions; cf. [63, §3.7]. Just as for Martin and Poisson boundaries, the resistance boundary essentially consists of different limiting behaviors of the (transient) random walk on the network, as the walker tends to infinity. It turns out that recurrent networks have no resistance boundary, and transient networks with no nontrivial harmonic functions have exactly one boundary point (corresponding to the fact that the monopole at x is unique; see Definition 2.15). In particular, the integer lattices (Zd , 1) each have 1 boundary point for d ≥ 3 and 0 boundary points for d = 1, 2. Further examples are discussed in §6. Outline. §2 recalls basic definitions and some previously obtained results. In particular, we give precise definitions for the Laplace operator ∆, the energy space HE , the reproducing kernel {vx }, monopoles wx , the monopolar domain M, and we discuss the Royden decomposition of HE into the finitely supported functions and the harmonic functions. §3 states the discrete Gauss-Green identity and gives the definition of the boundary sum bd G u ∂∂v , as a limit of sums. Some implications of the discrete Gauss-Green identity are given, including several characterizations of transience of the random walk on the network. §4 gives the definition of effective resistance, and discusses how this metric can be extended to infinite networks in different ways the free resistance RF (x, y) and wired resistance RW (x, y). §5 discusses the boundary sum representation for elements of Harm as introduced in (1.8). This section also gives an overview of the theory of Gel’fand triples, Minlos’ theorem, and Wiener’s theorem, and how these enable one to obtain a Gaussian probability measure on the space SE
alluded to in (1.4). §5 gives the boundary integral representation of elements of Harm: an integral version of (1.8) which is an HE -analogue of (1.2). §6 contains several examples which illustrate our results. Boundary theory is a well-established subject; the deep connections between harmonic analysis, probability, and potential theory have led to several notions of boundary and we will not attempt to give complete references. However, we recommend [58] for introductory material on Martin boundary and [70, 72] for a more detailed discussion. Introductory material on resistance networks may be found in [12] and [41], and [36] gives a detailed investigation of resistance forms (a potentialtheoretic generalization of resistance networks). More specific background appears in [4, 42] and the foundational paper [50]. With regard to infinite graphs and finite-energy functions, see [5, 10, 19, 24–27, 53, 55, 57, 63, 64, 66, 69, 70, 72]. Applications to analysis on fractals can be found in [35, 65]. For papers studying fractals as boundaries of networks or Markov processes, see [6–8, 28, 30, 37, 39, 53].

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2. The energy space HE We now proceed to introduce the key notions used throughout this paper: resistance networks, the energy form E, the Laplace operator ∆, the energy space HE , the reproducing kernel {vx }, and their elementary properties. Definition 2.1. A resistance network is a connected graph (G, c), where G is a graph with vertex set G0 , and c is the conductance function which defines adjacency by x ∼ y  iff cxy > 0, for x, y ∈ G0 . We assume cxy = cyx ∈ [0, ∞), and write c(x) := y∼x cxy . We require c(x) < ∞ but c(x) need not be a bounded function on G0 , and note that vertices of infinite degree are allowed. The notation c may be used to indicate the multiplication operator (cv)(x) := c(x)v(x), i.e., the diagonal matrix with entries c(x) with respect to the (vector space) basis {δx }. As the letters x, y, z always denote vertices, it causes no confusion to write x, y, z ∈ G instead of x, y, z ∈ G0 . Similarly, u and v will always denote functions which map vertices to scalars, e.g., u : G0 → C. In Definition 2.1, “connected” means simply that for any x, y ∈ G, there is a finite sequence {xi }ni=0 with x = x0 , y = xn , and cxi−1 xi > 0, i = 1, . . . , n. Conductance is the reciprocal of resistance, so one can think of (G, c) as a network of nodes G0 connected by resistors of resistance c−1 xy . We may assume there is at (1)

(2)

most one edge from x to y, as two conductors cxy and cxy connected in parallel (1) (2) can be replaced by a single conductor with conductance cxy = cxy + cxy . Also, we assume cxx = 0 so that no vertex has a loop, as electric current will never flow along a conductor connecting a node to itself. Definition 2.2. An exhaustion of G is an increasing sequence of finite and connected  subgraphs {Gk }∞ Gk . Since any vertex or edge k=1 , so that Gk ⊆ Gk+1 and G = is eventually contained in some Gk , there is no loss of generality in assuming they are contained in G1 , for the purposes of a specific computation. Definition 2.3. The notation

 x∈G

:= lim

k→∞



(2.1)

x∈Gk

is used whenever the limit is independent of the choice of exhaustion {Gk } of G. This is clearly justified, for example, whenever the sum has only finitely many nonzero terms, or is absolutely convergent as in the definition of E just below. Definition 2.4. The energy of functions u, v : G0 → C is given by the (closed, bilinear) Dirichlet form 1 E(u, v) := cxy (u(x) − u(y))(v(x) − v(y)), (2.2) 2 x∈G y∈G

with the energy of u given by E(u) := E(u, u). The domain of the energy is dom E = {u : G0 → C ... E(u) < ∞}.

(2.3)

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Since cxy = cyx and cxy = 0 for nonadjacent vertices, the initial factor of 12 in (2.2) implies there is exactly one term in the sum for each edge in the network. Definition 2.5. Let 1 denote the constant function with value 1 and recall that ker E = C1. The energy form E is symmetric and positive definite on dom E. Then dom E/C1 is a vector space with inner product and corresponding norm given by u, vE := E(u, v) and uE := E(u, u)1/2 .

(2.4)

The energy Hilbert space is defined to be dom E dom E HE := = . (2.5) ker E C1 Thus, HE consists of potentials: we are not interested in values u(x) as much as differences u(x) − u(y). In other words, if u and v are both elements of dom E and there is some constant k ∈ C such that u(x) − v(x) = k for all x ∈ G, then u and v are both representatives of the same element (equivalence class) of HE . Definition 2.6. Let vx be defined to be the unique element of HE for which vx , uE = u(x) − u(o),

for every u ∈ HE .

(2.6)

The collection {vx }x∈G forms a reproducing kernel for HE ; cf. [20, Cor. 2.7]. We call it the energy kernel and (2.6) shows its span is dense in HE . Note that vo corresponds to a constant function, since vo , uE = 0 for every u ∈ HE . Therefore, vo (or o) may often ignored or omitted. Definition 2.7. A dipole is any v ∈ HE satisfying the pointwise identity ∆v = δx −δy for some vertices x, y ∈ G. The elements of the energy kernel are all dipoles: one can check that ∆vx = δx − δo as in [20, Lemma 2.13]. Remark 2.8. To minimize cumbersome notation, let {x ∈ G} be the default index set from now on. That is, we use {vx } to denote the energy kernel {vx }x∈G , and span{vx } to denote the set of all linear combinations of elements of {vx }, etc. 2.1. The finitely-supported functions and the harmonic functions Definition 2.9. For v ∈ HE , one says that v has finite support iff there is a finite set F ⊆ G0 for which v(x) = k ∈ C for all x ∈ / F , i.e., the set of functions of finite support in HE is span{δx }, where δx is the Dirac mass at x, i.e., the element of HE containing the characteristic function of the singleton {x}. Define Fin to be the closure of span{δx } with respect to E. Remark 2.10. The usual candidate for an orthonormal basis (onb) in 2 (G0 ) would be the collection of Dirac masses {δx }; however, this is not an onb in HE . One can compute from (2.2) that δx , δy E = E(δx , δy ) = −cxy ,

for x = y,

and

E(δx ) = c(x),

(2.7)

so that δx ⊥ δy with respect to E. Moreover, Theorem 2.12 shows that {δx } is, in general, not even dense in HE . It is immediate from (2.7) and Definition 2.1 that δx ∈ HE .

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Definition 2.11. The harmonic subspace of HE is denoted Harm := {v ∈ HE ... ∆v(x) = 0, for all x ∈ G}.

(2.8)

Note that this is independent of choice of representative for v in virtue of (1.5). The following result is sometimes called the “Royden Decomposition” since [73, Thm. 4.1], in reference to Royden’s analogous result for Riemann surfaces; see [63, §VI], [41, §9.3]1 . It follows immediately from [20, Lemma 2.11], which states that δx , uE = ∆u(x) for any x ∈ G; cf. [20, Thm. 2.15]. Theorem 2.12 (Royden decomposition). HE = Fin ⊕ Harm. Definition 2.13. Let fx = PFin vx denote the image of vx under the (orthogonal) projection to Fin. Similarly, let hx = PHarm vx denote the image of vx under the projection to Harm. Remark 2.14 (Reproducing kernels for Fin and Harm). The reproducing kernel property behaves well with respect to orthogonal projections, and consequently, {fx } is a reproducing kernel for Fin, and {hx } is a reproducing kernel for Harm. While we will need complex-valued functions for some results obtained via spectral theory, it will usually suffice to consider R-valued functions because the reproducing kernels elements vx , fx , hx all have R-valued representatives [20, Lemma 2.24]. In fact, we must restrict attention to R-valued functions at one point; see Remark 5.10. 2.2. Monopoles Definition 2.15. A monopole at x ∈ G is an element wx ∈ HE which satisfies ∆wx (y) = δxy , where δxy is Kronecker’s delta. In case the network supports monopoles (that is, if the above Dirichlet equation admits finite-energy solutions), let wo always denote the unique energy-minimizing monopole at the origin. With vx and fx = PFin vx as above, we indicate the distinguished monopoles wxv := vx + wo

and

wfx := fx + wo .

(2.9)

Remark 2.16. Note that wo ∈ Fin, whenever it is present in HE , and similarly that wfx is the energy-minimizing monopole at x. To see this, suppose wx is any monopole at x. Since wx ∈ HE , write wx = f + h by Theorem 2.12, and get E(wx ) = E(f ) + E(h). Projecting away the harmonic component will not affect the monopole property, so wfx = PFin wx is the unique monopole of minimal energy. The Green function is g(x, y) = wyo (x), where wyo is the representative of wfy which vanishes at ∞. Definition 2.17. The dense subspace of HE spanned by monopoles (and dipoles) is M := span{vx } + span{wxv , wfx }.

(2.10)

Let ∆M be the closure of the Laplacian when taken to have the dense domain M. In general, we assume ∆ is unbounded as an operator on HE . 1 This

name is also sometimes associated with the corresponding (nonorthogonal) decomposition for the “grounded energy form”; see Remark 3.11.

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Since ∆ agrees with ∆M pointwise, we may suppress reference to the domain for ease of notation. It is shown in [20, Lemma 3.5] that ∆M is Hermitian with u, ∆M uE ≥ 0 for all u ∈ M. When given a pointwise identity ∆u = v, there is an associated identity in HE , but one must use the adjoint: ∆u(x) = v(x) for all x ∈ G if and only if v = ∆∗M u in HE [20, Lemma 3.7]. Note that ∆M may have defect vectors; such an object is an element of the Hilbert space HE (though clearly not an element of M) which has a representative u satisfying ∆M u(x) = −u(x), ∀x ∈ G,

and

u ∈ dom ∆∗M .

See [22, §4.2] or [18, §13.4]. While it is always the case that the (possibly unbounded) operator ∆M is Hermitian (i.e., ∆M ⊆ ∆∗M ), this shows that ∆M may fail to be self-adjoint (i.e., ∆M = ∆∗M ). Remark 2.18 (Monopoles and transience). The presence of monopoles in HE is equivalent to the transience of the simple random walk on the network with transition probabilities p(x, y) = cxy /c(x): note that if wx is a monopole, then the current induced by wx is a unit flow to infinity with finite energy. It was proved in [42] that the network is transient if and only if there exists a unit current flow to infinity; see also [41, Thm. 2.10]. Moreover, it is shown in [20, Lemma 3.6] that when the network is transient, M contains the spaces span{vx }, span{fx }, and span{hx }, where fx = PFin vx and hx = PHarm vx . When Harm = 0 (in particular, when the network is not transient), fx = vx and so M = span{vx } = span{fx } trivially. See also [19].

3. The discrete Gauss-Green formula In Theorem 3.3, we establish a discrete version of the Gauss-Green formula which extends Proposition 1.2 to the case of infinite graphs; the scope of validity of this formula is given in terms of the space M of Definition 2.15. The appearance of a somewhat mysterious boundary term alluded to in (1.7) prompts several questions which are discussed in Remark 3.6. 3.1. Relating ∆ to E Definition 3.1. If H is a subgraph of G, then the boundary of H is bd H := {x ∈ H ... ∃y ∈ H  , y ∼ x}.

(3.1)

The interior of a subgraph H consists of the vertices in H whose neighbours also lie in H: int H := {x ∈ H ... y ∼ x =⇒ y ∈ H} = H \ bd H. (3.2) For vertices in the boundary of a subgraph, the normal derivative of v is  ∂v (x) := cxy (v(x) − v(y)), for x ∈ bd H. (3.3) ∂ y∈H

Thus, the normal derivative of v is computed like ∆v(x), except that the sum extends only over the neighbours of x which lie in H.

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Definition 3.1 will be used primarily for subgraphs that form an exhaustion of G, in the sense of Definition 2.2. Definition 3.2. A boundary sum is computed in terms of an exhaustion {Gk } by   := lim , (3.4) k→∞

bd G

bd Gk

whenever the limit is independent of the choice of exhaustion, as in Definition 2.3. Theorem 3.3 (Discrete Gauss-Green Formula). If u ∈ HE and v ∈ M, then   u, vE = u∆v + u ∂∂v . (3.5) G



bd G

 Corollary 3.4. For all u ∈ dom ∆M , G ∆u = − bd G ∂∂u . Thus, the discrete Gauss-Green formula (3.5) is independent of choice of representatives. Remark 3.5. The proof of Theorem 3.3 follows from taking limits of   u(x)∆v(x) + u(x) ∂∂v (x). x∈Gk

x∈bd Gk

Thus, the decomposition (3.5) is true for all u, v ∈ HE , but is meaningless if it takes the form ∞ − ∞. A key point of Theorem 3.3 is that for u, v in the specified domains, the two sums are both finite and independent of choice of exhaustion. However, the specific value of each sum is dependent on the choice of representative for u; this motivates Definition 3.9. It is also clear that (3.5) remains true much more generally than under the  specified conditions; certainly the formula holds whenever x∈G |u(x)∆v(x)| < ∞. The requirement v ∈ M is the most explicit condition we could find that ensures convergence of this sum on any network. A formula similar to (3.5) appears in [11, Prop 1.3]; however, these authors apparently do not pursue the extension of this formula to infinite networks. Another similar result appears in [34, Thm. 4.1], where the authors give some conditions under which (1.6) extends to infinite networks. The main differences here are that the scope of Kayano and Yamasaki’s theorem is limited to a subset of what we call Fin, and that Kayano and Yamasaki are interested in when the boundary term vanishes; we are more interested in when it is finite and nonvanishing; see Theorem 3.10, for example. Since Kayano and Yamasaki do not discuss the structure of the space of functions they consider, it is not  clear how large the scope of their result is; their result requires the hypothesis x∈G |u(x)∆v(x)| < ∞, but it is not so clear what functions satisfy this. By contrast, we develop a dense subspace of functions on which to apply the formula. Furthermore, in the forthcoming paper [23], we show that these functions are relatively easy to compute.  ∂v Remark 3.6. We refer to as the “boundary term” by analogy with bd G u ∂ classical PDE theory. This terminology should not be confused with the notion of boundary that arises in the discussion of the discrete Dirichlet problem, where the

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boundary is a prescribed subset of G0 . As the boundary term may be difficult to contend with, it is extremely useful to know when it vanishes, for example: (i) when the network is recurrent (Theorem 3.10), (ii) when v is an element of the energy kernel [20, Lemma 5.8], (iii) when u, v, ∆u, ∆v lie in 2 [20, Lemma 5.12], and (iv) when either u or v has finite support [20, Lemma 5.16]. 3.2. More about monopoles and the space M This section studies the role of the monopoles with regard to the boundary term of Theorem 3.3, and provides several characterizations of transience of the network, in terms of the operator-theoretic properties of ∆M . that if h ∈ Harm satisfies the  hypotheses of Theorem 3.3, then E(h) =  Note ∂h h . On the other hand, E(u) = bd G ∂ G u∆u for all u ∈ HE iff the network is recurrent, as stated in Theorem 3.10. With respect to HE = Fin ⊕ Harm, this shows that the energy of finitely supported functions comes from the sum over G, and the energy of harmonic functions comes from the boundary sum. However,for a monopole wx , the representative specified by wx (x) = 0 satisfies but the representative specified by wx (x) = E(wx ) satisfies E(w) =  bd G w ∂w ∂ E(w) = G w∆w. Roughly speaking, a monopole is therefore “half of a harmonic function” or half-way to being a harmonic function. A further justification for this comment is given by Corollary 3.8 (the proof shows that a harmonic function can be constructed from two monopoles at the same vertex, see [20, Cor. 4.4]). The general theme of this section is the ability of monopoles to “bridge” the finite and the harmonic. Theorem 3.7 ([63, Thm. 1.33]). Let u be a nonnegative function on a recurrent network. Then u is superharmonic if and only if u is constant. It follows from Theorem 3.7 that Harm = 0 implies the existence of a monopole in HE , i.e., the transience of the network; cf. [20, Cor. 4.3]. However, it turns out that a nontrivial harmonic function can only exist when there is more than one monopole. Corollary 3.8. Harm = 0 iff there are at least two linearly independent monopoles at one (equivalently, every) vertex x. Definition 3.9. The phrase “the boundary term is nonvanishing” indicates that (3.5) holds with nonzero boundary sum when applied to u, vE , for every representative of u except one; namely, the one specified by u(x) = u, wxv E . Recall from Remark 2.18 that the network is transient iff there are monopoles in HE . From the Discrete Gauss-Green theorem, we obtain three more criteria for transience of the random walk. Theorem 3.10. The random walk on the network (G, c) with transition probabilities cxy p(x, y) = c(x) is transient if and only if any of the following equivalent conditions are satisfied:

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(i) the boundary term is nonvanishing, (ii) fk := (εk + ∆)−1 δx is weak-∗ convergent for some sequence εk → 0, or (iii) (ran ∆∗M )c
= Fin. Note that on any network, (ran ∆M ) cf. [20, Lemma 4.8].

clo

⊆ Fin and hence Harm ⊆ ker ∆∗M ;

Remark 3.11. An alternative approach to studying the space of finite-energy functions comes by considering the grounded inner product u, vo := u(o)v(o) + u, vE , which makes dom E into a Hilbert space D which we call the grounded energy space. This approach is discussed in [41], [63] and [32, 34, 49, 73]. Let D0 be the closure of span{δx } in D. If PD0 is the projection to D0 , and it is applied to the constant function 1 then PD0 1 = 1 if and only if the network is recurrent. In fact, when the network is transient, then (modulo additive constants) ⊥ both PD0 1 and PD 1 are scalar multiples of monopoles at o. The space D⊥ 0 is 0 spanned by monopoles and harmonic functions. See [20, §4.1] for more details.

4. Effective resistance There is a natural notion of distance on finite networks, which is defined in terms of resistance. Consider each edge of the network to be an electrical resistor of resistance c−1 xy . The effective resistance metric R(x, y) is the voltage drop between the vertices x and y if a current of one amp is inserted into the network at x and withdrawn at y. It is a bit surprising that this actually gives a metric, and there are several other equivalent formulations, most of which are well known. The essential reference for effective resistance is [36], but the reader may also find the excellent treatments in [63] and [41] to be helpful. Theorem 4.1. The resistance R(x, y) has the following equivalent formulations: R(x, y) = {v(x) − v(y) ... ∆v = δx − δy }

(4.1)

= {E(v) ∆v = δx − δy }

(4.2)

. ..

= 1/ min{E(v) v(x) = 1, v(y) = 0, v ∈ dom E}

(4.3)

= min{κ ≥ 0 |v(x) − v(y)| ≤ κE(v), v ∈ dom E}

(4.4)

= sup{|v(x) −

(4.5)

. ..

. ..

2

v(y)|2 ...

E(v) ≤ 1, v ∈ dom E}.

Remark 4.2 (Resistance distance via network reduction). Let G be a finite planar network and pick any x, y ∈ G0 . Then G may be reduced to a trivial network consisting only of these two vertices and a single edge between them via the use of three basic transformations: (i) series reduction, (ii) parallel reduction, and (iii) the ∇-Y transform [13, 67]. The effective resistance between x and y may be interpreted as the resistance of the resulting single edge; see Figure 1. See also [35] or [65] for the ∇-Y transform.

Resistance Boundaries of Infinite Networks x

x

x

Ω1

Ω1

Ω2

1

R(x,z) = Ω1+ Ω -1+Ω -1

y

y

2

3

1

Ω3

z

123

Ω2-1+Ω3-1 z

z

Figure 1. Effective resistance as network reduction to a trivial network. This basic example uses parallel reduction followed by series reduction; see Remark 4.2. 4.1. Resistance metric on infinite networks There are challenges in extending the notion of effective resistance to infinite networks. The existence of nonconstant harmonic functions h ∈ dom E implies the nonuniqueness of solutions to ∆u = f in HE , and hence (4.1) and (4.2) are no longer well defined. This issue is studied in detail in [23], and in [36] (by very different methods). There are also accounts in [41] and the literature on “uniqueness of currents” in infinite networks, e.g., [64, 66]. Two natural choices for extension lead to the free resistance RF and the wired resistance RW . In general, one has RF (x, y) ≥ RW (x, y) with equality iff Harm = 0. Both of these correspond to the selection of certain solutions to ∆u = δx − δy (in fact, these can be interpreted as Neumann and Dirichlet boundary conditions, respectively; see [23, Rem. 2.23]). Also, both are given in terms of limits computed with respect to certain networks associated to an exhaustion, in the sense of Definition 2.2. The notation {Gk }∞ k=1 always denotes an exhaustion of the infinite network (G, c), as in Definition 2.2. Since x and y are contained in all but finitely many Gk , we may always assume that x, y ∈ Gk , ∀k. Definition 4.3. If H is a finite subnetwork of G which contains x and y, define RH (x, y) to be the resistance distance from x to y as computed within H. In other words, compute RH (x, y) by any of the equivalent formulas of Theorem 4.1, but extremizing over only those functions whose support is contained in H. Definition 4.4. Let H 0 ⊆ G0 . Then the full subnetwork on H 0 has all the edges of G for which both endpoints lie in H 0 , with the same conductances. That is, cH = cG |H 0 ×H 0 . 4.1.1. Free resistance. Definition 4.5. For any subset H 0 ⊆ G0 , the free subnetwork H F is just the full subnetwork with vertices H 0 . That is, all edges of G with endpoints in H 0 are edges of H F , with the same conductances. Thus, we will denote H F by H to reduce notation. Let RH (x, y) denote the effective resistance between x and y as computed

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in H, as in Definition 4.3. The free resistance between x and y is defined to be RF (x, y) := lim RGk (x, y),

(4.6)

k→∞

where {Gk } is any exhaustion of G. The name “free” comes from the fact that this formulation is free of any boundary conditions or considerations of the complements of the Gk ; see [41, §9]. Theorem 4.6 is the free extension of Theorem 4.1 to infinite networks. Theorem 4.6 ([23, Thm. 2.14]). For an infinite network G, the free resistance RF (x, y) has the following equivalent formulations: RF (x, y) = v(x) − v(y), = E(v),

v = vx − vy

v = vx − vy

= min{D(I) I ∈ F(x, y) and I = . ..

(4.7) 

(4.8) ξγ χγ} −1

(4.9)

= (min{E(u) ... u ∈ HE , |u(x) − u(y)| = 1})

(4.10)

= inf{κ ≥ 0 |v(x) − v(y)| ≤ κE(v), ∀v ∈ HE }

(4.11)

= sup{|v(x) −

(4.12)

. ..

2

v(y)|2 ...

v ∈ HE , vE ≤ 1}

Fix x, y ∈ G and define the operator Lxy on HE by Lxy v := v(x) − v(y). Then (4.11)–(4.12) are equivalent to RF (x, y) = Lxy . 4.1.2. Wired resistance. Definition 4.7. Given a finite full subnetwork H of G, define the wired subnetwork H W by identifying all vertices in G0 \ H 0 to a single, new vertex labeled ∞. Thus, the vertex set of H W is H 0 ∪ {∞H }, and the edge set of H W includes all the edges of H, with the same conductances. However, if x ∈ H 0 has a neighbour y ∈ G0 \ H 0 , then H W also includes an edge from x to ∞ with conductance  cx∞ := cxy . (4.13) H

y∼x, y∈H 

The identification of vertices in Gk may result in parallel edges; then (4.13) corresponds to replacing these parallel edges by a single edge according to the usual formula for resistors in parallel. Let RH W (x, y) denote the effective resistance between x and y as computed in H W , as in Definition 4.3. The wired resistance is then defined to be RW (x, y) := lim RGW (x, y), k k→∞

where {Gk } is any exhaustion of G.

(4.14)

Resistance Boundaries of Infinite Networks

p

p

p

p

H3

H4

H5

H6

p H3X

23

p

24

H4X

p

25

H5X

125

p

26

H6X

Figure 2. Comparison of free and wired exhaustions for the example of the binary tree; see Definition 4.5 and Definition 4.7. Here, the vertices of Gk are all those which lie within k edges (“steps”) of the origin. If the edges of G all have conductance 1, then so do all the edges of each GF k and GW k , except for the edges incident upon ∞k = ∞Gk , which have conductance 2. The wired subnetwork is equivalently obtained by “shorting together” all vertices of H  , and hence it follows from Rayleigh’s monotonicity principle that RW (x, y) ≤ RF (x, y); cf. [12, §1.4] or [41, §2.4]. Theorem 4.8 ([23, Thm. 2.20]). The wired resistance may be computed by any of the following equivalent formulations: RW (x, y) = f (x) − f (y), = E(f ),

f = fx − fy

(4.15)

f = fx − f y

(4.16) −1

= (min{E(v) |v(x) − v(y)| = 1, v ∈ Fin})

(4.17)

= inf{κ ≥ 0 |v(x) − v(y)| ≤ κE(v), ∀v ∈ Fin}

(4.18)

= sup{|v(x) −

(4.19)

. ..

. ..

2

v(y)|2 ...

v ∈ Fin, vE ≤ 1}

Note that (4.15) and (4.16) are equivalent to RW (x, y) = min{v(x) − v(y) ... ∆v = δx − δy , v ∈ dom E} = min{E(v) ∆v = δx − δy , v ∈ dom E}. ...

(4.15
) (4.16
)

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4.2. von Neumann construction of the energy space HE Let R = RF or R = RW . The discussion of the effective resistance is important in this paper in two respects. (i) Theorem 4.11 shows that HE is the natural Hilbert space for studying the metric space (G, R). (ii) The function R(x, y) allows us to construct a probability measure in Theorem 5.15. Both of these results stem from the fact that (free or wired) effective resistance is a negative semidefinite function on G0 × G0 , as is shown in [23, Thm. 5.4]. Definition 4.9. A function M : X × X → R is negative semidefinite iff for any  f : X → R satisfying x∈X f (x) = 0, one has  f (x)M (x, y)f (y) ≤ 0, (4.20) x,y∈F

where F is any finite subset of X. One can think of M as a matrix and (4.20) as matrix multiplication. von Neumann and Schoenberg [2, 3, 59, 60, 68] showed that (4.20) is precisely the condition that allows one to embed a metric space into a Hilbert space. This theorem also has a form of uniqueness which may be thought of as a universal property. Theorem 4.10 (von Neumann). Suppose (X, d) is a metric space. There exists a Hilbert space H and an embedding w : (X, d) → H sending x → wx and satisfying d(x, y) = wx − wy H

(4.21)

2

if and only if d is negative semidefinite. Furthermore, if there is another Hilbert space K and an embedding k : H → K, with kx − ky K = d(x, y) and {kx }x∈X dense in K, then there exists a unique unitary isomorphism U : H → K. Theorem 4.11 ([23, Thm. 5.4]). (G, RF ) may be isometrically embedded in a Hilbert space, and this Hilbert space is unitarily equivalent to HE . Under this embedding, x is mapped to the energy kernel element vx . Moreover, (G, RW ) may be isometrically embedded in a Hilbert space, and this Hilbert space is unitarily equivalent to Fin. Under this embedding, x is mapped to fx = PFin vx . By this theorem, we see that HE is the natural choice of Hilbert space for studying the metric spaces (G, RF ) and (G, RW ).

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5. A boundary integral representation for the harmonic functions We are motivated by the following result, which follows readily from Theorem 3.3 and may be found in [20, Cor. 3.14]. Theorem 5.1 (Boundary representation of harmonic functions). For u ∈ span{hx },  x u(x) = u ∂h + u(o). (5.1) ∂ bd G

Proof. Note that u(x) − u(o) = vx , uE = u, vx E =

 bd G




x u ∂h by (2.6). ∂

Formula (5.1) begs comparison with the Poisson integral formula. Recall the classical result of Poisson that gives a kernel k : Ω×∂Ω → R from which a bounded harmonic function can be given via  u(x) = u(y)k(x, dy), y ∈ ∂Ω. (5.2) ∂Ω

One would like to obtain a (probability) measure space to serve as the boundary of G. We use some techniques from the theory of stochastic integration for which it was shown in [51] that a Hilbert space does not suffice; see [17, §3.1]. The workaround is to build a Gel’fand triple S ⊆ H ⊆ S
(a more precise definition appears just below), and construct a suitable probability measure on S
. In §5.1, we briefly describe the general theory of Gel’fand triples as they apply in the current context. In §5.2, we use an unbounded Laplacian ∆ to construct a Gel’fand triple for HE . Then in §5.3, we apply the general theory to the Gel’fand triple SE ⊆ HE ⊆ SE
and obtain a Gaussian probability measure P on SE
, and an isometric embedding HE → L2 (SE
, P). This allows us to study the boundary bd G as a subset of SE
. For the general theory of analysis in Hilbert space, see [15, 16]. 5.1. Gel’fand triples and duality In a little more detail, a Gel’fand triple (also called a rigged Hilbert space) is S ⊆ H ⊆ S
,

(5.3)




where S is dense in H and S is the dual of S. While S is a dense subspace of H with respect to the Hilbert norm, it also comes equipped with a strictly finer “test function” topology, and it is required that the inclusion mapping of S into H is continuous with respect to these topologies. Therefore, when the dual S
is taken with respect to this finer topology, one obtains a strict containment HE  S
. It turns out that S
is large enough to support a (Gaussian!) probability measure. We will give a “test function topology” as a Fr´echet topology defined via a specific sequence of seminorms. It was Gel’fand’s idea to formalize this construction abstractly using a system of nuclearity axioms [14, 44, 45]. This presentation is adapted from quantum mechanics. Remark 5.2 (Tempered distributions and the Laplacian). There is a concrete situation when the Gel’fand triple construction is especially natural: H = L2 (R, dx) and S is the Schwartz space of functions of rapid decay. That is, each f ∈ S is a

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C ∞ smooth function which decays (along with all its derivatives) faster than any polynomial as x → ±∞. In this case, S
is the space of tempered distributions and the seminorms defining the Fr´echet topology on S are pm (f ) := sup{|xk f (n) (x)| ... x ∈ R, 0 ≤ k, n ≤ m},

m = 0, 1, 2, . . . ,

where f (n) is the nth derivative of f . Then S
is the dual of S with respect to this Fr´echet topology. One can equivalently express S as ˜ 2 )n f ∈ L2 (R), ∀n}, S := {f ∈ L2 (R) ... (P˜ 2 + Q

(5.4)

d ˜ : f (x) → xf (x) are Heisenberg’s operators. The where P˜ : f (x) → 1
dx and Q 2 2 ˜ ˜ operator P + Q is often called the quantum mechanical Hamiltonian, but some others (e.g., Hida, Gross) would call it a Laplacian, and this perspective tightens the analogy with the present context. In this sense, (5.4) could be rewritten S := dom ∆∞ ; compare to (5.9) just below. Scattering theory in the present (discrete) context is studied in [26] and some interpolation theory results are given in [24].

The duality between S and S
allows for the extension of the inner product on H to a pairing of S and S
: ·, ·H : H × H → C

to

·, ·H˜ : S × S
→ R.

(5.5)

In other words, one obtains a Fourier-type duality restricted to S. The proof of Theorem 5.15 will require Minlos’ generalization of Bochner’s theorem from [46, 62]. This important result states that a cylindrical measure on the dual of a nuclear space is a Radon measure iff its Fourier transform is continuous. In this context, however, the notion of Fourier transform is infinitedimensional; cf. [40]. Theorem 5.3 (Minlos). Given a Gel’fand triple S ⊆ H ⊆ S
, there is a bijective correspondence between the positive definite functions f on S and the Radon probability measures on S
, determined uniquely by the identity  f (s) = e
s,ξH˜ dPf (ξ), ∀s ∈ S, (5.6) S

where ·, ·H˜ is the extended pairing on S × S
as in (5.5). Formula (5.6) may be interpreted as defining the Fourier transform of P. We apply Minlos’ theorem in the standard manner for white noise constructions, and obtain the following corollary. Corollary 5.4 (White noise). Given a Gel’fand triple S ⊆ H ⊆ S
, there is a probability measure P on S
satisfying  1 e
s,ξH˜ dP(ξ). (5.7) e− 2 s,sH = S

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In the proof of Theorem 5.15, we show that P in (5.7) is actually a Gaussian measure on SE
. The function on the left-hand side of (5.7) plays a special role in stochastic integration, and its use in quantization. To see that it is a positive definite function on S, we appeal to a famous result of Schoenberg which may be found in [3, 61]. Theorem 5.5 (Schoenberg). Let X be a set and let Q : X × X → R be a function. Then the following are equivalent. 1. Q is negative semidefinite. 2. ∀t ∈ R+ , the function pt (x, y) := e−tQ(x,y) is positive definite on X × X. 3. There exists a Hilbert space H and a function f : X → H such that Q(x, y) = f (x) − f (y)2H . In the proof of Theorem 5.15, we apply Schoenberg’s Theorem with t = 12 to the g(u, v) = u − v2E , which is negative semidefinite on HE × HE for the same reason that the resistance metric RF (x, y) = vx − vy 2E is negative semidefinite on G × G. See §4.2 and [23, Thm. 2.13]. 5.2. A Gel’fand triple for HE To apply Minlos’ Theorem, we first need to construct a Gel’fand triple for HE ; we begin by identifying a certain subspace of M = dom ∆M (as given in Definition 2.17) to act as the space of test functions, which we denote SE . We present here the construction of a Gel’fand triple which is convenient for the case when ∆ is unbounded on HE . See [21] for a less explicit construction which allows one to deal with the general case. ∗ M be a self-adjoint extension of the unbounded operator ∆M ; Definition 5.6. Let ∆ since ∆M is Hermitian and commutes with conjugation (since c is R-valued), a theorem of von Neumann’s states that such an extension exists. p ∗ M u := (∆ ∗ M∆ ∗M . . . ∆ ∗ M )u be the p-fold product of ∆ ∗ M applied to u ∈ HE . Let ∆ p ∗ M ) inductively by Define dom(∆ p p−1 ∗ M ) := {u ... ∆ ∗ M u ∈ dom(∆ ∗ M )}. dom(∆

(5.8)

Definition 5.7 (Test functions). The (Schwartz) space of potentials of rapid decay is ∗∞ dom(∆ M)

(∞

∗∞ SE := dom(∆ M ), p

where := consists of all u ∈ HE for which for any p. Since ∆M is unbounded, SE  HE . ∗ p=1 dom(∆M )

(5.9) p

∗ Mu ∆

∈ HE

Definition 5.8 (Distributions). For each p ∈ N, there is a seminorm on SE defined by p ∗ M uE . up := ∆ (5.10) p ∗ Since (dom ∆M ,  · p ) is a Hilbert space for each p ∈ N, the system of seminorms P = { · p }p∈N defines a Fr´echet topology on SE . The space SE
of Schwartz distributions or tempered distributions is the (dual) space of P-continuous linear functionals on SE .

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Remark 5.9. If deg(x) is finite for each x ∈ G0 , or if c < ∞, then one has  vx ∈ SE . In the first case, this can be proved from the identity δx = c(x)vx − y∼x cxy vy ∗M which is given in [20, Lem. 2.22]. In the second case, the bound on c implies ∆ is bounded and hence everywhere-defined. When SE contains {vx }, it should be noted that span{vx } is dense in SE with respect to E, but not with respect to the Fr´echet topology induced by the seminorms (5.10), nor with respect to the graph norm. One has the inclusions  )  )  ) vx s u ⊆ ⊆ (5.11) ∗ Ms ∗ Mu ∆M vx ∆ ∆ where s ∈ SE and u ∈ HE . The second inclusion is dense but the first is not. Remark 5.10. Note that SE and SE
consist of R-valued functions. This technical # ˜ detail is important because we do not expect the integral S  e
u,·W dP from (5.6) to converge unless it is certain that u, · is R-valued. This is the reason for the last conclusion of Theorem 5.13. Definition 5.11. Let χ[a, b] denote the usual indicator function of the interval [a, b] ⊆ ∗ M , and let R, and let S be the spectral transform in the spectral representation of ∆ E be the associated projection-valued measure. Then define En to be the spectral truncation operator acting on HE by  n ∗χ En u := S [ n1 , n]Su = E(dt)u. 1/n

Lemma 5.12. With respect to E, SE is a dense analytic subspace of HE . Proof. This essentially follows immediately once it is clear that En maps HE into SE . For u ∈ HE , and for any p = 1, 2, . . . ,  n p 2 ∗ M En uE = ∆ λ2p E(dλ)u2E ≤ n2p u2E , (5.12) 1/n

So En u ∈ SE . It follows that u − En uE → 0 by standard spectral theory. Theorem 5.13. SE ⊆ HE ⊆ extends to a pairing on SE ×

SE
SE





is a Gel’fand triple, and the energy form ·, ·E defined by

p −p ∗ M u, ∆ ∗ M ξE , u, ξW := ∆

(5.13)

where p is any integer such that |ξ(u)| ≤ K∆ uE for all u ∈ SE . This pairing on SE × SE
is equivalently given by p

u, ξW = lim ξ(En u), where the limit is taken in the topology

n→∞ of SE
.

(5.14)

˜n : S
→ HE defined via u, E ˜n ξE := Corollary 5.14. En extends to a mapping E E ξ(En u). Thus, we have a pointwise extension of · , ·W to HE × SE
given by ˜n ξE . u, ξW = lim u, E (5.15) n→∞

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5.3. The Wiener embedding and the space SE With Theorem 5.13, we have a Gel’fand triple and we are now ready to apply the white noise construction of Cor. 5.4. Note that in Theorem#5.15, expectations are taken with respect to the variable ξ ∈ SE
, that is, E(f ) := S  f (ξ) dP(ξ). E

Theorem 5.15 (Wiener embedding). The Wiener transform W : HE → L2 (SE
, P) is given by W : v → v˜,

v˜(ξ) := v, ξW ,

(5.16)

and is an isometry. The extended reproducing kernel {˜ vx }x∈G0 is a system of Gaussian random variables which gives the resistance distance by RF (x, y) = E((˜ vx − v˜y )2 ). Moreover, for any u, v ∈ HE , the energy inner product extends directly as    u, vE = E u ˜v˜ = u ˜v˜ dP. SE

(5.17)

(5.18)

Proof. Since g(u, v) = u−v2E is negative semidefinite by the same computation as in [23, Thm. 5.4], we may apply Schoenberg’s theorem and deduce that exp(− 12 u− v2E ) is a positive definite function on HE × HE . Consequently, an application of the Minlos correspondence (Theorem 5.3) to the Gel’fand triple established in Lemma 5.12 yields a Gaussian probability measure P on SE
. Moreover, (5.6) gives E(e
u,ξW ) = e− 2 uE , 1

2

(5.19)

provided that ξ is R-valued (so that the integral converges). Therefore, we give the proof for the R-valued subspace of SE (and of SE
), and then complexify in the last step via the standard decomposition into real and complex parts: u = u1 + u2 where ui is a R-valued elements of HE , etc. From (5.19), one computes   1 1 1 + u, ξW − u, ξ2W + · · · dP(ξ) = 1 − u, uE + · · · . (5.20)  2 2 SE Now it follows that E(˜ u2 ) = E(u, ξ2W ) = u2E for every u ∈ SE , by comparing the terms of (5.20) which are quadratic in u. Therefore, W : HE → SE
is an isometry, and (5.20) gives E(|˜ vx − v˜y |2 ) = E(vx − vy , ξ2 ) = vx − vy 2E ,

(5.21)

whence (5.17) follows from (4.8). Note that by comparing the linear terms, (5.20) implies E(1) = 1, so that P is a probability measure, and E(u, ξ) = 0 and E(u, ξ2 ) = u2W , so that P is actually Gaussian.

132

P.E.T. Jorgensen and E.P.J. Pearse Finally, use polarization to compute  1 u, vE = u + v2E − u − v2E 4    1  2 2 = E |˜ u + v˜| − E |˜ u − v˜| 4 1 2 2 = |˜ u + v˜| (ξ) − |˜ u − v˜| (ξ) dP(ξ) 4 SE  = u ˜(ξ)˜ v (ξ) dP(ξ).

by (5.21)

SE

This establishes (5.18) and, upon complexification, completes the proof.




Remark 5.16. Observe that Theorem 5.15 was carried out for the free resistance, but all the arguments go through equally well for the wired resistance; note that RW is similarly negative semidefinite by Theorem 5.5 and [23, Cor. 5.5]. Thus, there is a corresponding Wiener transform W : Fin → L2 (SE
, P) defined by W : v → f˜,

f = PFin v and f˜(ξ) = f, ξW .

(5.22)

Again, {f˜x }x∈G0 is a system of Gaussian random variables which gives the wired resistance distance by RW (x, y) = E((f˜x − f˜y )2 ). Remark 5.17. For u ∈ Harm and ξ ∈ SE
, let us abuse notation and write u for u˜ so as to avoid unnecessary tildes. That is, u(ξ) := u ˜(ξ) = u, ξW . Remark 5.18. The polynomials are dense in L2 (SE
, P): let ϕ(t1 , t2 , . . . , tk ) denote an ordinary polynomial in k variables. Then   ϕ(ξ) := ϕ u1 (ξ), u2 (ξ), . . . , un (ξ) (5.23) is a polynomial on SE
and   Polyn := {ϕ u1 (ξ), u2 (ξ), . . . , uk (ξ) , deg(ϕ) ≤ n,

. ..

uj ∈ HE , ξ ∈ SE
}

(5.24)

is the collection of polynomials of degree at most n, and {Polyn }∞ n=0 is an increasing family whose union is all of SE
. One can see that the monomials u, ξW are in L2 (SE
, P) as follows: compare like powers of u from either side of (5.20) to see  that E u, ξ2n+1 = 0 and W    (2n)! E u, ξ2n = |u, ξW |2n dP(ξ) = n u2n (5.25) W E , 2 n! SE and then apply the Schwarz inequality. 2
To see why the polynomials {Polyn }∞ n=0 should be dense in L (SE , P) observe that the sequence {PPolyn }∞ of orthogonal projections increases to the identity, n=0 and therefore, {PPolyn u ˜} forms a martingale, for any u ∈ HE (i.e., for any u ˜ ∈ 2
L (SE , P)).

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133

Denote the “multiple Wiener integral of degree n” by Hn := (cl span{u, ·nW ... u ∈ HE }) " {u, ·kW ... k < n, u ∈ HE }, for each n ≥ 1, and H0 := C1 for a vector 1 with 12 = 1. Then we have an orthogonal decomposition of the Hilbert space L2 (SE
, P) =

∞ *

Hn .

(5.26)

n=0

See [17, Thm. 4.1] for a more extensive discussion. A physicist would call (5.26) the Fock space representation of L2 (SE
, P) with “vacuum vector” 1. Note that Hn has a natural (symmetric) tensor product structure: Hn ∼ = HE⊗n , the n-fold symmetric tensor product of HE with itself. Observe that 1 is orthogonal to Fin and Harm, but is not the zero element of L2 (SE
, P). Familiarity with these ideas is not necessary for the sequel, but the decomposition (5.26) is helpful for understanding two key things: (i) The Wiener isometry W : HE → L2 (SE
, P) identifies HE with the subspace H1 of L2 (SE
, P), in particular, L2 (SE
, P) is not isomorphic to HE . In fact, it is the second quantization of HE . (ii) The constant function 1 is an element of L2 (SE
, P) but does not correspond to any element of HE . In particular, 1 is not equivalent to 0 in L2 (SE
, P) (as it was in HE ). It is somewhat ironic that we began this story by removing the constants (via the introduction of HE ), only to reintroduce them with a certain amount of effort, much later. Recall that we began with a comparison of the Poisson boundary representation for bounded harmonic functions with the boundary sum representation recalled in Theorem 5.1:   x u(x) = u(y)k(x, dy) ↔ u(x) = u ∂h + u(o). ∂ ∂Ω

bd G

In this section, we replace the sum with an integral and complete the parallel. Corollary 5.19 (Boundary integral representation for harmonic functions). For any u ∈ Harm and with hx = PHarm vx ,  u(x) = u(ξ)hx (ξ) dP(ξ) + u(o). (5.27) SE

Proof. Starting with (2.6), compute u(x) − u(o) = hx , uE = u, hx E =

 SE

uhx dP,

(5.28)

where the last equality comes by substituting v = hx in (5.18). It is shown in [20, Lem. 2.24] that hx = hx .


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Remark 5.20 (A Hilbert space interpretation of bd G). In view of Corollary 5.19, we are now able to “catch” the boundary between SE and SE
: the boundary bd G may be thought of as (a presumably proper subset of) SE
/HE . In parallel to the construction of the Martin boundary, one expects that SE
/HE is larger than necessary, and that P is probably supported on a much smaller set, comparable to the minimal Martin boundary; cf. [72, §7] or [70, Ch. IV]. Corollary 5.19 suggests that
(x, dξ) := hx (ξ)dP is the discrete analogue in HE of the Poisson kernel k(x, dy), and comparison of (1.8) with (5.27) gives a way of understanding a boundary integral as a limit of Riemann sums:   x u hx dP = lim u(x) ∂h (x). (5.29) ∂ k→∞

SE

bd Gk

(We continue to omit the tildes as in Remark 5.17.) By a theorem of Nelson, P is fully supported on those functions which are H¨ older-continuous with exponent α = 12 , which we denote by Lip( 12 ) ⊆ SE
; see [51, 52]. Recall from [23, Cor. 2.16] that HE ⊆ Lip( 12 ). Current research focuses on determining the precise relationship between bd G and these other spaces (Martin boundary, Lip( 12 )), and an explicit representation of bd G in terms of paths in G and/or cocycles. We expect that bd G will have applications in the analysis of self-similar fractals, by understanding the fractal as a boundary of a resistance network; see [53].

6. Examples In this section, we introduce the most basic family of examples that illustrate our technical results and exhibit the properties (and support the types of functions) that we have discussed above. Example 6.1 (Geometric integer model). For a fixed constant c > 1, let (Z, cn ) denote the network with integers for vertices, and with geometrically increasing conductances defined by cn−1,n = cmax{|n|,|n−1|} so that the network under consideration is ···

c3

−2

c2

−1

c

0

c

1

c2

2

c3

3

c4

···

Fix o = 0. On this network, the energy kernel is given by   k ≤ 0, 0, 1−r k+1 vn (k) = 1 ≤ k ≤ n, n > 0, 1−r ,   1−r n+1 1−r , k ≥ n, and similarly for n < 0. Furthermore, the function wo (n) = ar|n| ,

a :=

r 2(1 − r)

(6.1)

defines a monopole, and h(n) = sgn(n)(1 − wo (n)) defines an element of Harm.

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135

Example 6.2 (Geometric half-integer model). It is also interesting to consider (Z+ , cn ), as this network supports a monopole, but has Harm = 0. 0

c

1

c2

2

c3

3

c4

···

r The monopole can be obtained by rescaling (6.1); just take a := (1−r) . There cannot be any nontrivial harmonic functions on this network by [20, Lem. 5.5], which states that if h ∈ Harm\{0}, then h has at least two different limiting values at ∞. That is, there exist infinite paths γ1 = (x1 , x2 , . . . ) and γ2 = (y1 , y2 , . . . ) with limj→∞ h(xj ) = limj→∞ h(yj ). For k = 2, 3, . . . , the network (Z+ , k n ) can be thought of as the “projection” of the homogeneous tree of degree k (Tk , k1 1) under a map which sends x to n ∈ Z iff there are n edges between x and o.

Remark 6.3. One can consider  −1 more general integer networks, and in this case, Harm = 0 for (Z, c) iff cxy < ∞. In this case, Harm is spanned by a single bounded function; details appear in [20]. Networks of this form have been discussed elsewhere in the literature (for example, [34, Ex. 3.12, Ex. 4.9] and [32, Ex. 3.1, Ex. 3.2]), but the authors appear to assume that ∆ is self-adjoint. This is not generally the case when c is unbounded; in fact, the Laplacian is not self-adjoint for Example 6.2 or Example 6.1; see [22, §4.2] or [18, §13.4] for further discussion and the explicit computation of defect vectors. Example 6.4 (Star networks). Let (Sm , cn ) be a network constructed by conjoining m copies of (Z+ , cn ) by identifying the origins of each; let o be the common origin. Recall from Theorem 3.10 that the boundary term is nontrivial precisely when bd G = ∅; the presence of a monopole indicates that bd G contains at least one point. If Harm = 0, then there are at least two boundary points; see [20, Lem. 5.5] and Corollary 3.8. Example 6.4 shows how to construct a network which has a boundary with cardinality m. Note that these boundary points can be distinguished by monopoles, by constructing a monopole which is constant everywhere except on one branch. Example 6.5 (Networks of integer lattices). For d ≥ 3, let {Zd(k) }m k=1 be a collection of m copies of the d-dimensional integer lattice Zd with edges between nearest neighbours, and let ok denote the origin of Zd(k) . Let Zm be the Cayley graph of the cyclic group of order m, and denote its elements by {1, 2, . . . , m}. Now define (Zd  Zm , 1) by identifying ok ∈ Zd(k) with k ∈ Zm , thus conjoining all the copies of Zd . Since Zd is transient, each copy Zd(k) supports a monopole, and hence Harm has dimension m−1 for this network. This is essentially a variation of Example 6.4 where (Z+ , cn ) is replaced by Zd . Note that this is not the same as the Cayley graph of the wreath product Zd  Zm , which is instead a Diestel-Leader graph; cf. [71]. Example 6.6 (One-sided infinite ladder network). Consider two copies of the nearest-neighbour graph on the nonnegative integers Z+ , one with vertices labelled by {xn }, and the other with vertices labelled by {yn }. Fix two positive numbers

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α > 1 > β > 0. In addition to the edges cxn ,xn−1 = αn and cyn ,yn−1 = αn , we also add “rungs” to the ladder by defining cxn ,yn = β n : x0

α

α2

β

1

y0

x1

α

y1

x2 β

α2

α3

2

y2

x3 β

α3

α4

···

αn

3

y3

xn β

α4

···

αn

αn+1

···

αn+1

···

(6.2)

n

yn

This network was suggested to us by Agelos Georgakopoulos. In [21], we show that this example is a one-ended network with nontrivial Harm, by explicitly constructing a formula for a harmonic function of finite energy on this network. Example 6.7 (The reproducing kernel on the tree). Let (T , 1) be the binary tree network as in the top of Figure 2 with constant conductance c = 1. Figure 3 depicts the embedded image of a vertex vx , as well as its decomposition in terms of Fin and Harm. We have chosen x to be adjacent to the origin o; the binary label of this vertex would be x1 . In Figure 3, numbers indicate the value of the function at that vertex; artistic liberties have been taken. If vertices s and t are the same distance from o, then |fx (s)| = |fx (t)| and similarly for hx . Note that hx provides an example of a nonconstant harmonic function in HE . It is easy to see that limz→±∞ hx (z) = 12 ± 12 , whence hx is bounded. We can use hx of Figure 3 to describe an infinite forest of mutually orthogonal harmonic functions on the binary tree. Let z ∈ T be represented by a finite binary sequence: the root o corresponds to the empty sequence ∅, and the two vertices connected to it are 0 and 1. The neighbours of 0 are ∅, 00 and 01; the neighbours of 01 are 0, 010, and 011, etc. Define a mapping ϕz : T → T by prepending, i.e., ϕz (x) = zx. This has the effect of “rigidly” translating the tree so that the image lies on the subtree with root z. Then hz := hx ◦ ϕz is harmonic and is supported only on the subtree with root z. The supports of hz1 and hz2 intersect if and only if Im(ϕzi ) ⊆ Im(ϕzj ). For concreteness, suppose it is Im(ϕz1 ) ⊆ Im(ϕz2 ). If they are equal, it is because z1 = z2 and we don’t care. Otherwise, compute the dissipation of the induced currents  dhz1 , dhz2 D = 12 Ω(x, y)dhz1 (x, y), dhz2 (x, y). (x,y)∈ϕz1 (G1 )

Note that dhz2 (x, y) always has the same sign on the subtree with root z1 = o, but dhz1 (x, y) appears in the dissipation sum positively signed with the same multiplicity as it appears negatively signed. Consequently, all terms cancel and 0 = dhz1 , dhz2 D = hz1 , hz2 E shows hz1 ⊥ hz2 . This family of harmonic functions can be heuristically described by analogy with Haar wavelets.2 Consider the boundary of the tree as a copy of the unit interval with hx as the basic Haar mother wavelet; via the “shadow” cast by 2 Compare

to the “wavelet basis of eigenfunctions” discussed in [37] (and [38, 54]). These references were brought to our attention by a reader of [18].

Resistance Boundaries of Infinite Networks 1

1

1 x

137

vx

RF(x,o) = (vx) = 1 0 o 1 4

1 8

x

0

0

0

1 16

fx 1 2

o 3 4

1 2

hx o

1 4

RH(x,o) = (hx) =

1 4

15 16

7 8

x

3 4

1 16

1 8

1 4

RW(x,o) = (fx) =

1 8

1 16

Figure 3. The reproducing kernel on the tree with c = 1. For a vertex x which is adjacent to the origin o, this figure illustrates the elements vx , fx = PFin vx , and hx = PHarm vx ; see Example 6.7. limn→±∞ hx (xn ) = ±1 (this can be formalized in terms of cocycles). Then hz is a Haar wavelet localized to the subinterval of the support of its shadow, etc. Of course, this heuristic is a bit misleading, since the boundary is actually isomorphic to {0, 1}N with its natural cylinder-set topology. Acknowledgement The authors are grateful to Ecaterina Sava and Wolfgang Woess for organizing the Boundaries 2009 workshop and to Florian Sobieczky for organizing the 2009 Alp Workshop on Spectral Theory and Random Walks, thereby making it possible for us to meet and exchange ideas with a multitude of top-tier researchers. We are grateful to the participants of these workshops for their ideas and comments, suggestions and general mathematical stimulation (there were a lot of great talks!). In particular, we benefitted from conversations with Donald Cartwright, Agelos Georgakopolous, Vadim Kaimanovich, Matthias Keller, Jun Kigami, Massimo Picardello, Elmar Teufl, Wolfgang Woess, and Radek Wojciechowski.

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Palle E.T. Jorgensen University of Iowa Iowa City IA 52246-1419, USA e-mail: [email protected] Erin P.J. Pearse University of Oklahoma Norman OK 73019-0315, USA e-mail: [email protected]

Progress in Probability, Vol. 64, 143–161 c 2011 Springer Basel AG


Brownian Motion and Negative Curvature Marc Arnaudon and Anton Thalmaier Abstract. It is well known that on a Riemannian manifold, there is a deep interplay between geometry, harmonic function theory, and the long-term behaviour of Brownian motion. Negative curvature amplifies the tendency of Brownian motion to exit compact sets and, if topologically possible, to wander out to infinity. On the other hand, non-trivial asymptotic properties of Brownian paths for large time correspond with non-trivial bounded harmonic functions on the manifold. We describe parts of this interplay in the case of negatively curved simply connected Riemannian manifolds. Recent results are related to known properties and old conjectures. Mathematics Subject Classification (2000). Primary 58J65; Secondary 60H30, 31C12, 31C35. Keywords. Harmonic function, Poisson boundary, Cartan-Hadamard manifold, Conjecture of Greene-Wu, Dirichlet problem at infinity.

1. Introduction In complex analysis the desire to understand how geometry of a complex manifold influences its complex structure has been a guiding inspiration for decades. A typical problem in this direction is the following question. Question 1 (cf. Wu [33] p. 98). If a simply connected complete K¨ ahler manifold has sectional curvature ≤ −c < 0, is it biholomorphic to a bounded domain in Cn ? Analogous questions may be asked for Riemannian manifolds. The Riemannian counterpart to the above question concerns richness of the space of bounded harmonic functions on simply-connected negatively curved Riemannian manifolds. Question 2 (cf. Wu [33] p. 139). If M is a simply-connected complete Riemannian manifold with sectional curvature ≤ −c < 0, do there exist n bounded harmonic functions (n = dim M ) which give global coordinates on M ? Even if complete answers to these questions is not in sight, it seems fair to say that such problems have directly or indirectly inspired a huge part of work

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done in these areas. Concerning Question 2, under the given assumptions, it is still not known in general whether there exist non-trivial bounded harmonic functions at all. This leads us to the famous conjecture of Greene and Wu which claims existence of non-trivial bounded harmonic functions under slightly more refined curvature conditions. Conjecture (cf. Greene-Wu [13] p. 767). Let M be a simply-connected complete Riemannian manifold of non-positive sectional curvature and x0 ∈ M such that −2 SectM x ≤ −c r(x)

for all x ∈ M \ K

for some K compact, c > 0 and r = dist(x0 , ·). Then M carries non-constant bounded harmonic functions. From a probabilistic point of view, the conjecture of Greene and Wu concerns the eventual behaviour of Brownian motion on Cartan-Hadamard manifolds as time tends to infinity. We briefly sketch the relation with Brownian motion. Indeed, for any Riemannian manifold, we have the following well-known probabilistic characterization. Lemma 1.1. For a Riemannian manifold (M, g) the following two conditions are equivalent: i) There exist non-constant bounded harmonic functions on M . ii) Brownian on (M, g) has non-trivial exit sets, i.e., if X is a Brownian motion ˆ of M on M then there exist open subsets U in the 1-point compactification M such that P{Xt ∈ U eventually} = 0 or 1. More precisely, Brownian motion X on M may be realized on the space ˆ ) of continuous paths with values in the 1-point compactification M ˆ of C(R+ , M M , equipped with the filtration Ft = σ{Xs = prs |s ≤ t} generated by the coordinate projections prs up to time t. Let ζ = sup{t > 0 : Xt ∈ M } ˆ ). be the lifetime of X and let Finv denote the shift-invariant σ-field on C(R+ , M Then there is a canonical isomorphism between the space Hb (M ) of bounded harmonic functions on M and the set bFinv of bounded Finv -measurable random variables up to equivalence, given as follows: ∼ Hb (M ) −→ bFinv /∼ ,

u −→ lim(u ◦ Xt ). t↑ζ

(1.1)

(Bounded shift-invariant random variables are considered as equivalent, if they agree Px -a.e., for each x ∈ M . Here Px denotes the law of Brownian motion starting at x.) Note that the isomorphism (1.1) is well defined by the martingale convergence theorem; the inverse map to (1.1) is given by taking expectations: bFinv /∼ # H −→ u ∈ Hb (M ) where u(x) := Ex [H].

(1.2)

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145

In particular, u(x) := Px {Xt ∈ U eventually} defines a bounded harmonic function on M , which is non-constant if and only if U is a non-trivial exit set. Clearly {Xt ∈ U eventually} ∈ Finv . Conversely, to an element B ∈ Finv one can associate an exit set U such that {Xt ∈ U eventually} = B. For instance, one may take U = {x ∈ M : u(x) > 1/2}, where u is the harmonic map defined by u(x) = Px (B).

2. Brownian motion on rotationally symmetric manifolds In this section we determine the asymptotic behaviour of Brownian motion on rotationally symmetric Riemannian manifolds, see [26, 14]. Such manifolds play an important role as comparison models for more general manifolds. 2.1. Rotationally symmetric manifolds Let (M, g) be a Riemannian manifold of dimension n ≥ 2 and x ∈ M such that the exponential map expx defines a diffeomorphism between Tx M and M . The manifold M is then diffeomorphic to Rn ; in geodesic polar coordinates on M \{0} = ]0, ∞[ × S n−1 about 0 ∈ M the metric takes the form g = dr ⊗ dr + hr with Riemannian metrics hr on S n−1 depending on the radius r. We consider the following two particular cases: (a) g = dr ⊗ dr + f 2 (r, ·) h where f : ]0, ∞[ × S n−1 → ]0, ∞[ is a scalar function and h a fixed Riemannian metric on S n−1 (independent of r); (b) g = dr ⊗ dr + f 2 (r) dϑ2 where dϑ2 is the standard metric on S n−1 and f : ]0, ∞[ → ]0, ∞[. Case (b) corresponds to the case of rotationally invariant models, see [13]. Let q : M \{0} → S n−1 ,

(r, ϑ) → ϑ.

(2.1)

The following Lemma is immediately verified. Lemma 2.1. Let q : M \{0} → S n−1 be the angular map (2.1). (i) In situation (a) the map q : (M \{0}, g) → (S n−1 , h) is harmonic if and only if (n − 3) grad fr = 0 where grad fr denotes the gradient vector field to fr = f (r, ·) on (S n−1 , h). (ii) In situation (b) the map q : (M \{0}, g) → (S n−1 , dϑ2 ) is affine, and moreover a harmonic morphism with dilatation f −1 , i.e., ∆M (ϕ ◦ q) = f −2 (∆S n−1 ϕ) ◦ q,

ϕ ∈ C ∞ (S n−1 ),

where ∆M and ∆S n−1 are the Laplacians on (M, g), resp., (S n−1 , dϑ2 ). We can give a complete description of the behaviour of Brownian motion on rotationally invariant manifolds, see [14] for details.

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Theorem 2.2 (Brownian motion on rotationally invariant models). Let M be a rotationally invariant model with center 0 ∈ M and metric g = dr ⊗ dr + f 2 (r) dϑ2 . Let X be Brownian motion on (M, g) with X0 = x0 (x0 = 0) which is decomposed according to M \{0} = ]0, ∞[ × S n−1 into its radial and angular process X = (r(X), ϑ(X)). (i) For the radial process, we have r(X)t → ∞ a.s. (i.e., X is transient) if and only if  ∞ f 1−n (r) dr < ∞. 1

(ii) The lifetime ζ of X is either a.s. finite or a.s. infinite, and a.s. finite if and only if  ∞ )  ∞ n−1 1−n f (r) f (ρ) dρ dr < ∞. 1

r

(iii) The angular process ϑ(X) converges on S n−1 for t $ ζ a.s., if and only if  ∞ )  ∞ n−3 1−n f (r) f (ρ) dρ dr < ∞. 1

r

The latter is equivalent to M being a non Liouville manifold. Sketch of Proof. (1) Denote by Rt := r(X)t and Θt := ϑ(X)t the radial part, respectively angular part of X. Then dR = dW +

1 (∆r ◦ X) dt 2

  for some one-dimensional Brownian motion W . Since (∆r)(X) = (n−1) f
/f (R), the radial process R solves the SDE dR = dW +

 1 (n − 1) (f
/f ) ◦ R dt. 2

Hence R is a one-dimensional diffusion on ]0, ∞[ with infinitesimal generator 0 1/ 2 D + (n − 1) (f
/f ) D , 2

D := d/dt.

We may calculate the Riemannian quadratic variation of X (see for instance [12]) using the relation d[X, X] = d[R, R] + f 2 (R) d[Θ, Θ]. Taking into account that d[X, X] = (dim M ) dt = n dt and d[R, R] = dt, we arrive at d[Θ, Θ] = (n − 1) (f −2 ◦ R) dt.

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According to the Martingale Convergence Theorem of Darling-Zheng (see [12]) the #ζ process Θ converges as t $ ζ on S n−1 a.s. if and only if 0 f −2 (Rt ) dt < ∞ a.s. Let  t

T (t) =

f −2 (Rs ) ds ,

t<ζ,

0

and consider the continuous time change τt := T −1 (t) ≡ inf{s ∈ R+ : T (s) ≥ t},

t < Tζ := T (ζ).

˜ t := Rτt , Obviously Tζ is the maximal lifetime of the time-changed radial process R ˜ is with the consequence that Θ converges as t $ ζ if and only if the lifetime of R finite a.s. (2) Note that by Lemma 2.1, the angular map q is a harmonic morphism with dilatation f −1 . Hence X decomposes as Xt = (Rt , BT (t) ) for some Brownian motion B on (S n−1 , dϑ2 ). In other words, by means of the time change (τt ), radial and angular part of X decouple as Xτt = (Rτt , Bt ),

t < Tζ .

In the new time scale the angular component is just Brownian motion on (S d−1 , dϑ2 ) which runs up to time Tζ . In particular, the angular part converges if and only if Tζ is finite a.s. (3) Since R is a one-dimensional diffusion on ]0, ∞[ with generator 0 1/ 2 D + (d − 1) (f
/f ) D , 2 ˜ is generated by the time-changed radial process R 0 1 2/ 2 f D + (d − 1) (f
/f ) D . 2 ˜ can now All questions of interest concerning transience and lifetime of R, resp. R, be answered by means of the classical Feller theory of one-dimensional diffusions.
Recall that a Riemannian manifold M is called Liouville manifold , if all bounded harmonic functions on M are constant. Note that in dimension 2, by Theorem 2.2, a rotationally invariant model M is Liouville if and only if Brownian motion on M is recurrent. The radial curvature of M at a point x is by definition the restriction of the sectional curvature to radial planes E, i.e., planes E ⊂ Tx M containing the radial vector (∂/∂r)x . For rotationally invariant models M the radial curvature at any point x obviously depends only on the radius r = r(x). We denote kM (r) := radial curvature at x ∈ M if r(x) = r.

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For a rotationally invariant model M with metric g = dr ⊗ dr + f 2 (r) dϑ2 , one verifies immediately kM = −f

/f. 2.2. Curvature conditions for the Liouville property The conditions of Theorem 2.2 are easily rewritten in terms of radial curvature, see [14] for instance. It is interesting to note that no lower curvature bounds are required for the existence of non-trivial bounded harmonic functions. Theorem 2.3 (Bounded harmonic functions on rotationally symmetric manifolds). Let (M, g) be a rotationally symmetric model with center 0 ∈ M and Riemannian metric g = dr ⊗ dr + f 2 (r) dϑ2 . Let kM (r) = −f

(r)/f (r) be the radial curvature function of (M, g). Suppose that kM (·) ≤ 0. Further let c = 1 in case of n = 2, resp., c = 1/2 in case of n ≥ 3. A. If for some ε > 0, (c + ε) kM (r) ≤ − 2 , for r sufficiently large, r log r then M carries non-constant bounded harmonic functions. B. If however for some ε > 0, (c − ε) kM (r) ≥ − 2 , for r sufficiently large, r log r then M is a Liouville manifold. Remark 2.4. Note that constant negative curvature outside a compact set is not sufficient for existence of non-constant bounded harmonic functions, not even for hyperbolicity (i.e., transience of Brownian motion), as the following example shows. Let M = R2 be equipped with a rotationally symmetric metric, for instance, given by the radial function f (r) = exp(−r) for r > 1 and a differentiable interpolation for 0 ≤ r ≤ 1, such that f (0) = 0 and f
(0) = 1 holds. Then (M, g) has constant negative curvature outside the unit disk, but according to Theorem 2.2, Brownian motion on (M, g) is recurrent; the manifold M is hence Liouville.

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3. Brownian motion on Cartan-Hadamard manifolds of pinched negative curvature Let (M, g) be a Cartan-Hadamard manifold, i.e., a simply connected, metrically complete Riemannian manifold of non-positive sectional curvature. (All manifolds in the sequel are supposed to be connected.) In terms of the exponential map ∼ expx0 : Tx0 M −→ M at a fixed base point x0 ∈ M , we identify ∼ ρ : Rn ∼ = Tx0 M −→ M.

Via pullback of the metric g on M to Rn , we obtain an isometric isomorphism (M, g) ∼ = (Rn , ρ∗ g); in particular, M \ {x0 } ∼ = ]0, ∞[ × S n−1 . In terms of such global polar coordinates, Brownian motion X on M will be decomposed into its radial and angular part, Xt = (r(Xt ), ϑ(Xt )) where r(Xt ) = dist(x0 , Xt ) and where ϑ(Xt ) takes values in S n−1 . 3.1. Boundaries at infinity For a Cartan-Hadamard manifold M of dimension n there is a natural geometric compactification which is given by attaching an ideal boundary M (∞) at infinity. Formally the points of the boundary are given as equivalence classes of geodesics which stay in bounded distance, ¯ = M ∪˙ M (∞) M where M (∞) = {γ : R → M | γ geodesics }/∼ and γ1 ∼ γ2 ⇔ lim sup dist(γ1 (t), γ2 (t)) < ∞. t→∞

Then M ∪ M (∞) equipped with the cone topology is homeomorphic to the closed unit ball B ⊂ Rn with boundary ∂B = S n−1 , cf. [6, 11], whereas M itself is diffeomorphic to the open unit ball. In particular, S∞ (M ) := M (∞) ≡ {γ(∞) | γ geodesics} is a (n − 1)-sphere which serves as horizon at infinity. In terms of polar coordinates on M , a sequence (rn , ϑn )n∈N of points in M converges to a point on the sphere at infinity S∞ (M ) if and only if rn → ∞ and ϑn → ϑ ∈ S n−1 . Definition 3.1. Let M be a Cartan-Hadamard manifold and f : S∞ (M ) → R be a continuous function defined on the sphere at infinity. The Dirichlet problem at infinity is to find a harmonic function h : M → R which extends continuously to S∞ (M ) and coincides there with the given function f , i.e., h|S∞ (M ) = f. The Dirichlet problem at infinity is said to be solvable if this is possible for every such function f .

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Solvability of the Dirichlet problem at infinity provides a rich class of nontrivial bounded harmonic functions on M , given as solutions of the Dirichlet problem at infinity for various boundary functions. 3.2. Angular convergence and solvability of the Dirichlet problem at infinity First results concerning angular convergence have been obtained by J.-J. Prat [29, 30]. He proved that on a Cartan-Hadamard manifold with sectional curvature bounded from above by a negative constant −b2 , b > 0, Brownian motion is transient, i.e., almost surely all paths of the Brownian motion exit M at the sphere at infinity [30]. If in addition the sectional curvatures are bounded from below by a constant −a2 , a > b, he showed that the angular part ϑ(Xt ) of the Brownian motion almost surely converges as t → ζ. Y. Kifer [21] presented a stochastic proof that on Cartan-Hadamard manifolds with sectional curvature pinched between two strictly negative constants and satisfying a certain additional technical condition, the Dirichlet problem at infinity has a unique solution. The proof there was given in explicit terms for the two-dimensional case. D. Sullivan [31] finally gave a complete stochastic proof of the fact that on a Cartan-Hadamard manifold with pinched negative curvature the Dirichlet problem at infinity is uniquely solvable. The crucial open point has been to show that the harmonic measure class is non-trivial in this case. Theorem 3.2 (J.-J. Prat [30], Y. Kifer [21]). Let (M, g) be a Cartan-Hadamard manifold of dimension n such that −a2 ≤ SectM ≤ −b2 < 0. Let Xt = (r(Xt ), ϑ(Xt )) be a Brownian motion on M decomposed in its radial and angular part. Then a.s., as t → ∞, r(Xt ) → ∞,

and

ϑ(Xt ) converges on S n−1 .

Denoting Θ∞ := limt→∞ ϑ(Xt ) and µx := Px ◦ Θ−1 ∞ , for every Borel set U ⊂ S∞ (M ), the function u(x) = Px {Θ∞ ∈ U } ≡ µx (U )

is harmonic on M .

Remark 3.3. By the maximum principle, the Poisson hitting measure at infinity are equivalent, µx ∼ µy for x, y ∈ M, and define a measure class on the sphere at infinity. Indeed, for any Borel set U ⊂ S∞ (M ), the assignment hU : x → µx (U ) defines a bounded harmonic function on M , which by the maximum principle is either identically equal to 0 or 1 or takes its values in ]0, 1[. For the solvability of the Dirichlet problem at infinity this measure class µ must be shown to be non-trivial.

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Theorem 3.4 (D. Sullivan [31], M. T. Anderson [3]). Let (M, g) be a CartanHadamard manifold of dimension n such that −a2 ≤ SectM ≤ −b2 < 0. The harmonic measure class on S∞ (M ) is positive on each non-void open set. In fact, if xi in M converges to x∞ in S∞ (M ), then the Poisson hitting measures µxi tend weakly to the Dirac mass at x∞ . Corollary 3.5 (Solvability of the Dirichlet problem at infinity). Let (M, g) be a ndimensional Cartan-Hadamard manifold with sectional curvature pinched between to negative constants, −a2 ≤ SectM ≤ −b2 < 0. For f ∈ C(S∞ (M )) let

 u(x) = Ex f (Θ∞ )].

Then ¯ ), u ∈ C(M

∆u = 0 on M

and

u|S∞ (M ) = f.

Denoting by Hb (M ) the Banach space of bounded harmonic functions on M , a complete description of bounded harmonic functions on M has been given by M.T. Anderson [3]. It is an interesting feature of the pinched curvature case that any bounded harmonic function on M comes from a solution of the Dirichlet problem at ∞ (for some bounded measurable f ). Theorem 3.6 (M.T. Anderson [3]). Let (M, g) be a Cartan-Hadamard manifold of 2 dimension n ≥ 2, whose sectional curvatures satisfy −a2 ≤ SectM x ≤ −b < 0 for all x ∈ M . Then the linear mapping P : L∞ (S∞ (M ), µ) → Hb (M ), f → P (f ),

 P (f )(x) :=

f dµx

(3.1)

S∞ (M )

is a norm-nonincreasing isomorphism onto Hb (M ). In the case of a Cartan-Hadamard manifold of pinched negative curvature the limiting angle Θ∞ = limt→∞ ϑ(Xt ) generates the shift-invariant σ-field of X, Finv = σ(Θ∞ ). Thus all shift-invariant properties which allow to distinguish Brownian paths for large times, are expressible in terms of the angular projection of X onto S∞ (M ). Remark 3.7 (Probabilistic conditions for the solvability of the Dirichlet problem at ∞). The proof of solvability of the Dirichlet problem at ∞ by probabilistic methods proceeds in several steps: 1. Prove transience of Brownian motion, i.e., r(Xt ) → ∞ for the radial part of Brownian motion.

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2. Establish convergence of the angular part ϑ(Xt ) of Brownian motion; this provides entrance measures µx into the sphere at ∞. 3. Show that the harmonic measure class at infinity is non-trivial and all boundary points are regular, i.e., µxn → δx∞ in probability, as xn → x∞ ∈ S∞ (M ). The upper curvature bound in this approach guarantees a certain escape rate for Brownian motion, whereas the lower curvature bound controls the angular oscillations. The constant curvature bounds for solvability of the Dirichlet problem have been relaxed in various directions, e.g., Hsu-March [18], Hsu [16], V¨ ah¨ akangas [32], Holopainen-V¨ah¨ akangas [15]. It is interesting to note that the curvature may well tend to −∞ as long the oscillation of curvature (in the sense of the quotient of lower and upper bounds) is bounded. Theorem 3.8 (V¨ ah¨ akangas [32], Holopainen-V¨ ah¨ akangas [15]). Let M be a simplyconnected complete Riemannian manifold of non-positive sectional curvature such that c SectM ∀x ∈ M \ K x ≤ − 2 r (x) for some K compact in M , c > 0 and r = dist(x0 , ·). Suppose that there exists C > 1 such that for each x ∈ M \ K and all radial planes E, E
⊂ Tx M M
|SectM x (E)| ≤ C |Sectx (E )|.

Then the Dirichlet problem at ∞ is solvable. 3.3. The role of lower curvature bounds The discussion above raises the question to what extent lower curvature bounds are necessary. In the special case of a Riemannian surface M of negative curvature bounded from above by a negative constant, Kendall [20] gave a simple stochastic argument that the Dirichlet problem at infinity is uniquely solvable. He used the fact that every geodesic on the Riemannian surface connecting two different points on the sphere at infinity divides the surface into two totally geodesic disjoint half-parts. Starting from a point x on M , with non-trivial probability, Brownian motion will eventually stay in one of the two half-parts up to its lifetime. As this is valid for every geodesic and every starting point x, the non-triviality of the harmonic measure class on S∞ (M ) easily follows. Choi [8] provided a general criterion, the so-called convex conic neighbourhood condition, which is sufficient for solvability of the Dirichlet problem at infinity. The ¯ by condition requires that distinct points x, y ∈ S∞ (M ) can be separated in M convex sets.

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Definition 3.9. A Cartan-Hadamard manifold M satisfies the convex conic neighbourhood condition at x ∈ S∞ (M ) if for any y ∈ S∞ (M ), y = x, there exist ¯ , containing x and y respectively, such that Vx and Vy are subsets Vx and Vy of M ¯ disjoint open sets of M in terms of the cone topology and Vx ∩ M is convex with C 2 -boundary. If this condition holds for all x ∈ S∞ (M ), M is said to satisfy the convex conic neighbourhood condition. Theorem 3.10 (H. I. Choi [8]). Let M be a Cartan-Hadamard manifold satisfying the convex conic neighbourhood condition, with SectM ≤ c for some constant c < 0. Then the Dirichlet problem at infinity for M is solvable. Intuitively the convex conic neighbourhood condition looks as a plausible assumption suggesting that lower curvature bounds might be dispensable. It has been shown by Ancona [2] that this is however not the case. In [2] he constructed an example of a complete, simply connected Riemannian manifold M of dimension 3, with SectM ≤ −1, and a point ∞M ∈ S∞ (M ) such that (i) Brownian motion on M has a.s. infinite lifetime; (ii) With probability 1, any Brownian motion on exits from M at ∞M . ¯ contains all of M . In addition, the convex hull of any neighbourhood of ∞ in M M

Remark 3.11. It is obvious that on Ancona’s manifold the Dirichlet problem at infinity is not solvable. Indeed, let ¯ ) such that ∆u = 0 on M and u|S∞ (M ) = f. f ∈ C(S∞ (M )), u ∈ C(M Then, for any x ∈ M ,

 u(x) = Ex f (X∞ )] = f (∞M )

which shows that the function u must be constant. Hence only the constant harmonic functions have a continuous continuation to the boundary at ∞. Nevertheless the above discussion does not answer the question whether there exist non-trivial bounded harmonic functions on Ancona’s manifold, since there may be non-trivial bounded harmonic functions which do not come from boundary values at infinity. Ancona did not address this problem in his paper [2]. It turns out that there are indeed plenty of non-trivial bounded harmonic functions with no continuation to the boundary at infinity, as will be discussed in the next section. Borb´ely [7] constructed a similar example of a Cartan-Hadamard manifold with curvature bounded above by a strictly negative constant, on which the Dirichlet problem at infinity is not solvable. Borb´ely does not discuss the behaviour of Brownian motion on his manifold. However it turns out that from a probabilistic point of view, the manifolds of Ancona and Borb´ely share similar properties. Borb´ely’s construction is however technically simpler; in the next section we shall give a detailed probabilistic analysis of his manifold.

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4. Brownian motion on Cartan-Hadamard manifolds of unbounded negative curvature In this section we discuss examples of negatively curved manifolds where the potential theoretic boundary does not coincide with the geometric boundary at infinity. To see the full Poisson boundary, certain points at infinity need to be blown up in a non-trivial way. Such examples indicate that the situation concerning the conjecture of Greene and Wu about existence of non-trivial bounded harmonic functions on Cartan-Hadamard manifolds may be more complicated than expected. 4.1. Description of the manifold The construction of the following example is basically due to Borb´ely [7]. Let H2 = {(x, y) ∈ R2 | y > 0} be the hyperbolic half-plane equipped with the hyperbolic metric ds2H2 of constant curvature −1. Denote by L = {(0, y) | y > 0} the positive y-axis in H2 and let H denote one component of H2 \ L. We shall define a Riemannian manifold M as the warped product: M := (H ∪ L) ×g S 1 , with Riemannian metric ds2M = ds2H2 + g ds2S 1 , where g : H ∪ L → R+ is a positive C ∞ -function which has to be chosen appropriately. By identifying points ( , α1 ) and ( , α2 ) with ∈ L and α1 , α2 ∈ S 1 , we make M a simply connected space. L(−∞)

L(0)

H L(s) r

α

p = (r, s, α) ←− γsα = (·, s, α)

L(∞)

Figure 1. Fermi coordinates (r, s, α) for M

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We introduce Fermi coordinates (r, s, α) on M , where for a point p ∈ M the coordinate r is the hyperbolic distance between p and the geodesic L, i.e., the hyperbolic length of the perpendicular on L through p. The coordinate s is the parameter on the geodesic {L(s) : s ∈ ]−∞, ∞[ }, i.e., the length of the geodesic segment on L joining L(0) and the orthogonal projection L(s) of p onto L. Furthermore, α ∈ [0, 2π[ is the parameter on S 1 when using the parameterization eiα of S 1 . Theorem 4.1. Let M = (H ∪ L) ×g S 1 be equipped with the metric ds2M = ds2H2 + g ds2S 1 . There exists a choice for a smooth wedge function g : R+ × R → R+ such that the following two properties hold true: (i) (M, ds2M ) is Cartan-Hadamard with SectM ≤ −1; (ii) Xt → L(+∞) a.s. for each Brownian motion X on M . Remark 4.2. Typical geodesics in M are the ones of the type γsα : r → (r, s, α). We get the sphere at infinity S∞ (M ) by following these geodesics to infinity, together with the endpoints L(−∞), L(+∞) of the vertical geodesic L which serves as symmetry axis: / 0 S∞ (M ) = γsα (∞) | s ∈ R, α ∈ [0, 2π[ ∪ {L(±∞)}.

L(−∞)

γs0

γsα

α

(r, s, α)

L(+∞)

γsα (+∞)

(r, s, 0) γs0 (+∞)

S∞ (M ) Figure 2. Sphere at infinity S∞ (M )

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4.2. Asymptotic behaviour of Brownian motion The behaviour of Brownian motion on M may be investigated in the given coordinates, by studying the corresponding stochastic differential equations. It turns out that Brownian motion almost surely exits the manifold at a distinguished point of the sphere at infinity, independently of its starting point x at time 0. Theorem 4.3 (Arnaudon-Thalmaier-Ulsamer [5]). Writing Brownian motion on M in terms of the Fermi coordinates Xt = (Rt , St , At ), we have the following asymptotic properties, as t approaches the lifetime ζ of X, (i) Rt → +∞ a.s.

(ii) St → +∞ a.s.

(iii) At → Aζ a.s.,

where the limiting variable Aζ is a non-trivial random variable taking values in S 1 . Moreover, for any x ∈ M , the support of Px ◦ (Aζ )−1 equals all of S 1 . Remark 4.4. An immediate consequence of Theorem 4.3 is that, for each nonconstant f ∈ C(S 1 ; R), the formula h(x) = Ex [f (Aζ )]

(4.1)

defines a non-trivial bounded harmonic function h on M . The variable Aζ gives the asymptotic direction on the sphere at ∞ along which Brownian motion approaches the point ∞M = L(+∞) ∈ S∞ (M ). The question whether M carries further non-trivial bounded harmonic functions besides functions of the type (4.1), turns out to be highly non-trivial. To answer this question we go back to equations of Brownian motion in the specified Fermi coordinates  1 1  dR = F (R, S) dt + dM  dS = F 2 (R, S) dt + dM 2   dA = dM 3 with M 1 , M 2 and M 3 local martingales. After an appropriate change of time (which turns the drift of R into a deterministic linear motion) this system writes as  ˆ= ˆ1  dR dt + dM   ˆ S) ˆ dt + dM ˆ2 dSˆ = f (R,    ˆ ˆ3 dA = dM ˆ i converge as t → ζ. where the local martingale parts M t The idea now is to compare the asymptotic behaviour of (R, S) with the deterministic dynamical system in the (r, s)-plan given by   drt = dt r˙t = 1 i.e., dst = f (rt , st ) dt s˙ t = f (t, st ).

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It seems reasonable to expect that ˆ t , Sˆt , ) ≈ (rt , st ), (R

for t & 0,

where (rt , st ) is an integral curve of the vector field ∂ ∂ + f (r, s) . ∂r ∂s Taking into account the fact that there is a good approximation q = q(r) such that f (r, s) ≈ q(r) for large r, V =

we consider Γs0 (t) =

  t t, s0 + q(u) du 0

and let Γs0 := {Γs0 (t) : t ≥ 0} . Then (Γs0 )s0 ∈R defines a foliation of H. The change of coordinates: Φ : R+ × R → R+ × R,  #r  (r, s) → r, s − 0 q(u) du =: (r, z) transforms integral curves of the dynamical system to straight lines. s

z Φ(Γs0 )

s0 Γs0

Φ −→

L(+∞)

r

Φ(L(+∞))

r

Figure 3. The coordinate transformation Φ

4.3. Description of the Poisson boundary The family (Γs0 )s0 ∈R when rotated about the symmetry axis (given by the vertical geodesic L) defines a foliation of M . If the idea is correct that Brownian motion exits M asymptotically along the leaves of this foliation, then the z-component of Brownian motion (using the coordinates (r, z, α) as above) should have a large time limit.

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Theorem 4.5 (Arnaudon-Thalmaier-Ulsamer [5]). Considering Brownian motion on M in the new coordinates: X = (R, S, A) −→ X = (R, Z, A) where

 Zt := St −

Rt

q(u) du, 0

we find the following asymptotic behaviour: almost surely, as t → ζ,  R →∞   t Zt → Zζ ∈ R   At → Aζ ∈ S 1 . Moreover, the induced measures Px ◦ Zζ−1 on R, respectively, Px ◦ A−1 on S 1 have ζ full support. Remark 4.6. For each f ∈ Cb (R × S 1 ), f = const,   u(x) = Ex f (Zζ , Aζ ) defines a non-trivial bounded harmonic function on M . Indeed, it can be shown that Finv = σ(Zζ , Aζ ). (4.2) The proof of (4.2) is far from being obvious and uses time reversal arguments [28], coupling arguments for the time-reversed process, as well as a 0/1 law for the time-reversed process, see [5] for details. Theorem 4.7 (Arnaudon-Thalmaier-Ulsamer [5]). Let B(R × S 1 ; R) be the set of bounded measurable functions on R × S 1 , with the equivalence relation f1 ' f2 if f1 = f2 Lebesgue-a.e. 1. The map (B(R × S 1 ; R)/ ') −→ Hb (M )   f −→ x → E [f (Zζ (x), Aζ (x))] is one to one. 2. The inverse map is given as follows. For x ∈ M , letting K(x, ·, ·) be the density of (Zζ (x), Aζ (x)) with respect to the Lebesgue measure on R × S 1 , for any h ∈ Hb (M ) there exists a unique f ∈ B(R × S 1 ; R)/ ' such that  ∀x ∈ M, h(x) = K(x, z, a) f (z, a) dzda. (4.3) R×S1

Moreover, for x ∈ M , the kernel K(x, ·, ·) in (4.3) is strictly positive a.e. Note that the space of bounded harmonic functions Hb (M ) on the constructed manifold M is as rich as in the pinched curvature case −a2 ≤ SectM ≤ −b2 < 0,

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where one has Finv = σ(Θ∞ ). The rich space of harmonic functions on M comes however from completely different reasons. There is no contribution from the limiting angle of Brownian motion, since Brownian motion degenerates to a single point at infinity. To see the true Poisson boundary, the point ∞M at infinity has to be blown up to a twodimensional space. Remark 4.8. From the point of view of Harnack inequalities harmonic functions in the constructed manifold share an interesting feature: on any neighbourhood in M of the point ∞M ∈ S∞ (M ), a bounded harmonic function on M attains each value between its global minimum and global maximum. Remark 4.9. The constructed manifold has another interesting property in contrast to the pinched curvature case: it is easy to show that the σ-field of terminal events is strictly larger than the σ-field of shift-invariant events. This implies that the space of bounded harmonic functions is a proper subspace of the bounded space-time harmonic functions. 4.4. Absolute continuity of the harmonic measure class It is interesting to note that on the constructed manifold, despite of diverging curvature, the harmonic measure has a density (Poisson kernel) with respect to the Lebesgue measure on the Poisson boundary R × S 1 . In the pinched curvature case the harmonic measure may well be singular with respect to the surface measure on the sphere at infinity (see [21], [25], [19], [10] for results in this direction). Typically it is the fluctuation of the geometry at infinity which prevents harmonic measure from being absolutely continuous. It is well known that pinched curvature alone does in general not allow to bound the angular derivative of the Riemannian metric, when written in polar coordinates. 4.5. Outlook The example discussed in this section can be modified in such a way that, with probability 1, every point on S∞ (M ) is a cluster point of Xt (when t → ∞). This can be achieved by properly moving the attracting point ∞M on the sphere at infinity. Such examples may turn out to be possible candidates for disproving Greene-Wu’s conjecture.

References [1] A. Ancona, Negatively curved manifolds, elliptic operators, and the Martin boundary, Ann. of Math. (2) 125 (1987), no. 3, 495–536. [2] , Convexity at infinity and Brownian motion on manifolds with unbounded negative curvature, Rev. Mat. Iberoamericana 10 (1994), no. 1, 189–220. [3] M.T. Anderson, The Dirichlet problem at infinity for manifolds of negative curvature, J. Differential Geom. 18 (1983), no. 4, 701–721 (1984).

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[4] M.T. Anderson, R. Schoen, Positive harmonic functions on complete manifolds of negative curvature, Ann. of Math. (2) 121 (1985), no. 3, 429–461. [5] M. Arnaudon, A. Thalmaier, S. Ulsamer, Existence of non-trivial harmonic functions on Cartan-Hadamard manifolds of unbounded curvature, Math. Z. 263 (2009), 369– 409. [6] R.L. Bishop, B. O’Neill, Manifolds of Negative Curvature, Trans. Amer. Math. Soc. 145 (1968), 1–49. [7] A. Borb´ely, The nonsolvability of the Dirichlet problem on negatively curved manifolds, Differential Geom. Appl. 8 (1998), no. 3, 217–237. [8] H.I. Choi, Asymptotic Dirichlet problems for harmonic functions on Riemannian manifolds, Trans. Amer. Math. Soc. 281 (1984), no. 2, 691–716. [9] M. Cranston, On specifying invariant σ-fields. Seminar on Stochastic Processes 1991, Birkh¨ auser, Progr. Probab., vol. 29 (1992), 15–37. [10] M. Cranston, C. Mueller, A review of recent and older results on the absolute continuity of harmonic measure, Geometry of random motion (Ithaca, N.Y., 1987), Contemp. Math., vol. 73, Amer. Math. Soc., Providence, RI, 1988, pp. 9–19. [11] P. Eberlein, B. O’Neill, Visibility Manifolds. Pacific J. Math. 46 (1973), no. 1, 45– 109. ´ [12] M. Emery, Stochastic calculus in manifolds, Universitext, Springer-Verlag, Berlin, 1989, With an appendix by P.-A. Meyer. [13] R.E. Greene, H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. [14] W. Hackenbroch, A. Thalmaier, Stochastische Analysis. Eine Einf¨ uhrung in die Theorie der stetigen Semimartingale, B. G. Teubner, Stuttgart, 1994. [15] I. Holopainen, A. V¨ ah¨ akangas, Asymptotic Dirichlet problem on negatively curved spaces. International Conference on Geometric Function Theory, Special Functions and Applications (R.W. Barnard and S. Ponnusamy, eds.), J. Analysis 15 (2007), 63–110. [16] E.P. Hsu, Brownian motion and Dirichlet problems at infinity, Ann. Probab. 31 (2003), no. 3, 1305–1319. [17] P. Hsu, W.S. Kendall, Limiting angle of Brownian motion in certain two-dimensional Cartan-Hadamard manifolds, Ann. Fac. Sci. Toulouse Math. (6) 1 (1992), no. 2, 169– 186. [18] P. Hsu, P. March, The limiting angle of certain Riemannian Brownian motions, Comm. Pure Appl. Math. 38 (1985), no. 6, 755–768. [19] A. Katok, Four applications of conformal equivalence to geometry and dynamics, Ergodic Theory Dynam. Systems 8∗ (1988), no. Charles Conley Memorial Issue, 139–152. [20] W.S. Kendall, Brownian motion on a surface of negative curvature, Seminar on probability, XVIII, Lecture Notes in Math., vol. 1059, Springer, Berlin, 1984, pp. 70– 76. [21] Ju.I. Kifer, Brownian motion and harmonic functions on manifolds of negative curvature, Theor. Probability Appl. 21 (1976), no. 1, 81–95.

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[23] [24] [25] [26] [27] [28]

[29] [30]

[31] [32] [33]

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, Brownian motion and positive harmonic functions on complete manifolds of nonpositive curvature, From local times to global geometry, control and physics (Coventry, 1984/85), Pitman Res. Notes Math. Ser., vol. 150, Longman Sci. Tech., Harlow, 1986, pp. 187–232. H. Le, Limiting angle of Brownian motion on certain manifolds, Probab. Theory Related Fields 106 (1996), no. 1, 137–149. , Limiting angles of Γ-martingales, Probab. Theory Related Fields 114 (1999), no. 1, 85–96. F. Ledrappier, Propri´ et´e de Poisson et courbure n´egative, C. R. Acad. Sci. Paris S´er. I Math. 305 (1987), no. 5, 191–194. P. March, Brownian motion and harmonic functions on rotationally symmetric manifolds, Ann. Probab. 11 (1986) 793–801. F. Mouton, Comportement asymptotique des fonctions harmoniques en courbure n´egative, Comment. Math. Helv. 70 (1995), 475–505. ´ Pardoux, Grossissement d’une filtration et retournement du temps d’une diffusion, E. S´eminaire de Probabilit´es, XX, 1984/85, Lecture Notes in Math., vol. 1204, Springer, Berlin, 1986, pp. 48–55. ´ J.-J. Prat, Etude asymptotique du mouvement brownien sur une vari´et´e riemannienne a ` courbure n´egative, C. R. Acad. Sci. Paris S´er. A-B 272 (1971), A1586–A1589. ´ , Etude asymptotique et convergence angulaire du mouvement brownien sur une vari´et´e ` a courbure n´ egative, C. R. Acad. Sci. Paris S´er. A-B 280 (1975), A1539– A1542. D. Sullivan, The Dirichlet problem at infinity for a negatively curved manifold, J. Differential Geom. 18 (1983), no. 4, 723–732 (1984). A. V¨ ah¨ akangas, Dirichlet problem at infinity for A-harmonic functions, Potential Anal. 27 (2007), no. 1, 27–44. H.H. Wu, Function theory on noncompact K¨ ahler manifolds, Complex differential geometry, DMV Sem., vol. 3, Birkh¨ auser, Basel, 1983, pp. 67–155.

Marc Arnaudon Laboratoire de Math´ematiques et Applications, CNRS: UMR6086 Universit´e de Poitiers, T´el´eport 2 – BP 30179 F-86962 Futuroscope Chasseneuil Cedex, France e-mail: [email protected] Anton Thalmaier Unit´e de Recherche en Math´ematiques, FSTC Universit´e du Luxembourg 6, rue Richard Coudenhove-Kalergi L-1359 Luxembourg, Grand-Duchy of Luxembourg e-mail: [email protected]

Progress in Probability, Vol. 64, 163–179 c 2011 Springer Basel AG


Stochastically Incomplete Manifolds and Graphs Rados
law Krzysztof Wojciechowski Abstract. We survey geometric properties which imply the stochastic incompleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. In particular, we completely characterize stochastic incompleteness for spherically symmetric graphs and show that, in contrast to the case of Riemannian manifolds, there exist examples of stochastically incomplete graphs of polynomial volume growth. Mathematics Subject Classification (2000). Primary 39A12; Secondary 58J65. Keywords. Stochastic incompleteness, explosion, heat kernel, manifolds, graphs, curvature, volume growth.

1. Introduction A diffusion process whose lifetime is almost surely infinite is said to be stochastically complete (or conservative or non-explosive). If this fails to occur, that is, if the total probability of the particle undergoing the diffusion to be in the state space is less than one at some time, the process is said to be stochastically incomplete. A trivial way for stochastic incompleteness to occur is to impose a killing boundary condition. It is the objective of this article to survey the geometric properties, in the case when no such killing condition is present, that cause the stochastic incompleteness of the minimal diffusion process associated to the Laplacian on manifolds and graphs. We draw heavily from the survey article of A. Grigor
yan [14] for the case of Riemannian manifolds and then present some recent results for graphs. Examples of geodesically complete but stochastically incomplete manifolds were first given by R. Azencott [1]. Specifically, if M is a geodesically complete, This work was completed with the support of FCT grant SFRH/BPD/45419/2008 and FCT project PTDC/MAT/101007/2008.

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simply connected, negatively curved, analytic Riemannian manifold and k(r) denotes the infimum, in absolute value, of the sectional curvatures at distance r, then Azencott showed that M is stochastically incomplete if k(r) ≥ Cr2+ for  > 0 and all r large [1, Proposition 7.9]. In these examples, the large negative curvature forces the particle to infinity in a finite time and explosion occurs. Furthermore, Azencott showed that such a manifold is stochastically complete if the sectional curvature is uniformly bounded from below. In 1978, S.T. Yau showed that a geodesically complete manifold whose Ricci curvature is bounded from below is stochastically complete [32]. A different type of criterion for stochastic completeness in terms of volume growth was given by Grigor
yan in 1986. In particular, 2 Grigor
yan’s result implies that if V (r) ≤ ecr , where V (r) denotes the volume of a geodesic ball of radius r, then a geodesically complete manifold is stochastically complete [13]. The examples of Azencott, or the case of model manifolds, show that Grigor
yan’s criterion is sharp. The corresponding question for graphs was explicitly addressed by J. Dodziuk and V. Mathai in 2006. By analyzing bounded solutions of the heat equation they show that graphs of bounded valence are stochastically complete [7, Theorem 2.10]. Examples of stochastically incomplete graphs were given in [30, 31]. These examples are trees branching rapidly in all directions from a fixed vertex. More specifically, letting k+ (r) denote the minimum number of outward pointing edges, where the minimum is taken over all vertices  on a sphere of radius r in a tree, then ∞ the diffusion is stochastically incomplete if r=0 k+1(r) < ∞ [31, Theorem 3.4]. Therefore, in the case of graphs, the number of outward pointing edges plays the role of the negative sectional curvature in sweeping the particle out to infinity. The volume growth for such trees is factorial and, while smaller, at least comparable to the examples of Azencott. In this article, we give many more examples of stochastically incomplete graphs. In particular, in Theorem 4.8, we completely characterize the stochastic incompleteness of spherically symmetric graphs and use this to give examples of stochastically incomplete graphs with only polynomial volume growth. Thus, in the case of graphs, no direct analogue of Grigor
yan’s theorem holds.

2. Stochastic incompleteness In this section we give an overview of properties equivalent to stochastic incompleteness. Here, the manifold and graph settings are analogous so we do not distinguish between the two. To avoid trivial examples, we assume that all manifolds are geodesically complete and that all graphs are infinite. Furthermore, we assume that all underlying spaces are connected. We start by outlining a construction of the heat kernel. In both cases, one has a Laplacian acting on a dense subset of the space of L2 functions. We choose our sign convention so that the Laplacian is a positive operator. Therefore, in n ∂ 2 the case of Rn , we take ∆ = − i=1 ∂x 2 , while, for graphs, if f is a function on i  the vertices, then, pointwise, the Laplacian acts by ∆f (x) = y∼x (f (x) − f (y)),

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where y ∼ x indicates that the vertices x and y form an edge. We note that this sign convention is consistent with the one followed in [7, 31] but opposite of [14]. In both cases, the Laplacian is essentially self-adjoint and one uses the functional calculus to define a semigroup of operators e−t∆ . The action of the semigroup is given by a kernel pt (x, y), henceforth called the heat kernel, so that, for t > 0 and x ∈ M  −t∆ e u0 (x) = pt (x, y)u0 (y)dy M

where u0 is a continuous, bounded function. Alternatively, the heat kernel can be constructed by an exhaustion argument. In this construction, one takes a sequence of increasing subsets of the whole space, defines the heat kernel with Dirichlet boundary conditions for each subset in the exhaustion, and then passes to the limit. This is the approach taken in the case of manifolds in [4] and for graphs in [29, 30] and the equivalence of the two constructions is demonstrated there. The heat kernel is positive, symmetric, satisfies the semigroup property and has total integral less than or equal to 1. In particular, pt (x, y) gives the transition density for a diffusion process on the underlying space, referred to as the minimal diffusion process associated to the Laplacian (see [1, 3, 14]). Definition 2.1. The minimal diffusion process associated to the Laplacian is stochastically incomplete if  pt (x, y)dy < 1 M

for some (equivalently, all) t > 0 and x ∈ M . We will follow convention and say that the space is stochastically incomplete if this is the case. Note that, another way of writing this is e−t∆ 1 < 1 where 1 indicates the constant function whose value is 1. The following theorem gives some equivalent formulations of stochastic incompleteness. In particular, stochastic incompleteness is equivalent to the nonuniqueness of bounded solutions of the heat equation. These criteria originate in the works of R.Z. Has
minski˘ı [15] and W. Feller [8] for Euclidean spaces and Feller [9] and G.E.H. Reuter [27] in the discrete setting. For a full historical overview and proof in the case of manifolds see [14, Theorem 6.2], for a proof in the case of the graph Laplacian [31, Theorem 3.1], for more general operators which are generators of regular Dirichlet forms on discrete sets with an arbitrary measure of full support [22, Theorem 1]. Theorem 2.2. The following statements are equivalent: # (1) M pt (x, y)dy < 1 for some (equivalently, all) t > 0 and x ∈ M . (2) For some (equivalently, all) λ > 0, there exists a bounded, positive function v satisfying (∆ + λ)v = 0. (3) For some (equivalently, all) λ > 0, there exists a bounded, non-negative, nonzero function v satisfying (∆ + λ)v ≤ 0.

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(4) There exists a non-zero, bounded function u : M × (0, ∞) → R satisfying  ∆u(x, t) + ∂u for x ∈ M, t > 0 ∂t (x, t) = 0 limt→0+ u(·, t) ≡ 0. Remark 2.3. The condition (3) is sometimes referred to by saying that M is λmassive. More generally, an open subset Ω of M is called λ-massive, if there exists a bounded, non-negative function v satisfying (∆ + λ)v ≤ 0 on M such that v|M \Ω ≡ 0 and v is non-zero on Ω. By a maximum principle argument, it is easy to see that this property is preserved by enlarging Ω or by removing a compact subset from Ω [14, Proposition 6.1]. Therefore, M is stochastically incomplete if it contains a λ-massive subset and, furthermore, stochastic incompleteness is preserved under the operation of removing a compact subset from M . Remark 2.4. For another formulation of stochastic completeness in terms of a weak form of the Omori-Yau maximum principle see [26, Theorem 1.1].

3. Stochastically incomplete manifolds In this section we survey some of the known examples of stochastically incomplete manifolds. As mentioned in the introduction, the discovery of such examples goes back to the work of Azencott. In particular, letting K(r) and k(r) denote the supremum and infimum of the absolute value of the sectional curvatures at distance r from a fixed point on a negatively curved, simply connected, analytic manifold M , Azencott [1, Proposition 7.9] proved that: #r (i) If 1r 0 K(s)ds ≤ C for r large, then M is stochastically complete. (ii) If k(r) ≥ Cr2+ for  > 0 and all r ≥ r0 , then M is stochastically incomplete. These results are achieved by applying criteria for the explosion time of the diffusion in terms of the coefficients of the operator. In [32], the heat kernel is analyzed and, in particular, it is shown that, if the # Ricci curvature of M is bounded from below, then M pt (x, y)dy = 1. This was reproved by using a maximum principle argument to show uniqueness of bounded solutions of the heat equation under the same assumption in [4, Theorem 4.2]. Therefore, any manifold whose Ricci curvature is bounded from below is stochastically complete. This was extended by P. Hsu who #showed that, if κ(r) denotes any ∞ 1 function satisfying κ2 (r) ≥ − inf x∈Br Ric(x), then κ(r) dr = ∞ implies stochastic completeness [16]. Related results were also given by K. Ichihara [17, Theorem 2.1] by comparing with the case of model manifolds, and M. Murata [25, Theorem A] who studied the uniqueness of non-negative solutions of the heat equation. A different type of criterion for stochastic completeness was given by M.P. Gaffney in [11]. Letting r(x) denote the distance from x to a fixed reference point, Gaffney proves that M is stochastically complete if e−cr(·) is integrable on M for all positive constants c. This gives rise to a volume criterion for stochastic completeness. If V (r) denotes the Riemannian volume of a geodesic ball in M ,

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then L. Karp and P. Li, in an unpublished article, showed that V (r) ≤ ecr implies stochastic completeness by studying solutions of the heat equation [20]. A better volume growth condition given by Grigor
yan states that if  ∞ r dr = ∞, (3.1) log V (r) then M is stochastically complete [13] (see also [14, Theorem 9.1]). This criterion was extended to a general setting of local Dirichlet spaces by K.T. Sturm [28, Theorem 4]. That Grigor
yan’s criterion (3.1) is sharp can be seen by considering the case of spherically symmetric or model manifolds Mσ . These are manifolds, homeomorphic to Rn = R+ × S n−1 , which, following the removal of some number of points, have well-defined polar coordinates (r, θ1 , . . . , θn−1 ), and whose Riemannian metric is given by g = dr2 + σ 2 (r)gS n−1 . Here, gS n−1 denotes the standard Euclidean metric on S n−1 and σ is a smooth function satisfying σ(0) = 0 and σ
(0) = 1. In particular, the area of a geodesic sphere is given by S(r) = ωn σ n−1 (r) where ωn is the area of the sphere in Rn . See [14, Section 3] or [12] for details. It can be shown [14, Corollary 6.8] that model manifolds are stochastically complete if and only if  ∞ V (r) dr = ∞. (3.2) S(r) In particular, if one chooses σ(r) so that V (r) ≥ er stochastically incomplete.

2+

for  > 0, then Mσ is

Remark 3.1. Grigor
yan asked [14, Problem 9 on page 238] if the condition (3.2) could replace (3.1) in implying stochastic completeness for general manifolds. In a recent paper, C. B¨ ar and G.P. Bessa give examples of connected sums of model manifolds which satisfy (3.2) but are stochastically incomplete [2, Theorem 1.3].

4. Stochastically incomplete graphs We now begin our discussion of stochastically incomplete graphs. We consider G = (V, E) an infinite, locally finite, connected graph with vertex set V and edge set E. We do not consider the case of multiple edges or loops. We use the notation x ∼ y to indicate that the vertices x and y form an edge and say that x and y are adjacent or neighbors if this is the case. We let m(x) = |{y | y ∼ x}| denote the valence or degree of x, that is, the number of neighbors of x. We equip the graph with the usual metric, that is, d(x, y), the distance between the vertices x and y, is defined as the number of edges in the shortest path connecting x and y. We then fix a vertex x0 and let Sr = Sr (x0 ) = {x | d(x, x0 ) = r} denote the sphere of radius r about x0 . We let S(r) denote thearea of a sphere r of radius r, defined as the number of vertices in Sr , and V (r) = i=0 S(i) denote the volume of a ball of radius r, in analogy with the case of Riemannian manifolds.

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We consider real-valued functions on the vertices of G and, if f is such a function, we define the Laplacian by    ∆f (x) = f (x) − f (y) = m(x)f (x) − f (y). y∼x

y∼x

As mentioned previously, ∆ is positive and essentially self-adjoint on the space of finitely supported functions on V , which is a dense subset of the Hilbert space of square summable functions on V . Furthermore, the Laplacian is a bounded operator if and only if m(x) ≤ K for all vertices x. See, for example, [5,7,21,29,30] for some of the basic properties of the Laplacian. Remark 4.1. Many authors (e.g., [24]) consider a different Laplacian which acts 1 ! (x) = f (x) − 1  on functions on vertices by ∆f y∼x f (y) = m(x) ∆f (x). The m(x) ! is bounded on the space of square summable functions, with a weighted operator ∆ !

inner product, and it can be shown that e−t∆ 1 = 1 for all t > 0, where 1 denotes the function which is 1 on all vertices of G. In particular, the minimal diffusion ! is always stochastically complete [30]. See [21] for other process associated to ∆ ! and [22, 23] for a unified framework. differences between ∆ and ∆ In [7], Dodziuk and Mathai study the heat equation using the maximum principle as in [4]. They show that, if m(x) ≤ K for all vertices x, then bounded solutions of the heat equation on G are uniquely determined by initial data. In particular, all such graphs are stochastically complete. This result was extended in several ways. In [6], Dodziuk applied this technique to a Laplacian with weights. More specifically, consider a weighted graph, that is, a graph where each edge x ∼ y is assigned a positive, symmetric weight  ax,y .  The resulting weighted Laplacian A acts on functions by Af (x) = y∼x ax,y f (x)−  f (y) . In [6, Theorem 4.1], it is shown that, if m(x) ≤ K1 for all vertices x and ax,y ≤ K2 for all edges x ∼ y, then the heat equation involving this Laplacian has unique bounded solutions and, as such, e−tA 1 = 1 for all t ≥ 0. It should be noted that, under these assumptions, the corresponding operators are bounded on the appropriate Hilbert spaces, and it was recently shown by M. Keller and D. Lenz [23, Corollary 27] that generalized Laplacians which are bounded on weighted 2 spaces generate stochastically complete processes. In [29], A. Weber replaced the assumption m(x) ≤ K with a different curvature condition on the graph. Specifically, letting, r(x) = d(x, x0 ) where x0 is a fixed reference vertex, if is shown that ∆r(x) ≥ −C for C ≥ 0 implies that the graph is stochastically complete [29, Corollary 4.15]. To give a geometric interpretation to this last result, let m± (x) = |{y | y ∼ x and r(y) = r(x) ± 1}| denote the number of vertices one step further and closer, respectively, from x0 then is x. It follows that ∆r(x) ≥ −C if and only if m+ (x) − m− (x) ≤ C. Hence, in particular, a graph will be stochastically complete if it is not expanding too much, relative to the number of incoming edges, in all directions.

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These results were obtained by applying a maximum principle to study bounded solutions of the heat equation. Furthermore, the geometric assumptions are imposed at each vertex of the graph. However, the stochastic completeness or incompleteness of a graph should be determined by geometric conditions at infinity as in the Riemannian setting. In [30, 31] a different technique is applied. Specifically, the same question is approached through the study of positive solutions to the difference equation (∆ + λ)v(x) = 0 for a positive constant λ. By (2) in Theorem 2.2, stochastic incompleteness is equivalent to the boundedness of the solution v. This fact was used to obtain a general criterion for stochastic completeness extending the result of Dodziuk and Mathai. Specifically, fixing a  vertex x0 and letting 1 K(r) = maxx∈Sr (x0 ) m(x), Theorem 3.2 in [31] states that, if ∞ r=0 K(r) = ∞, then G is stochastically complete. We now sharpen this result as follows: we let m+ (x) = |{y | y ∼ x and r(y) = r(x) + 1}| as above, and let K+ (r) = maxx∈Sr m+ (x). Theorem 4.2. If

∞  r=0

1 = ∞, K+ (r)

then G is stochastically complete. Proof. Let v > 0 satisfy (∆ + λ)v(x) = 0 for λ  > 0 and  x ∈ G. Let  w(r) = maxx∈Sr v(x). Then, (∆+λ)v(x0 ) = 0 implies that y∼x0 v(y)−v(x0 ) = λv(x0 ). Therefore, as w(0) = v(x0 ),      K+ (0) w(1) − w(0) ≥ v(y) − w(0) = λw(0) y∼x0

so that

λ w(0). (4.1) K+ (0) Now, choose xr ∈ Sr such that w(r) = v(xr ). Then, (∆ + λ)v(xr ) = 0 implies that       v(y) − v(xr ) = v(xr ) − v(y) + λv(xr ). (4.2) w(1) − w(0) ≥

y∼xr y∈Sr+1

y∼xr y ∈Sr+1

Using (4.1) and (4.2), it follows by induction that w(r + 1) − w(r) > 0 for all r ≥ 0. Therefore,      K+ (r) w(r + 1) − w(r) ≥ v(y) − w(r) y∼xr y∈Sr+1

   w(r) − v(y) + λw(r) > λw(r).

=

y∼x y ∈Sr+1

This implies that w(r + 1) − w(r) >

λ λ w(r) > w(0). K+ (r) K+ (r)

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  ∞  1 Therefore, ∞ r=0 K+ (r) = ∞ implies r=0 w(r + 1) − w(r) = ∞ so that w, and, therefore, v, is unbounded.
Remark 4.3. For trees, we have the following complementary result: ∞ if k+ (r) = minx∈Sr m+ (x) > 0, then the tree is stochastically incomplete if r=0 k+1(r) < ∞ [31, Theorem 3.4]. In particular, in the case of spherically symmetric trees Tk , that is, trees which contain a vertex x0 such that k(r) = k+ (r) = K+ (r), we have that Tk is stochastically incomplete if and only if ∞  1 < ∞. k(r) r=0

4.1. Stochastically incomplete subgraphs As mentioned in Remark 2.3, λ-massiveness is preserved under the operation of removing a compact (or finite, in the graph case) subset. In particular, this can be used to show that a graph which contains a stochastically incomplete subgraph with only finitely many vertices adjacent to vertices that are not in the subgraph " with G, considered as a graph on is stochastically incomplete. That is, if G ⊂ G its own, stochastically incomplete and ∂G = {x ∈ G | ∃ y ∼ x with y ∈ G} a finite " is stochastically incomplete. set, then G This result was extended by Keller and Lenz in [22, Theorem 3] where a gen" in terms of G by considering eral condition for the stochastic incompleteness of G " is given. In fact, it should be pointed out, that the Dirichlet Laplacian on G ⊂ G Keller and Lenz consider processes associated to much more general operators of the form L = B + C, where B is a weighted Laplacian and C is a potential, with L acting on an 2 space with an arbitrary measure of full support; furthermore, their graphs are not necessarily locally finite, but we specialize their results to " is a stochastically incomplete our setting. In particular, they show that, if G ⊂ G " is stochastically incomplete. subgraph and supx∈∂G |{y ∼ x | y ∈ G}| < ∞, then G Moreover, they show that every stochastically incomplete graph G is a sub" [22, Theorem 2]. They construct the graph of a stochastically complete graph G " by attaching, to each vertex x in G, m(x)c(x) copies of the graph supergraph G N with vertex set {0, 1, 2, . . .} and edges n ∼ n + 1 for n = 0, 1, 2, . . .. Here, m(x) denotes the valence of the vertex x in G and attaching means that the vertex x is identified with the vertex 0 in N. They show that, if c(x) is chosen so ∞ that r=0 c(x1r ) = ∞ for every sequence of vertices such that xr ∼ xr+1 for all " is stochastically complete. r = 0, 1, 2, . . ., then G We now show that this result is optimal by analyzing the stochastic completeness of the supergraph constructed this way in the case of spherically symmetric trees. As mentioned above, for spherically symmetric trees = k+ (r) = K+ (r) ∞ k(r) 1 and they are stochastically incomplete if and only if r=0 k(r) < ∞. Such trees are determined by the function k and we denote them by Tk . Now, instead of ˜ attaching the graphs N as in [22], we connect k(r) terminal vertices, by which we

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mean vertices of valence 1, to each vertex x ∈ Sr ⊂ Tk . As Keller and Lenz point out in [22], this construction has an equivalent effect (see Remark 4.7 below). We ˜ ˜ denote the resulting graph Tkk . Therefore, each x ∈ Tk ⊂ Tkk which is in Sr has ˜ ˜ k(r) neighbors in Sr+1 ⊂ Tk and k(r) terminal vertex neighbors in Tkk \ Tk . ˜

Theorem 4.4. Tkk is stochastically incomplete if and only if ∞ ˜  k(r) + 1 r=0

k(r)

< ∞. ˜

Proof. Let v > 0 satisfy (∆ + λ)v(x) = 0 for λ > 0 and x ∈ Tkk . For every ˜ y˜ ∈ Tkk \ Tk there exists a unique x ∼ y˜ such that  x ∈ Tk . Then, (∆ + λ)v(˜ y) =   1 v(˜ y ) − v(x) + λv(˜ y ) = 0, implies that v(˜ y ) = 1+λ v(x). In particular, for every ˜

y˜ ∈ Tkk \ Tk with x ∼ y˜ and x ∈ Tk , we have  λ v(x) − v(˜ y) = v(x) = αv(x) 1+λ

(4.3)

λ for α = 1+λ . Now, by (4.3) and by averaging v over spheres in Tk , it suffices to consider positive functions w defined on Tk whose value depends only on the distance from x0 and which satisfy   ˜ ∆Tk + αk(r) + λ w(r) = 0 (4.4) where ∆Tk denotes the Laplacian on Tk and r is the distance from x0 . The sto˜ chastic incompleteness of Tkk is then equivalent to such a function being bounded.     ˜ For r = 0, (4.4) states that k(0) w(0) − w(1) + αk(0) + λ w(0) = 0. Therefore, & ' ˜ αk(0) +λ w(1) − w(0) = w(0). (4.5) k(0)     ˜ + For r > 0, (4.4) states that k(r) w(r) − w(r + 1) + w(r) − w(r − 1) + (αk(r) λ)w(r) = 0. Therefore, & ' ˜  αk(r) +λ 1  w(r + 1) − w(r) = w(r) + w(r) − w(r − 1) . (4.6) k(r) k(r)

Applying (4.5) and (4.6), it follows by induction that w(r + 1) − w(r) > 0 for all r ≥ 0. Therefore, the increment w(r + 1) − w(r) can be estimated as follows: & ' & ' ˜ ˜ αk(r) +λ αk(r) +λ+1 w(r) ≤ w(r + 1) − w(r) < w(r). k(r) k(r) Therefore, &

˜ αk(r) +λ 1+ k(r)

'

& w(r) ≤ w(r + 1) <

˜ αk(r) +λ+1 1+ k(r)

' w(r)

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and iterating this down to r = 0 gives & ' & ' r r   ˜ +λ ˜ +λ+1 αk(i) αk(i) 1+ w(0) ≤ w(r + 1) < 1+ w(0). k(i) k(i) i=0 i=0 ˜

It follows that Tkk is stochastically incomplete if and only if ∞  ∞ ˜   ˜ k(r) +λ k(r) + 1 v is bounded ⇔ w is bounded ⇔ 1+ <∞⇔ < ∞. k(r) k(r) r=0 r=0
˜ Example 4.5. As an example, we let k(r) = (r + 1)2 and k(r) = r + 1 so that ˜ ˜ ˜ k k Tk is stochastically incomplete and Tk ⊂ Tk with Tk complete. Note that, on Tkk , 2 K+ (r) = (r + 1) + (r + 1) while k+ (r) = 0 for r > 0, so that neither of our general results concerning the stochastic completeness of trees apply in this case. ˜ Remark 4.6. If k(r) = 0 for all r ≥ 0, then we recover the result for spherically symmetric trees mentioned previously, that is, Tk is stochastically incomplete if ∞ 1 and only if r=0 k(r) < ∞. Remark 4.7. The only difference between this construction, where we connect terminal vertices to each vertex in Tk , and the one in [22], where one attaches λ paths to infinity, is in the constant α = 1+λ in (4.3). Namely, when one attaches a path to a vertex x ∈ Tk , by identifying that vertex with the vertex 0 in the graph N, then a calculation with difference equations gives that  λ + λ(λ + 2)  v(x) − v(1) = v(x) = βv(x) λ + 2 + λ(λ + 2) and one replaces the constant α in (4.3) with the constant β. We mention this fact to make it clear, as Keller and Lenz in [22] do, that it is not necessary to add terminal vertices to a graph in order to construct the complete supergraph. 4.2. Spherically symmetric graphs We now give a necessary and sufficient condition for the stochastic incompleteness of spherically symmetric graphs and illustrate the result with several examples. We consider graphs with a vertex x0 such that m± (x) = |{y | y ∼ x and r(y) = r(x)±1}| where r(y) = d(y, x0 ), depend only on the distance between x and x0 . We write k± (r) for the common values of m± (x) on Sr . We call such graphs spherically symmetric and denote them by Gk± . For the special case of spherically symmetric trees Tk discussed previously, it follows that k+ (r) = k(r) and k− (r) = 1. Note, that we make no assumptions concerning m0 (x) = |{y | y ∼ x and r(y) = r(x)}| and it will become clear in the course of the proof of the next theorem that these quantities play no role in the stochastic incompleteness of Gk± . This is somewhat surprising as one might expect that by choosing m0 (x) large and making the spheres highly connected one might slow the diffusion down, but this is not the case and we give an interpretation of this in Remark 4.10 below.

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We recall that S(r) denotes the number of vertices in the sphere of radius r about x0 , while rV (r) denotes the number of vertices in the ball of radius r, so that, V (r) = i=0 S(i). Theorem 4.8. Gk± is stochastically incomplete if and only if ∞  r=0

V (r) < ∞. k+ (r)S(r)

Proof. Let v > 0 satisfy (∆+λ)v(x) = 0 for λ > 0 and x ∈ Gk± . By averaging over spheres, it suffices to consider functions depending only of the distance from x0 . That is, if v satisfies the conditions above, then,  using the identities k+ (r)S(r) = 1 k− (r + 1)S(r + 1), it follows that w(r) = S(r) x∈Sr v(x) satisfies (∆ + λ)w(r) = 0 for all r ≥ 0. Note that, it is at this point that the terms involving m0 (x) cancel out. Therefore, we consider positive functions w satisfying (∆ + λ)w(r) = 0 for λ > 0 and r ≥ 0. We claim that r  λ w(r + 1) − w(r) = S(i)w(i). (4.7) k+ (r)S(r) i=0   This follows by induction. For r = 0, we have (∆ + λ)w(0) = k+ (0) w(0) − w(1) + λw(0) = 0 which implies that w(1) − w(0) =

λ w(0). k+ (0)

  Now, for r > 0, (∆ + λ)w(r) = 0 implies that k (r) w(r + 1) − w(r) = +   k− (r) w(r) − w(r − 1) + λw(r). Therefore, if (4.7) holds for w(r) − w(r − 1), then, applying k+ (r − 1)S(r − 1) = k− (r)S(r), we obtain  k− (r)  λ w(r + 1) − w(r) = w(r) − w(r − 1) + w(r) k+ (r) k+ (r) & ' r−1  k− (r) λ λ = S(i)w(i) + w(r) k+ (r) k+ (r − 1)S(r − 1) i=0 k+ (r) =

r−1  λ λ S(i)w(i) + w(r) k+ (r)S(r) i=0 k+ (r)

=

r  λ S(i)w(i) k+ (r)S(r) i=0

establishing (4.7). In particular, (4.7) implies that w(r + 1) > w(r) for all r ≥ 0 so that the increments w(r + 1) − w(r) can be estimated as follows: λV (r) λV (r) w(0) ≤ w(r + 1) − w(r) ≤ w(r). k+ (r)S(r) k+ (r)S(r)

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  ∞  V (r) Therefore, if ∞ r=0 k+ (r)S(r) = ∞, then r=0 w(r + 1) − w(r) = ∞ and w is unbounded. On the other hand,  r   λV (r) λV (i) w(r) ≤ 1+ w(0) w(r + 1) ≤ 1 + k+ (r)S(r) k+ (i)S(i) i=0 so that, if

∞

V (r) r=0 k+ (r)S(r)

< ∞, then w is bounded.




We now illustrate the theorem by giving several examples. Example 4.9. For the case of spherically symmetric trees, k+ (r) = k(r), the branching number, and k− (r) = 1. Furthermore, k+ (r)S(r) = S(r + 1), so that Theorem 4.8 implies that Tk is stochastically incomplete if and only if ∞  r=0

For such trees, S(r) = (4.8) is equivalent to

$r−1 i=0

V (r) < ∞. S(r + 1)

(4.8)

k+ (i) and, by the limit comparison test for series, ∞  r=0

1 <∞ k+ (r)

(4.9)

which was obtained as Corollary 3.8 in [31] and, as a special case, in Theorem 4.4. One can extend these examples by connecting any number of vertices on a particular sphere Sr in Tk and this has no effect on the stochastic completeness. That is, let Tk ⊂ Gk , where Gk is obtained by connecting some number of vertices on each sphere Sr . Then, Theorem 4.8 shows that Gk is stochastically incomplete if and only if ∞  V (r) < ∞. (4.10) S(r + 1) r=0 Furthermore, by applying heat kernel comparison theorems, it follows that the heat kernels on Tk and Gk are equal. Specifically, it is easy to see that the heat kernel on Tk depends only on the distance from x0 , that is, pTt k (x0 , x) = pTt k (r) for Tk k all x ∈ Sr . Then, applying Theorem 3.11 in [31], it follows that pG t (x0 , x) = pt (r) Gk for all x ∈ Sr , where pt (x0 , x) denotes the heat kernel on Gk . This gives another proof of the fact that Gk is stochastically incomplete if and only if Tk is, since Gk and Tk have the same set of vertices. The graphs Gk are an analog of model manifolds and the criterion (4.10) # ∞ V (r) corresponds to the condition S(r) dr < ∞ in (3.2). Specifically, dV (r) := V (r + 1) − V (r) = S(r + 1) plays the role of V
(r) = S(r) from the manifold case. The essential point for the correspondence, apart from the spherical symmetry, is that each vertex in Gk has a unique shortest path connecting it to the origin vertex x0 , which is also the case for model manifolds.

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Remark 4.10. It is surprising that the criteria (4.8) and (4.9) for stochastic incompleteness apply to Gk as well as to Tk . For example, take Tk with k+ (r) = k(r) = (r + 1)2 so that Tk is stochastically incomplete with S(r) = (r!)2 . Now, connect each vertex x ∈ Sr to every other vertex in Sr to obtain Gk . Then, at x ∈ Sr ⊂ Gk , m+ (x) = k+ (r) = (r + 1)2 , while m0 (x) = (r!)2 − 1 so, probabilistically, the particle is much more likely to remain on the sphere Sr than to continue ˜ outwards. On the other hand, Theorem 4.4 shows that adding only k(r) = r+1 terminal vertices at x does have the effect of trapping the particle. This contrast can be understood in light of the following model for the diffusion process governed by the heat kernel. The direction of each jump is chosen randomly with probabilities as in the case of the simple random walk on the graph; however, the holding time of the particle at a vertex is a random variable whose exponential distribution depends on the valence of the vertex. Specifically, if the particle undergoing the diffusion is at a vertex x at time s, then it will jump, after 1 a random time, to one of the neighbors of x with probability m(x) . Furthermore, the probability that it has not jumped from the vertex x at time s + t is e−tm(x) . See [3, 23] for more details on this model of the diffusion process. Therefore, in the example above, although the particle will most likely jump to another vertex on the same sphere, the factorial valence at that vertex implies that the particle is not expected to remain there for long. In the case of adding terminal vertices, the holding time of the particle at a vertex of valence one is expected to be much longer then at a vertex of high valency and this explains why, in this case, the particle is slowed down and explosion does not occur. Example 4.11. Let S(r) be given with S(0) = 1. We then connect each vertex in Sr to every vertex in Sr+1 for all r ≥ 0. Such graphs are spherically symmetric with k+ (r) = S(r + 1) and k− (r) = S(r − 1) and we denote them by GS . Theorem 4.8 then implies that GS is stochastically incomplete if and only if r ∞  i=0 S(i) < ∞. (4.11) S(r + 1)S(r) r=0 This allows us to create many examples of stochastically incomplete graphs with polynomial volume growth. For example, letting S(r) = (r + 1)3 then, by 2 2 (4.11), GS is stochastically incomplete with V (r) = (r+1) 4(r+2) . Moreover, these ∞ 1 examples show that the condition r=0 k+ (r) < ∞ which is sufficient for the stochastic incompleteness of trees is not, in general, sufficient by considering GS with k+ (r) = S(r + 1) = (r + 2)2 . Furthermore, given any graph, not necessarily spherically symmetric, whose spheres satisfy (4.11) for some vertex x0 , one can create a spherically symmetric stochastically incomplete graph by connecting all vertices in Sr to all vertices in Sr+1 for r ≥ 0. This operation, where we add edges but do not change the vertex set to obtain a stochastically incomplete graph from a complete one, is complementary to the procedure from [22] described in Section 4.1, where we added vertices and

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edges to obtain a complete graph from an incomplete one. On the other hand, in Example 4.9 we discussed how adding edges along the same sphere has, in this case, no effect on stochastic completeness. Example 4.12. One can also apply our techniques to the case of the weighted Laplacian as found in [6] and, as a special case, in [22, 23]. For example, consider the weighted graph Na with vertex set V = {0, 1, 2, . . .}, edges r ∼ r + 1 for r = 0, 1, 2, . . ., and edge weights a(r) = ar,r+1 . Consider a function v on the vertices of Na satisfying (A + λ)v(r) = 0 for r = 0, 1, 2, . . . where A denotes λ the weighted Laplacian. It follows that v(1) − v(0) = a(0) v(0) and one shows by induction that r λ  v(r + 1) − v(r) = v(i). a(r) i=0 Therefore, estimating as in Theorem 4.8, we have that v will be bounded if and only if ∞  r <∞ a(r) r=0 which is a special case of Theorem 4.8 with S(r) = 1 and k+ (r) = a(r).

5. Concluding remarks 5.1. Volume growth We have shown that, with respect to volume growth, stochastically incomplete manifolds and graphs exhibit quite different behavior. However, we still believe that a general criterion for stochastic completeness of graphs in terms of volume growth, in analogy to Grigor
yan’s result, should exist. On the other hand, as mentioned previously, in [2], it is shown that the criterion for stochastic completeness of # ∞ V (r) model manifolds, that is, S(r) dr = ∞, does not imply stochastic completeness # ∞ V (r) of general manifolds. Furthermore, it is also shown in [2] that S(r) dr < ∞ does not, in general, imply stochastic incompleteness. We are already in a position to prove the analogous statements for graphs. ∞ V (r) First, we show that the condition r=0 S(r+1) = ∞ does not imply stochastic completeness. For this, take GS with S(r) = (r+1)3 in Example 4.11. This example is stochastically incomplete but V (r) > S(r + 1) for r large. ∞ V (r) To show that r=0 S(r+1) < ∞ does not imply stochastic incompleteness, ˜ ˜ consider Example 4.5 where we take T k with k(r) = (r + 1)2 and k(r) = r + 1. ˜

k

By Theorem 4.4, Tkk is stochastically complete. Let S(r) and V (r) denote the " area of the sphere and volume of the ball in Tk while S(r) and V" (r) denote the r ˜ k 2 corresponding quantities in Tk . Then S(r) = (r!) and V (r) = i=0 S(i) and, ∞ V (r) ˜ " = from Theorem 4.8, we know that r=0 S(r+1) < ∞. For Tkk , we have that S(r)

Stochastic Incompleteness " S(r) r + rS(r − 1) where we set S(−1) = 0. Therefore, V (r) = i=0 iS(i − 1) and we have that  ∞ ∞   V (r) + ri=0 iS(i − 1) V" (r) = " S(r + 1) + (r + 1)S(r) r=0 S(r + 1) r=0 <

∞ 

177 r i=0

" = V (r) + S(i)

∞

 rV (r − 1) V (r) + < ∞. S(r + 1) r=1 (r + 1)S(r) r=0

5.2. Discretization and rough isometries There is a well-known discretization procedure originating in the works of M. Kanai in which a graph is associated to a Riemannian manifold in such a way that many properties of the graph reflect those of the manifold [18, 19]. This procedure is used to show that certain properties of manifolds are preserved by rough isometries. For example, in [19, Theorem 1], Kanai shows that rough isometries preserve the property of a manifold to be parabolic. The proof uses a graph to approximate the manifold, shows that parabolicity is preserved during the discretization, and then shows that rough isometries preserve the parabolicity of graphs. The same technique is used in [18] to show that volume growth is preserved by rough isometries. However, a crucial assumption in Kanai’s construction is that the manifolds and graphs have bounded geometry. For manifolds, this means, in particular, that the Ricci curvature is bounded from below and for graphs this means that the valence is uniformly bounded from above. As we have mentioned previously, these assumptions automatically imply that the spaces in question are stochastically complete and Kanai’s rough isometry scheme does not apply in our case. In par˜ ticular, in Example 4.5 the graphs Tk and Tkk are roughly isometric but Tk is ˜ stochastically incomplete while Tkk is stochastically complete. 5.3. Quantum graphs As mentioned previously, Grigor
yan’s volume criterion (3.1) was extended to the more general setting of local Dirichlet spaces by Sturm [28]. In particular, metric or quantum graphs are covered by this extension. Therefore, we have shown that, with respect to volume growth, solutions of the heat equation behave differently on quantum and discrete graphs. This is also the case for solutions of the wave equation where, in the quantum graph setting, as on manifolds, solutions of the wave equation have finite propagation speed, in contrast to the case of discrete graphs. See [10] for details. Acknowledgment I would like to thank many people with whom I’ve had helpful and inspiring discussions while working on the material in this article. In particular, I would like to thank J´ ozef Dodziuk, Pedro Freitas, Sebastian Haeseler, Matthias Keller, Daniel Lenz, Erin Pearse, Florian Sobieczky, and Jean-Claude Zambrini.

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References [1] Robert Azencott, Behavior of diffusion semi-groups at infinity, Bull. Soc. Math. France 102 (1974), 193–240. MR 0356254 (50 #8725) [2] Christian B¨ ar and G. Pacelli Bessa, Stochastic completeness and volume growth (2009), available at arXiv:0908.4222v1[math.DG]. [3] Kai Lai Chung, Markov chains with stationary transition probabilities, Die Grundlehren der mathematischen Wissenschaften, Bd. 104, Springer-Verlag, Berlin, 1960. MR 0116388 (22 #7176) [4] J´ ozef Dodziuk, Maximum principle for parabolic inequalities and the heat flow on open manifolds, Indiana Univ. Math. J. 32 (1983), no. 5, 703–716. MR 711862 (85e:58140) [5] , Difference equations, isoperimetric inequality and transience of certain random walks, Trans. Amer. Math. Soc. 284 (1984), no. 2, 787–794. MR 743744 (85m:58185) [6] , Elliptic operators on infinite graphs, Analysis, geometry and topology of elliptic operators, World Sci. Publ., Hackensack, NJ, 2006, pp. 353–368. MR 2246774 (2008f:58019) [7] J´ ozef Dodziuk and Varghese Mathai, Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians, The ubiquitous heat kernel, Contemp. Math., vol. 398, Amer. Math. Soc., Providence, RI, 2006, pp. 69–81. MR 2218014 (2007c:81054) [8] William Feller, Diffusion processes in one dimension, Trans. Amer. Math. Soc. 77 (1954), 1–31. MR 0063607 (16,150d) [9] , On boundaries and lateral conditions for the Kolmogorov differential equations, Ann. of Math. (2) 65 (1957), 527–570. MR 0090928 (19,892b) [10] Joel Friedman and Jean-Pierre Tillich, Wave equations for graphs and the edge-based Laplacian, Pacific J. Math. 216 (2004), no. 2, 229–266. MR 2094545 (2005k:05142) [11] Matthew P. Gaffney, The conservation property of the heat equation on Riemannian manifolds., Comm. Pure Appl. Math. 12 (1959), 1–11. MR 0102097 (21 #892) [12] R.E. Greene and H. Wu, Function theory on manifolds which possess a pole, Lecture Notes in Mathematics, vol. 699, Springer, Berlin, 1979. MR 521983 (81a:53002) [13] A.A. Grigoryan, Stochastically complete manifolds, Dokl. Akad. Nauk SSSR 290 (1986), no. 3, 534–537 (Russian). MR 860324 (88a:58209) [14] Alexander Grigoryan, Analytic and geometric background of recurrence and nonexplosion of the Brownian motion on Riemannian manifolds, Bull. Amer. Math. Soc. (N.S.) 36 (1999), no. 2, 135–249. MR 1659871 (99k:58195) [15] R.Z. Hasminski˘ı, Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Teor. Verojatnost. i Primenen. 5 (1960), 196–214 (Russian, with English summary). MR 0133871 (24 #A3695) [16] Pei Hsu, Heat semigroup on a complete Riemannian manifold, Ann. Probab. 17 (1989), no. 3, 1248–1254. MR 1009455 (90j:58158) [17] Kanji Ichihara, Curvature, geodesics and the Brownian motion on a Riemannian manifold. II. Explosion properties, Nagoya Math. J. 87 (1982), 115–125. MR 676590 (84m:58166b)

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[18] Masahiko Kanai, Rough isometries, and combinatorial approximations of geometries of noncompact Riemannian manifolds, J. Math. Soc. Japan 37 (1985), no. 3, 391–413. MR 792983 (87d:53082) [19] , Rough isometries and the parabolicity of Riemannian manifolds, J. Math. Soc. Japan 38 (1986), no. 2, 227–238. MR 833199 (87e:53066) [20] Leon Karp and Peter Li, The heat equation on complete Riemannian manifolds, available at math.uci.edu/~pli/heat.pdf. [21] Matthias Keller, The essential spectrum of the Laplacian on rapidly branching tessellations, Math. Ann. 346 (2010), no. 1, 51–66, DOI 10.1007/s00208-009-0384-y. MR 2558886 [22] Matthias Keller and Daniel Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs (2009), available at arXiv:0904.2985v1[math.FA]. [23] , Unbounded Laplacians on graphs: Basic spectral properties and the heat equation, Math. Model. Nat. Phenom. (to appear). [24] Guia Medolla and Alberto G. Setti, Long time heat diffusion on homogeneous trees, Proc. Amer. Math. Soc. 128 (2000), no. 6, 1733–1742. MR 1694874 (2000j:58048) [25] Minoru Murata, Uniqueness and nonuniqueness of the positive Cauchy problem for the heat equation on Riemannian manifolds, Proc. Amer. Math. Soc. 123 (1995), no. 6, 1923–1932. MR 1242097 (95g:58231) [26] Stefano Pigola, Marco Rigoli, and Alberto G. Setti, A remark on the maximum principle and stochastic completeness, Proc. Amer. Math. Soc. 131 (2003), no. 4, 1283–1288 (electronic). MR 1948121 (2003k:58063) [27] G.E.H. Reuter, Denumerable Markov processes and the associated contraction semigroups on l, Acta Math. 97 (1957), 1–46. MR 0102123 (21 #918) [28] Karl-Theodor Sturm, Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp -Liouville properties, J. Reine Angew. Math. 456 (1994), 173–196. MR 1301456 (95i:31003) [29] Andreas Weber, Analysis of the Laplacian and the heat flow on a locally finite graph (2008), available at arXiv:0801.0812v3[math.SP]. [30] Radoslaw K. Wojciechowski, Stochastic completeness of graphs, Ph.D. Thesis, 2007, arXiv:0712.1570v2[math.SP]. [31] , Heat kernel and essential spectrum of infinite graphs, Indiana Univ. Math. J. 58 (2009), no. 3, 1419 –1442. [32] Shing Tung Yau, On the heat kernel of a complete Riemannian manifold, J. Math. Pures Appl. (9) 57 (1978), no. 2, 191–201. MR 505904 (81b:58041) Radoslaw Krzysztof Wojciechowski Grupo de F´ısica-Matem´ atica Complexo Interdisciplinar da Universidade de Lisboa Av. Prof. Gama Pinto, 2 PT–1649–003 Lisboa, Portugal e-mail: [email protected]

Progress in Probability, Vol. 64, 181–199 c 2011 Springer Basel AG


Generalized Solutions and Spectrum for Dirichlet Forms on Graphs Sebastian Haeseler and Matthias Keller Abstract. In the framework of regular Dirichlet forms we consider operators on infinite graphs. We study the connection of the existence of solutions with certain properties and the spectrum of the operators. In particular we prove a version of the Allegretto-Piepenbrink theorem which says that positive (super)-solutions to a generalized eigenvalue equation exist exactly for energies not exceeding the infimum of the spectrum. Moreover we show a version of Shnol’s theorem, which says that existence of solutions satisfying a growth condition with respect to a given boundary measure implies that the corresponding energy is in the spectrum. Mathematics Subject Classification (2000). Primary 47B39; Secondary 35P05.

1. Introduction The relation between spectrum and generalized solutions for elliptic operators has been studied for many decades. There are two statements which seem to hold in great generality: • For a given energy there exist positive (super)-solutions if and only if the energy does not exceed the infimum of the spectrum. • An energy for which solutions with certain growth restrictions exist is in the spectrum. The first statement is sometimes referred to as the Allegretto-Piepenbrink theorem and the second statement as Shnol’s theorem. In the present work we want to present and prove precise versions of the results mentioned above in the context of regular Dirichlet forms on infinite graphs. These forms lead to nearest neighbor operators with positive potentials on 2 spaces of weighted graphs with arbitrary measure. These operators are also discussed under the name of graph Laplacians or operators on networks. Let us emphasize that we do not need the restriction of local finiteness of the graph. For

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background and basic features we refer the reader to [12, 13]. As mentioned above some of the results proven in this paper are certainly known in some contexts. However to the best of our knowledge the results in the present general and complete form are not found elsewhere. We have tried to present the techniques of proof in a transparent and self-contained way to make the material accessible to readers from various backgrounds. Let us briefly review some of the history of the results mentioned above (see also the corresponding sections for further recent references). The AllegrettoPiepenbrink theorem was first studied in the context of second-order partial differential equations [1, 18, 20] where oscillatory behavior of solutions was investigated. In [3, 8, 27] the question was studied for Riemannian manifolds and a survey for Schr¨ odinger semi-groups is found in [24]. Recently in [16] a result of this type was proven in the general context of strongly local Dirichlet forms. For further discussion and references we refer to [17]. In contrast to the situations discussed so far Dirichlet forms on discrete spaces are not strongly local. The simplest examples of such non-local forms appear along with locally finite graphs and Laplace operators. In this situation the first results can be found in [4, 6] (without weights) and more recently in [7, 5], where weights of the edges and potentials bounded from below are allowed. For infinite matrices with non-negative entries and bounded spectral radius, as they appear for instance in the context of random walks, a corresponding result is often discussed under the name of Perron-Frobenius theorem. For references we refer to [19, 21, 28, 29, 30]. Let us finally mention [9] where the implication that positive solutions cannot exist above the infimum of the spectrum are proven for a very general class of non-local quadratic forms on Rd . Those forms are not required to be local and it is mentioned that the method also works for the graph case. Let us now turn to some of the history of Shnol’s theorem. The classical Shnol’ theorem, see [25], deals with the Schr¨ odinger equation in Rd with a potential bounded from below. It says that the existence of a solution which grows at most subexponentially implies that the corresponding energy lies in the spectrum. On the other hand there exist examples of generalized eigenfunctions where the energy does not belong to the spectrum due to exponential growth of the solution. Later the result was rediscovered in [23, 24]. In [2] a Shnol’ type theorem is proven for strongly local Dirichlet forms. There, the condition of subexponential growth is measured in terms of the so-called intrinsic metric. Confirm also [17] for further background and references. For quantum graphs a Shnol’ theorem is proven in [14]. There it is mentioned that an analogous theorem also holds for combinatorial graphs. The paper is structured as follows. In Section 2 we define the set-up and recall some basic facts about Dirichlet forms on graphs. A version of the AllegrettoPiepenbrink theorem is stated and proven in Section 3. The proof is divided into several parts some of which may be interesting on their own right. In Subsection 3.1 we provide a ground state representation for the forms in our context. Moreover we prove a Harnack inequality and a minimum principle on finite sets in Subsec-

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tion 3.2. Section 4 is devoted to the proof of two versions of Shnol’s theorem. To do so we introduce two notions of boundary in Subsection 4.1. The two versions are stated in the same section as the boundary is introduced and are proven in Subsection 4.2. In Subsection 4.3 we take a closer look at the special case where the Laplace part of the operator is bounded. In this situation it suffices to look for subexponentially bounded functions. This case arguably resembles the original Shnol’ theorem best. Finally in Section 5 we briefly discuss which of our results still hold in a setting of non-regular Dirichlet forms.

2. Definitions and preliminaries Let V be a countable set. For a measure m : V → (0, ∞) let 

2 (V, m) = {u : V → R | |u(x)|2 m(x) < ∞} x∈V

and denote the corresponding scalar product by ·, · and the corresponding norm by ·. Let b : V × V → [0, ∞) be such that (b1) b(x,  y) = b(y, x) for all x, y ∈ V (b2) y∈V b(x, y) < ∞ for all x ∈ V and let c : V → [0, ∞). We can associate a weighted graph (V, b) to b by letting V be the vertex set and saying of vertices x, y ∈ V is connected by an edge with weight b(x, y) whenever b(x, y) > 0. In this case we write x ∼ y. A graph is called locally finite if we have |{y ∈ V | x ∼ y}| < ∞ for all x ∈ V . We say a set W ⊆ V is connected if for all x, y ∈ W there exists a finite sequence of vertices x = x0 , . . . , xn = y in W such that xj ∼ xj+1 , j = 0, . . . , n − 1. We call such a sequence of vertices a path from x to y and n the length of the path. We denote by d(x, y) the infimum of lengths of paths connecting x, y ∈ V . This defines a metric if the graph is connected and is called the graph metric. Note that for the metric it is only relevant whether the values of b(x, y) are zero or non-zero. 2 Let Qmax := Qmax b,c,m : (V, m) → [0, ∞] be given by  1  Qmax (u) = b(x, y)(u(x) − u(y))2 + c(x)u(x)2 , 2 x,y∈V

x∈V

and D(Q ) = {u ∈ (V, m) | Q (u) < ∞}. By (b2) it follows that cc (V ) ⊆ D(Qmax ). Let Q be the restriction of Qmax to max

2

max

·Q



D(Q) = cc (V )

.

where ·Q := ·2 + Qmax (·) and cc (V ) is the space of finitely supported functions. By Fatou’s lemma Qmax is closed and hence every restriction is closable. Thus the form Q is closed by definition of D(Q). Moreover cc (V ) ⊆ D(Q) implies that Q is regular, i.e., D(Q)∩cc (V ) is dense in cc (V ) with respect to the supremum

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norm ·∞ and D(Q) with respect to ·Q . For a discussion in which situation this regularity assumption is needed we refer to Section 5. By polarization we get a bilinear form D(Q) × D(Q) → R which we denote also by Q and which has the mapping rule  1  Q(u, v) = b(x, y)(u(x) − u(y))(v(x) − v(y)) + c(x)u(x)v(x). 2 x,y∈V

x∈V

One can check that (Q, D(Q)) is a Dirichlet form (see [10, Theorem 3.1.1]), i.e., (Q, D(Q)) is closed, C(u) ∈ D(Q) and Q(C(u)) ≤ Q(u) for all u ∈ D(Q) and all normal contractions C. (A normal contraction is a mapping C : R → R satisfying C(0) = 0 and |C(x)− C(y)| ≤ |x−y|, x, y ∈ R.) For background on Dirichlet forms see [10]. By general theory there is a self-adjoint positive operator L : D(L) ⊆

2 (V, m) → 2 (V, m) which represents the form Q. Moreover we let  )  " F := w : V → R | b(x, y)|w(y)| < ∞ for all x ∈ V y∈V

and " Lw(x) :=

1  c(x) b(x, y)(w(x) − w(y)) + w(x), m(x) m(x) y∈V

where, for each x ∈ V , the sum exists by assumption on w. Then L is a restriction " (see [12]). of L In [12] the question is addressed whether the operator L is essentially selfadjoint. Essential self-adjointness on cc (V ) is proven under the geometric assumption (A) Every path (xn ) of pairwise distinct vertices has infinite measure, i.e.,  m(x n) = ∞ n " c (V ) ⊆ 2 (V, m). Note, inf x∈V m(x) > 0 implies (A). For and provided that Lc instance this is the case if m ≡ 1. Moreover (A) implies D(Q) = D(Qmax ). However for the results presented in this paper we do not need essential self-adjointness of the operator nor assumption (A). We call a function w ∈ F" a solution to E ∈ R, if for all x ∈ V " Lw(x) = Ew(x). " Moreover we call a function w ∈ F" a super-solution to E ∈ R whenever Lw(x) ≥ " appears Ew(x). In the following, we will always assume that w ∈ F" whenever Lw in an inequality. We denote the bottom of the spectrum of L by E0 := E0 (L) := inf σ(L). We call E0 the ground state energy. Moreover we call a non-trivial w : V → R " − E0 )w = 0 a ground state. The ground state energy can be obtained such that (L

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by the Rayleigh-Ritz variational formula (see for instance [22]) Q(u) inf 2 = 2, u∈cc (V ) u u where the last equality follows since Q is regular. E0 =

inf

Q(u)

u∈D(Q)

3. The Allegretto-Piepenbrink theorem The theorem we prove in this section connects the existence of positive (super)solutions for an energy with the bottom of the spectrum of the corresponding operator. The first part of the theorem says that if there is a positive supersolution, then the energy lies below or at the bottom of the spectrum. This part is proven by a ground state representation of the Dirichlet form which is presented in Subsection 3.1. The second part shows that for every energy not exceeding the infimum of the spectrum there is a positive super-solution. This will be proven by a Harnack inequality and a corresponding minimum principle which will be given in Subsection 3.2. We have already discussed some of the history of this theorem. Let us also mention that an alternative proof for the Laplacian of a graph without weights and measure m ≡ 1 was recently given in [31]. Theorem 3.1. (Allegretto-Piepenbrink theorem) Let V be connected and infinite and let E ∈ R. Then the following are equivalent: (i) E ≤ E0 . " − E)w ≥ 0. (ii) There exists a non-trivial w : V → [0, ∞) satisfying (L " − E)w ≥ 0. (iii) There exists w : V → (0, ∞) satisfying (L Moreover if the graph (V, b) is locally finite the above is also equivalent to: " − E)w = 0. (iv) There exists w : V → (0, ∞) satisfying (L It is clear that in the case where the graph is finite (i) and (iv) are not equivalent. One might ask whether (i) implies (iv) for infinite graphs which are not locally finite. The following example shows that this is in general not the case. Example. The example is a so-called star graph. Let V = N0 , m ≡ 1, c ≡ 0 and b satisfy (b1), (b2) such that b(k, n) > 0 if and only if either k = 0 or n = 0. Suppose w is a non-negative solution to E ∈ R \ {0}. Then we have for k > 0 ˜ Lw(k) = b(k, 0)(w(k) − w(0)) = Ew(k) and ˜ Lw(0) =

∞ 

b(0, k)(w(0) − w(k)) = Ew(0).

k=1

Thus combining both equalities and dividing by E = 0 we get ∞  w(k) w(0) = − k=1

which implies w ≡ 0 since we assumed w ≥ 0.

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3.1. The ground state representation In this section we show that we can transform the form Q with respect to a positive ground state. Additionally it is sufficient to have a positive super-solution for the ground state energy to prove a respective inequality. One should mention that a ground state representation can be proven in a far more general setting. The situation of certain non-local forms on Rd which is presented [9] can be carried over to our situation as is also pointed out there. Moreover the special case where the underlying graph is Zd is considered in [11]. Let w ∈ F" be positive and define bw : V × V → [0, ∞) by bw (x, y) = b(x, y)w(x)w(y). Obviously (b1) is satisfied and (b2) follows from w ∈ F". Moreover denote the operator of multiplication by a positive function by the function itself. −1 2 Define Qmax := Qmax ·) : 2 (V, m) → [0, ∞] and D(Qmax w w ) = {u ∈ (V, m) | bw ,0,m (w max Qw (u) < ∞} which gives 2  1  u(x) u(y) max Qw (u) = b(x, y)w(x)w(y) − . 2 w(x) w(y) x,y∈V

Since bw satisfies (b1), (b2) and w−1 (cc (V )) = cc (V ) we have cc (V ) ⊆ D(Qmax w ). By the definition and the discussion in the previous section the restriction Qw of Qmax to w D(Qw ) = cc (V )

·Qw

defines a symmetric, non-negative, closed, regular bilinear form. We will employ " and some energy. In this case Qw in the case where w is a (super)-solution to L the potential c : V → [0, ∞) enters the definition of Qw indirectly via w. Proposition 3.2. (Ground state representation) Let w : V → (0, ∞) and E ∈ R be " − E)w = 0. Then for u ∈ D(Q) ∩ D(Qw ) we have such that (L 2

Q(u) = Qw (u) + E u . " − E)w ≥ 0 one still has Q(u) ≥ Qw (u) + E u2 for u ∈ D(Q) ∩ D(Qw ). If only (L Proof. Let u, v ∈ cc (V ) such that u = wv. We employ the property that w is a " = Ew and write out the definition of Lw " solution to Lw    " Eu(x)2 m(x) = Lw(x) v(x)u(x)m(x) x∈V

x∈V

=



x,y∈V

b(x, y)(w(x) − w(y))w(x)v(x)2 +



c(x)u(x)2 .

x∈V

 1

 We write the first sum as two equal parts in the sense of 2 . . . + 12 . . .. In one of these sums we factor out −1 and exchange x and y by renaming and use b(x, y) = b(y, x) by (b1). Since w is a solution and therefore in F", all sums converge

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absolutely. Hence application of Fubini’s theorem is justified and after adding the two sums we obtain    1  ··· = b(x, y)(w(x) − w(y)) w(x)v(x)2 − w(y)v(y)2 + c(x)u(x)2 . 2 x,y∈V

x∈V

We continue with the terms in the first sum. We use v = u/w (w(x) − w(y))(w(x)v(x)2 − w(y)v(y)2 )  2 u(x) u(y) = (u(x) − u(y))2 − w(x)w(y) − . w(x) w(y) Multiplying both terms on the right-hand side by b(x, y) and summing each over x, y ∈ V we get by the calculations above for u ∈ cc (V ) E u2 = Q(u) − Qw (u). Since Q and Qw are regular the statement is true for all u ∈ D(Q) ∩ D(Qw ). In " − E)w ≥ 0 the very first equality in the the case where w is a super-solution (L calculation above turns into ‘≤’ and we also obtain the desired result.
" − E)w ≥ 0. Then Corollary 3.3. Let w : V → (0, ∞), E ∈ R be such that (L E ≤ E0 . Proof. This can be seen easily since Qw is non-negative, cc (V ) ⊆ D(Qw ) and by the preceding proposition E0 =

inf u∈cc (V ),u=1

Q(u) ≥ E +

inf u∈cc (V ),u=1

Qw (u) ≥ E.




3.2. A Harnack inequality In this section we show that the maximum of a non-negative solution on a finite set can be bounded by a constant times its minimum on this set, where the constant is independent of the function and depends continuously on the energy. This inequality then directly implies a minimum principle, which says that a nontrivial, non-negative super-solution is already positive and a pointwise bound for the super-solution. For operators on locally finite graphs a Harnack inequality can be found in [4, 5, 6, 7]. There are more involved Harnack inequalities which are used for heat kernel estimates and which require very restrictive boundedness assumptions on the operator. For a survey of those estimates see [15]. Proposition 3.4 (Harnack inequality). Let W ⊆ V be a finite and connected set. There exists a continuous, monotonically decreasing function CW : R → [0, ∞) " − E)w ≥ 0 we have such that for all super-solutions w : V → [0, ∞) satisfying (L max w(x) ≤ CW (E) min w(x). x∈W

x∈W

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Proof. Let w be a non-negative super-solution to E on W . A simple calculation shows that w is also a super-solution to all E
< E. Let I ⊂ R be the maximal interval such that there exists a non-negative, nontrivial super-solution on W to all values in I. For E
∈ I and w ∈ F" a non-negative, nontrivial super-solution on W , we proceed as follows: Since W is finite w takes its minimum at some y ∈ W and its maximum at some x ∈ W . Let x0 , . . . , xn ∈ W be a path of pairwise distinct " − E
)w(xj ) ≥ 0 vertices from y to x. We have for j = 1, . . . , n − 1 employing (L & '  1 0≤ b(xj , z)(w(xj ) − w(z)) + c(xj )w(xj ) − E
w(xj ) m(xj ) z∈V & ' & '  b(xj , xj+1 ) 1
≤ b(xj , z) + c(xj ) − E w(xj ) − w(xj+1 ), m(xj ) m(xj ) z∈V

where the last inequality follows from w ≥ 0. We conclude & '  1
w(xj+1 ) ≤ b(xj , z) + c(xj ) − m(xj )E w(xj ). b(xj , xj+1 ) z∈V  Note that E
< y (b(x, y) + c(x))/m(x) for all x ∈ V since otherwise by induction we conclude that w ≡ 0 on W , which is a contradiction. Now applying this inequality recursively along the path from x to y we obtain & ' n−1   1 w(x) ≤ b(xj , z) + c(xj ) − m(xj )E
w(y) ≤ CW (E
)w(y), b(xj , xj+1 ) j=0 z∈V

with


CW (E ) := max

min

x,y∈W x0 ∼···∼xn

n−1  j=0

1 b(xj , xj+1 )

&



' b(xj , z) + c(xj ) − m(xj )E




> 0,

z∈V

where the minimum is taken over all paths connecting x and y, x = y. (Since W is finite the maximum is taken over a finite set and thus CW (E
) is finite as well.) The statement that CW is monotonously decreasing on I follows from its definition and hence we can extend CW continuously to R \ I.
There are two corollaries of the Harnack inequality. For a similar minimum principle as the following see also [12, Theorem 8] and [13, Theorem 7]. Corollary 3.5. (Minimum principle) Let W ⊆ V be connected, E ∈ R and w : V → " − E)w ≥ 0 on W . Then either w > 0 or w ≡ 0 on W . [0, ∞) satisfying (L Proof. This is a direct consequence of the Harnack inequality combined with an exhausting argument.


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Remark 3.6. By Corollary 3.3 there are no positive solutions for energies above the infimum of the spectrum. Together with the preceding corollary every non-negative super-solution for E ∈ (E0 , ∞) is trivial. In this sense the Harnack inequality becomes trivial for E ∈ (E0 , ∞). Corollary 3.7. Let V be connected, x0 ∈ V and I ⊂ (−∞, E0 ] be bounded. For all x ∈ V there exists Cx := Cx (x0 , I) > 0 such that for all w : V → [0, ∞) satisfying " − E)w ≥ 0 for some E ∈ I we have w(x0 ) = 1 and (L Cx−1 ≤ w(x) ≤ Cx . Proof. Let x ∈ V be arbitrary and W = {x0 , . . . , xn } a set of vertices which form a path from x0 to x. By Harnack’s inequality there is CW (E) such that for all w satisfying the assumption of the lemma we get CW (E)−1 w(x0 ) ≤ w(x) ≤ CW (E)w(x0 ) = CW (E). Since CW is monotonously decreasing by Proposition 3.4, it takes its maximum at inf I. Hence Cx := CW (inf I) satisfies the assertion.
3.3. Two limiting procedures In this subsection we introduce two limiting procedures. First we show how to obtain a (super)-solution on V given a sequence of (super)-solutions on an exhausting sequence of finite sets. Secondly we show that whenever we have a converging sequence of (super)-solutions and a converging sequence of energies the limiting function is a (super)-solution with respect to the limiting energy. We call a sequence (Wn ) of finite, connected subsets of V an exhausting  sequence if Wn−1 ⊂ Wn for n ∈ N and V = n Wn . Lemma 3.8. Let V be connected, (Wn ) an exhausting sequence and x0 ∈ W0 . For " − E)wn ≥ 0 E ∈ (−∞, E0 ] let wn : V → [0, ∞) be such that wn (x0 ) = 1 and (L " " − E)w ≥ 0 on Wn for all n ∈ N. Then there exists a positive w ∈ F such that (L on V . If in addition the graph is locally finite and the sequence (wn ) satisfies " − E)wn = 0 on Wn for all n ∈ N, then there exists a positive w ∈ F" such that (L " − E)w = 0 on V . (L Proof. Since Cx−1 ≤ wn (x) ≤ Cx for all x ∈ V by Corollary 3.7 the set {wn } is relatively compact in the topology of pointwise convergence. Hence there exists a subsequence wnk converging pointwise to some w. For each x ∈ V there are only " − E)wn (x) ≥ 0 does not hold. By Fatou’s lemma we finitely many n ∈ N where (L get that w is a super-solution. We have w(x0 ) = 1 since wn (x0 ) = 1 for all n ∈ N and thus by Corollary 3.5 we get that w is positive. Suppose the graph is locally finite. By similar arguments as above we obtain for a sequence of solutions (wn ) a positive solution in the limit since we are allowed to interchange limits and sums due to local finiteness. More precisely this is possible because in locally finite graphs all involved sums sum only over finitely many non-zero terms.


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S. Haeseler and M. Keller We now turn to the second limiting procedure.

Lemma 3.9. Let (En ) be a sequence of real numbers converging to E, (wn ) a sequence of non-negative functions on V converging pointwise to a function w " − En )wn ≥ 0 on a subset W ⊆ V . Then (L " − E)w ≥ 0 on W . If in and assume (L " − En )wn = 0 addition the graph is locally finite and the sequence (wn ) satisfies (L " − E)w = 0 on W . on W for all n ∈ N, then (L " − En )wn (x) ≥ 0 for x ∈ W after multiplying by m(x) Proof. We obtainfrom (L and subtracting y b(x, y)wn (y) that     b(x, y)wn (y) ≤  b(x, y) + c(x) − m(x)En  wn (x). y∈V

y∈V

Since all terms are positive we obtain the first statement by Fatou’s lemma. If (wn ) are solutions we have equality in the equation above. Moreover if the graph is locally finite we can interchange the limit and the sum since b(x, y) > 0 for only finitely many y ∈ V . This yields the second statement for every fixed x ∈ V .
3.4. Proof of the Allegretto-Piepenbrink theorem In the proof of Theorem 3.1 we now put the pieces together. The direction which is not immediate is (i)⇒(ii). The idea is to construct a sequence of super-solutions by applying the resolvent (L − E)−1 to non-negative functions which are zero on an exhausting sequence of V . We then take the limits along this sequence to obtain super-solutions and let E tend to E0 . By a diagonal sequence argument we then obtain the result. Proof of Theorem 3.1. Clearly (ii) and (iii) are equivalent by the minimum principle from Corollary 3.5. The implication (ii) ⇒ (i) is proven in Corollary 3.3. We next show that (i) implies (iii). Let (Wn ) be an exhausting sequence of finite subsets of V , (ϕn ) a sequence of non-negative, nontrivial functions in 2 (V, m) with support in V \ Wn . Let x0 ∈ V . As the resolvent is positivity improving it follows that (L − E)−1 ϕn (x0 ) > 0. Define 1 wn(E) := (L − E)−1 ϕn (L − E)−1 ϕn (x0 ) (E)

(E)

for E ∈ (−∞, E0 ). Obviously wn (x0 ) = 1. Moreover wn ∈ D(L) for all E ∈ " (−∞, E0 ), since (−∞, E0 ) lies in the resolvent set of L. Since L is a restriction of L (E) the function wn is a super-solution and since the resolvent is positivity preserving (E) (E) (even positivity improving) we have wn ≥ 0 (even wn > 0). By Lemma 3.8 (E) (E) " − E)w (x) ≥ 0 for all x ∈ V and there is w : V → (0, ∞) such that (L w(E) (x0 ) = 1. This yields the existence of non-negative nontrivial super-solutions on (−∞, E0 ). Now let Em be a sequence in (−∞, E0 ) converging to E0 and denote wm = w(Em ) . By Corollary 3.7 there exists Cx = Cx (x0 , {Em }) for each x ∈ V such that Cx−1 ≤ wm (x) ≤ Cx for all m ∈ N. Hence the set {wm } is relatively

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compact in the topology of pointwise convergence. By the Bolzano-Weierstrass theorem and a diagonal sequence argument there exists a subsequence converging " − E0 )w ≥ 0 on V . Moreover pointwise to a function w. By Lemma 3.9 we have (L w is non-trivial since wm (x0 ) = 1 for all m ∈ N0 . Obviously (iv) implies (iii). We now assume that the graph is locally finite to show that (i) implies (iv). Note that by arguing as in the implication (i) ⇒ (ii), Lemma 3.8 and Lemma 3.9 enable us to construct the desired solution to E.


4. Shnol’s theorem The main idea of the theorems presented in this section is that the spectral values of an operator can be determined by existence of suitable solutions. Clearly for eigenvalues there are corresponding solutions in 2 (V, m), which are even in D(L). However for an energy to be in the spectrum the existence of a Weyl sequence is sufficient. Such a sequence can be constructed whenever a solution satisfies a growth restriction with respect to a ‘boundary measure’ along a sequence of growing sets. To this end we will introduce the ‘boundary measure’. Indeed we have two candidates at hand, one of which will lead to an 2 -condition and the other one to an 1 -condition. The 2 -condition seems to be more natural in view of operator theory. On the other hand the 1 -condition has the more natural definition of boundary from a geometric point of view. Finally in Subsection 4.3 we look at operators with bounded Laplace part and arbitrary positive potential. In this case it suffices to look for subexponentially growing solutions. 4.1. Two notions of boundary and Shnol’s theorem We start with a definition of the boundary. Let A ⊆ V . We set Ac := V \ A and define the boundary ∂A ⊆ A as ∂A := {y ∈ A | ∃x ∈ Ac : x ∼ y}. Note that ∂A∪∂Ac is the set of vertices which are contained in an edge connecting A and Ac . In what follows ∂Ac always means the boundary of Ac and not (∂A)c . Next we define the boundary measures. The first one yields the 2 -condition. Definition 4.1. For A ⊆ V we define the boundary measure   b(y, x)b(y, z) µA : ∂A → (0, ∞], x → . m(y) c y∈∂A z∈∂A

For w : V → R the boundary norm p of w with respect to A is defined as & ' 12   2 2 p(w, A) := w(x) µA (x) + w(x) µAc (x) . x∈∂A

x∈∂Ac

The boundary norm p is an 2 -norm. The measure is determined in some sense by the weight of the paths of length two into the complement of the set A and back. It leads to the following version of Shnol’s theorem.

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Theorem 4.2. (Shnol’s theorem – 2 -version) Let E ∈ R and w ∈ F" be a solution " − E)w = 0. Assume there exists a sequence An ⊆ V , n ∈ N, such to E, i.e., (L that wn := w · 1An ∈ D(Q) and p(w, An ) → 0, wn 

n → ∞.

Then E ∈ σ(L). As mentioned above there is a second notion of boundary which seems to be more natural in a geometric sense. However the corresponding boundary norm is an 1 -norm. Definition 4.3. Let A ⊆ V . We define the boundary measure   12  b(x, y)2  . νA : ∂A → (0, ∞], x →  m(y) c y∈∂A

For w : V → R the boundary norm q of w with respect to A is defined as   q(w, A) := |w(x)|νA (x) + |w(x)|νAc (x). x∈∂A

x∈∂Ac

The corresponding version of Shnol’s theorem is stated below. Theorem 4.4. (Shnol’s theorem – 1 -version) Let E ∈ R and w ∈ F" be a solution " − E)w = 0. Assume there exists a sequence An ⊆ V , n ∈ N such to E, i.e., (L that wn := w · 1An ∈ D(Q) and q(w, An ) → 0, wn 

n → ∞.

Then E ∈ σ(L). The proof consists of three parts. In the first step we estimate the ‘norm’ of " (L − E) applied to a restricted solution by the boundary norms. Second we recall a generalized Weyl criterion for Dirichlet forms. Finally we prove a lemma which allows us to connect the first and the second step. The proof of the theorem is essentially reduced to putting the pieces together. Before we turn to the proof let us briefly discuss the geometric interpretation of the boundary measures introduced above: A well-established way to measure the boundary of a set appears in the context of Cheeger constants, which is used to estimate the bottom of the spectrum, see [4, 5, 6, 13, 30, 31]. To obtain Cheeger’s constant an infimum over finite sets is taken where one divides the measure of the boundary divided by the volume of the set. Let us consider the case of a graph Laplacian without weights, i.e., b(x, y) ∈ {0, 1}, m ≡ 1 and c ≡ 0. Denote by deg the vertex degree of the graph. In this context the measure of the boundary |∂E A| is exactly the number of edges leaving the finite set A. It is a direct calculation that |∂E A| = Q(1A ). From this

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perspective it is reasonable to compare our boundary measures to |∂E A| by looking at q(1A , A) and p(1A , A). To do so let degA (x) be the number of edges which contain x ∈ V and connect A and Ac . Note that degA is zero outside of ∂A ∪ ∂Ac . We have with this notation  Q(1A ) = degA (x), x∈∂A

 p(1A , A) = 





 12 degA (y) ,

x∈∂A y∈∂Ac ,y∼x

q(1A , A) =



1

degA (x) 2 .

x∈∂A

An easy calculation shows that we always have q(1A , A) ≤ Q(1A ) ≤ p(1A , A)2 . As for the converse inequalities note degA (∂Ac )−1 p(1A , A)2 ≤ Q(1A ) ≤ degA (∂A)q(1A , A) where degA (B) := maxy∈B degA (x). In particular q(1A , A), Q(1A ) and p(1A , A)2 are all of the same order whenever there is a uniform upper bound on degA . 4.2. Proof of the Shnol’ type theorems The following lemma is the key estimate for the proof of the two versions of Shnol’s theorem. " − E)w = 0 and A ⊆ V such Lemma 4.5. Let E ∈ R, w ∈ F" be a solution, i.e., (L that wA := w · 1A ∈ D(Q). Then for all v ∈ cc (V ) we have  " − E)wA (x)v(x)|m(x) ≤ min{p(w, A), q(w, A)}v. |(L x∈V

Proof. A direct calculation shows that for x ∈ V   + b(x, y)w(y),  c 1 y∈∂A "  (L − E)wA (x) = b(x, y)w(y), m(x)  −

x ∈ A, x ∈ Ac .

(∗)

y∈∂A

We first show the asserted inequality with respect to p(w, A). By (∗) we have  2   1  " − E)wA 2 =  (L b(x, y)w(y) . m(x) c c B∈{A,A } x∈B

y∈∂B

Here B ∈ {A, Ac } of course means that B is either the set A or the set Ac . We continue to calculate with the inner terms of the sum. We get for B ∈ {A, Ac } and

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x ∈ B by expanding  2   b(x, y)w(y) = y∈∂B c



b(x, y)b(x, z)w(y)w(z).

y,z∈∂B c

Applying the inequality 2ab ≤ a2 + b2 to w(y)w(z) we obtain  1  ··· ≤ b(x, y)b(x, z)(w(y)2 + w(z)2 ) = b(x, y)b(x, z)w(y)2 , 2 c c y,z∈∂B

y,z∈∂B

where the equality follows from the symmetry of the terms after applying Fubini’s theorem. Since all terms are positive this and all further applications of Fubini’s theorem are justified. We also get for B ∈ {A, Ac } and x ∈ B by Fubini’s theorem and since b(x, y) = 0 for y ∈ ∂B and x ∈ B c \ ∂B  1   w(y)2 µB c (y) = b(x, y)b(x, z)w(y)2 m(x) c c x∈B

y∈∂B

y,z∈∂B

Putting this together into the calculation at the beginning we get   " − E)wA 2 ≤ (L w(y)2 µB (y) = p(w, A)2 . B∈{A,Ac } y∈∂B

The desired inequality associated with respect to p(w, A) now follows from the Cauchy-Schwarz inequality. For the inequality associated with respect to q(w, A) we get by (∗), the triangle inequality and Fubini’s theorem     " − E)wA (x)v(x)|m(x) ≤ |(L b(x, y)|w(y)v(x)|. B∈{A,Ac } x∈∂B y∈∂B c

x∈V

Applying the Cauchy-Schwarz inequality and the definition of q yield the statement & '1& ' 12    b(x, y)2 2  ··· ≤ |w(y)| |v(x)|2 m(x) = q(w, A)v. m(x) c c B∈{A,A } y∈∂B

x∈∂B

x∈∂B


The second ingredient for the proof of the Shnol’ theorems is the following Weyl-sequence criterion for Dirichlet forms. It is taken from [26], Lemma 1.4.4 and we include it for completeness. Proposition 4.6. Let h be a closed, semibounded form, H the associated self-adjoint operator and suppose that D0 ⊆ D(h) is dense with respect to  · h . Then the following assertions are equivalent: (i) E ∈ σ(H). (ii) There exists a sequence (un ) in D(h) with un → 1 and sup v∈D0 ,vh ≤1

|h(un , v) − Eun , v| → 0,

n → ∞.

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195

In [26] the proposition is stated with D0 = D(h). The extension to D0 dense in D(h) is of course immediate. The following lemma is a slight generalization of the second part of Proposition 3.3 in [12]. It provides us with the possibility to pair F" and cc (V ) via the form Q. Lemma 4.7. Let w ∈ F" and v ∈ cc (V ). Then we have   " " w(x)Lv(x)m(x) Lw(x)v(x)m(x) = x∈V

x∈V

 1  b(x, y)(w(x) − w(y))(v(x) − v(y)) + c(x)w(x)v(x) = 2 x,y∈V

x∈V

and all sums converge absolutely. In particular if w ∈ D(Q) ∩ F" the term on the right-hand side reads Q(w, v).  Proof. Since w ∈ F" we have y∈V b(x, y)|w(y)| < ∞ for all x ∈ V by definition. This yields for v ∈ cc (V )    |b(x, y)w(x)v(y)| = |v(y)| b(x, y)|w(x)| < ∞. x,y∈V

Moreover by (b2) 

y∈V

|b(x, y)w(x)v(x)| =

x,y∈V

 x∈V

x∈V

|w(x)||v(x)|



b(x, y) < ∞.

y∈V

Hence all sums which appear in the calculation converge absolutely. Now the Lemma is a direct consequence of Fubini’s theorem.
We are now in the position to prove Theorem 4.2 and Theorem 4.4. Proof of Theorem 4.2. Obviously we have wn = w · 1An ∈ F" whenever w is in F" . Moreover we assumed wn ∈ D(Q). Therefore we get by Lemma 4.7 and Lemma 4.5 for v ∈ cc (V )       " |(Q − E)(wn , v)| =  (L − E)wn (x)v(x)m(x) ≤ p(w, An )v ≤ p(w, An )vQ .   x∈V

Thus E ∈ σ(L) follows from Proposition 4.6 and our assumptions.




Proof of Theorem 4.4. By the same arguments as in the proof of Theorem 4.2 |(Q − E)(wn , v)| ≤ q(w, An )vQ and hence E ∈ σ(L) follows from Proposition 4.6 and our assumptions.




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4.3. Bounded Laplacians in the magnifying glass In this paragraph we want to take a closer look at the situation when the Laplace part of the operator L associated to the form Q is bounded. More precisely we do not assume that the potential c is bounded but that there exists Cb > 0 such that  b(x) := b(x, y) ≤ Cb m(x), y∈V

for all x ∈ V . One can show that this is equivalent to L ≤ 2Cb , whenever c ≡ 0. " is bounded on p (V, m) for all p ∈ [1, ∞] Moreover in this case the restriction of L (for details see Section 3 of [13]). Assuming such a finite Cb exists, any energy E that admits a subexponentially bounded solution belongs to the spectrum. Recall that d(·, ·) is the graph metric defined in Section 2. Theorem 4.8. (Shnol’s theorem – bounded Laplace version) Assume there is Cb > 0 such that b(x) ≤ Cb m(x) for x ∈ V . Let w ∈ F" be a solution for some E ∈ R. Assume further that w is subexponentially bounded with respect to the graph metric, i.e., e−αd(·,x0) w ∈ 2 (V, m) for all α > 0 and some fixed x0 ∈ V . Then E ∈ σ(L). For the proof we follow the ideas of [2] for strongly local Dirichlet forms and [14] for quantum graphs. A function J : [0, ∞) → [0, ∞) is said to be subexponentially bounded if for any α > 0 there exists a Cα ≥ 0 such that J(r) ≤ Cα eαr for all r > 0. For the proof of Theorem 4.8 we need the following auxiliary lemma. Lemma 4.9. Let J : [0, ∞) → [0, ∞) be subexponentially bounded and m > 0. Then for all δ > 0 there exist arbitrarily large numbers r > 0 such that J(r+m) ≤ eδ J(r). Proof. Assume the contrary. Then there exists an r0 ≥ 0 such that J(r0 ) = 0 and J(r + m) > eδ J(r) for all r ≥ r0 . By induction we get J(r0 + nm) > enδ J(r0 ) for n ∈ N. This is a contradiction to J(r) ≤ Cα eαr for α(m + 1) < δ and large n.
Proof of Theorem 4.8. By assumption on Cb we have in particular that b(x, y) ≤ Cb m(x) for all x, y ∈ V . This yields for all A ⊆ V and x ∈ V    b(x, y)b(y, z) µA (x) = ≤ Cb b(x, y) ≤ Cb2 m(x). m(y) c c y∈∂A z∈∂A

y∈∂A

Set wn := w·1Bn , where Bn is the distance-n-ball with respect to the graph metric. Note that ∂Bn ⊆ Sn and Sn+1 = ∂Bnc for all n ∈ N. We obtain from the definition of p and the estimate above   p(w, Bn )2 ≤ w(y)2 µBn (y) + w(y)2 µBnc (y) y∈Sn

≤ Cb2

y∈Sn+1



w(y)2 m(y)

y∈∂Sn ∪∂Sn+1

=

Cb2 (wn+1 2

− wn−1 2 ).

Generalized Solutions and Spectrum for Dirichlet Forms Moreover wn 2 =



|eαd(x,x0 ) e−αd(x,x0) w(x)|2 m(x)

x∈Bn

≤ e2αn

197



|e−αd(x,x0 ) w(x)|2 m(x)

x∈Bn

≤e

2αn

e−αd(x,x0) w2

which implies that the function n → wn 2 is subexponentially bounded as well. Let (δn ) be a positive sequence converging to zero. By the previous lemma, for all (n) (n) n there exists a sequence (jk ) with jk → ∞ for k → ∞ such that wj (n) +1 2 ≤ eδn wj (n) −1 2 . k

k

We pick a diagonal sequence which we denote by (nk ) and obtain 2 2 2 p(w, Bnk )2 2 wnk +1  − wnk −1  2 wnk −1  ≤ C ≤ C (eδnk − 1) → 0, k → ∞. b b wnk 2 wnk 2 wnk 2

Applying Theorem 4.2 gives E ∈ σ(L).




5. Non-regular Dirichlet forms – a short discussion The aim of this final section is to review briefly which of the results in this paper still hold when we drop the regularity assumption on (Q, D(Q)). Regularity is always needed when one wants to approximate quantities by functions in cc (V ) and it is therefore a reasonable assumption. We have left this discussion until the end for the sake of convenience and to avoid confusion. However many parts of the results do not depend on the regularity of the form. Clearly the discussion is only relevant when Q = Qmax (for an example which shows that this case can happen see [12, Section 4]). For the following let Q
be a Dirichlet form which is a closed extension of Q, L
the corresponding operator and E0
the corresponding ground state energy. In the Allegretto-Piepenbrink theorem (Theorem 3.1) the implication E ≤ E0
⇒ (ii), (iii) remains true for Q
along with the Harnack inequality (Proposition 3.4) and the minimum principle in (Corollary 3.5). Also the implication E ≤ E0
⇒ (iv) under the assumption of local finiteness still holds. The ground state representation (Proposition 3.2) still holds for u ∈ cc (V ) ∩ D(Q
) ∩ D(Q
w ). However if D(Q) = D(Q
) the space cc (V ) ∩ D(Q
) ∩ D(Q
w ) might be too small to conclude E ≤ E0
from (ii) or (iii). The proof of Shnol’s theorem (Theorem 4.2 and 4.4) still goes through if one additionally assumes wn ∈ D(L
). In order to make our proof work for wn ∈ D(Q
) one has to show the statement of Lemma 4.7 for functions v ∈ D(Q
).

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Acknowledgment The authors are grateful to Daniel Lenz for generously passing on his knowledge about Dirichlet forms as well as giving various hints in several discussions. The second author wants to thank Rupert Frank for sharing the ideas about the ground state transformation for graphs. Moreover the authors are indebted to Rados law Wojciechowski for pointing out some of the literature. The research of the second author was financially supported by a grant from the Klaus Murmann Fellowship Programme (sdw).

References [1] W. Allegretto. On the equivalence of two types of oscillation for elliptic operators. Pac. J. Math. 55, 319–328, 1974. [2] A. Boutet de Monvel, D. Lenz and P. Stollmann, Shnol’s theorem for strongly local Dirichlet forms. Israel J. Math. 173, 189–211, 2009. [3] S.Y. Cheng, S.T. Yau, Differential equations on Riemannian manifolds and their geometric applications. Comm. Pure Appl. Math. 28(3), 1975, 333–354. [4] J. Dodziuk, Difference equations, isoperimetric inequalities and transience of certain random walks, Trans. Am. Math. Soc. 284, 787–794, 1984. [5] J. Dodziuk, Elliptic operators on infinite graphs, Analysis, Geometry and Topology of Elliptic Operators: Papers in Honor of Krzysztof P. Wojciechowski, World Scientific Pub Co, 353–368 2006. [6] J. Dodziuk, L. Karp, Spectral and function theory for combinatorial Laplacians, Geometry of Random Motion, (R. Durrett, M.A. Pinsky ed.) AMS Contemporary Mathematics, Vol 73, 25–40, 1988. [7] J. Dodziuk, V. Mathai, Kato’s inequality and asymptotic spectral properties for discrete magnetic Laplacians. The ubiquitous heat kernel. Contemp. Math., 398, 69–81, Amer. Math. Soc., Providence, RI, 2006. [8] D. Fischer-Colbrie, R. Schoen, The structure of complete stable minimal surfaces in 3-manifolds of nonnegative scalar curvature. Comm. Pure Appl. Math. 33(2), 199– 211, 1980. [9] R.L. Frank, R. Seiringer, Non-linear ground state representations and sharp Hardy inequalities. J. Funct. Anal. 255(12), 3407–3430, 2008. ¯ [10] M. Fukushima, Y. O-shima, M. Takeda, Dirichlet forms and symmetric Markov processes. de Gruyter Studies in Mathematics, 19. Walter de Gruyter & Co., Berlin, 1994. x+392 pp. ISBN: 3-11-011626-X [11] R.L. Frank, B. Simon, T. Weidl, Eigenvalue bounds for perturbations of Schr¨ odinger operators and Jacobi matrices with regular ground states. Comm. Math. Phys. 282(1), 199–208, 2008. [12] M. Keller, D. Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs, preprint (2009), arXiv:0904.2985, to appear in: J. Reine Angew. Math. [13] M. Keller, D. Lenz, Unbounded Laplacians on graphs: Basic spectral properties and the heat equation, Math. Model. Nat. Phenom. 5, no. 4, 198–224, 2010.

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[14] P. Kuchment, Quantum graphs II. Some spectral properties of quantum and combinatorial graphs. J. Phys. A, 38(22), 4887–4900, 2005. [15] T. Kumagai, Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms. Publ. Res. Inst. Math. Sci. 40(3), 793–818, 2004 [16] D. Lenz, P. Stollmann, I. Veseli´c, The Allegretto-Piepenbrink theorem for strongly local Dirichlet forms. Doc. Math. 14, 167–189, 2009. [17] D. Lenz, P. Stollmann, I. Veseli´c, Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms. To appear in: OTAMP 2008 proceedings. (arXiv: 0909.1107) [18] W.F. Moss, J. Piepenbrink, Positive solutions of elliptic equations. Pacific J. Math. 75(1), 219–226, 1978. [19] B. Mohar, W. Woess, A survey on spectra of infinite graphs. Bull. London Math. Soc. 21(3), 209–234, 1989. [20] J. Piepenbrink, Nonoscillatory elliptic equations. J. Differential Equations, 15, 541– 550, 1974. [21] W.E. Pruitt, Eigenvalues of non-negative matrices. Ann. Math. Statist. 35, 1797– 1800, 1964. [22] M. Reed, B. Simon, Methods of modern mathematical physics. IV. Analysis of operators. Academic Press, New York-London, 1978. xv+396 pp. ISBN: 0-12-585004-2 [23] B. Simon, Spectrum and continuum eigenfunctions of Schr¨ odinger operators. J. Funct. Anal., 42, 347–355, 1981. [24] B. Simon, Schr¨ odinger semigroups. Bull. Amer. Math. Soc., 7(3), 447–526, 1982. [25] I.E. Shnol’, On the behaviour of the eigenfunctions of Schr¨ odinger’s equation. Mat. Sb., 42, 273–286, 1957. erratum 46(88), 259, 1957. [26] P. Stollmann, Caught by disorder: A Course on Bound States in Random Media, volume 20 of Progress in Mathematical Physics. Birkh¨ auser 2001. [27] D. Sullivan, Related aspects of positivity in Riemannian geometry. J. Differential Geom. 25(3), 327–351, 1987. [28] D. Vere-Jones, Ergodic properties of nonnegative matrices. I. Pacific J. Math. 22, 361–386, 1967. [29] D. Vere-Jones, Ergodic properties of nonnegative matrices. II. Pacific J. Math. 26, 601–620, 1968. [30] W. Woess, Random walks on infinite graphs and groups. Cambridge Tracts in Mathematics, 138. Cambridge University Press, Cambridge, 2000. xii+334 pp. ISBN: 0521-55292-3 [31] R.K. Wojciechowski, Heat kernel and essential spectrum of infinite graphs. Indiana Univ. Math. J. 58, 1419–1441, 2009. Sebastian Haeseler and Matthias Keller Mathematical Institute Friedrich Schiller University Jena, D-07743 Jena, Germany e-mail: [email protected] [email protected]

Progress in Probability, Vol. 64, 201–226 c 2011 Springer Basel AG


A Geometric Approach to Absolutely Continuous Spectrum for Discrete Schr¨ odinger Operators Richard Froese, David Hasler and Wolfgang Spitzer Abstract. We review a geometric approach to proving absolutely continuous (ac) spectrum for random and deterministic Schr¨ odinger operators developed in [9–12]. We study decaying potentials in one dimension and present a simplified proof of ac spectrum of the Anderson model on trees. The latter implies ac spectrum for a percolation model on trees. Finally, we introduce certain loop tree models which lead to some interesting open problems. Mathematics Subject Classification (2000). 82B44. Keywords. Absolutely continuous spectrum, transfer matrices, hyperbolic geometry, tree graphs.

1. Introduction The study of one-particle Schr¨ odinger operators of the form H = −∆ + q with kinetic energy −∆ and (random) potential q has caught the attention of many researchers over several decades. As an introduction to this topic we recommend the books by Cycon, Froese, Kirsch, Simon[6], by Stollmann [20], and the paper by Kirsch [14]. In the discrete setting, we choose the kinetic energy to be the negative of the adjacency matrix, ∆, of some graph G. The most important example is the d-dimensional regular graph, Zd . Since there are only very few examples of potentials, q, where the spectrum of H is known explicitly we would be content knowing, for instance, the existence of point and absolutely continuous (henceforth ac) spectrum of H, the level statistics of eigenvalues or the long-time behavior under the Schr¨ odinger time evolution. For example, from scattering theory it is well known that if q decays fast enough (that is, if q is integrable) then the spectrum of H = −∆ + q inside the spectrum of −∆ (on Zd , this is the interval [−2d, 2d]) is purely ac and outside this interval the spectrum is pure point.

202

R. Froese, D. Hasler and W. Spitzer

An important model in solid state physics concerns the case when q is a random potential. In the simplest scenario we assume that the values q(v) and q(w) for two different vertices v, w ∈ Zd are chosen independently from an a-priori given probability measure, ν. Let us multiply the potential, q, by the factor a > 0 and interpret a as the disorder parameter. Anderson discovered in 1958 that for large disorder or at large energy the spectrum of Ha = −∆ + a q is pure point. By now there is an extensive literature on this phenomenon which is known as Anderson localization. The proofs are based on the seminal work of Fr¨ ohlich and Spencer [8] and of Aizenman and Molchanov [1]. However, there is currently no proof of the existence of ac spectrum at small disorder (or delocalization) on Z3 . This is considered an outstanding open problem in Mathematical Physics, also known as the extended states conjecture. One valuable contribution to this conjecture might come from replacing the graph Zd by a simpler graph such as a tree and study there extended states (synonymous with ac spectrum) for random potentials. This has indeed been achieved first by Klein [16] in 1998 (and later by Aizenman, Sims, Warzel [2] in 2006) who proved the extended states conjecture on trees. Motivated by Klein’s result we first constructed novel examples of potentials on a tree that produce ac spectrum [9]. Then we reproved a variant of Klein’s result [10]. A simplified version of this proof is presented in Section 5. In order to move somewhat closer to the lattice Zd we consider a random potential on a tree that is strongly correlated instead of independently distributed [11]. We prove that for small correlations (a large part of) the ac spectrum is stable but it is well known that it disappears completely at maximum correlation, see Section 6. In Section 7, we present three models where we add loops to a (binary) tree. It is only the mean-field loop tree model where we can solve the spectrum of the new Laplacian. On top of this Laplacian we add a certain random potential and prove stability of a large ac component. After a short review of some spectral theory we discuss one-dimensional Schr¨ odinger operators. We reprove the stability of the ac spectrum with respect to an integrable potential, a Mourre estimate, and the stability with respect to a square integrable random potential. The proofs follow from simple geometric properties of the M¨ obius transformation (or transfer matrix) with respect to the Poincar´e metric, which controls the spreading of the Green function in terms of the potential. In Section 4, these M¨ obius transformations are generalized to general graphs (including, for instance, Zd and trees), and, like in one dimension, express the Green function as a limit of products of M¨obius transformations.

2. Setup A graph G = (V, E) consists here of a countably infinite set V called the vertex set. E ⊆ V × V is called the edge set and obeys (i) if (v, w) ∈ E then (w, v) ∈ E; (ii) supv∈V |{w ∈ V : (v, w) ∈ E}| < ∞. v, w ∈ V are called nearest neighbors if (v, w) ∈ E.

Absolutely Continuous Spectrum

203

The most important example is the d-dimensional regular lattice, Zd , but we may as well consider the graph with vertex set V = Nd0 (d ∈ N) and edge set  E = {(x, y) ∈ V × V : x − y1 := di=1 |xi − yi | = 1}. Another example of interest is the (rooted) regular tree, Tk , k ∈ N. Here, V = n≥0,0≤j≤kn −1 {(n, j)} ⊂ N20 . Two vertices v = (n, j) and w = (m, ) are nearest neighbors if m = n + 1 and j ∈ k1 { , + 1, . . . , + k − 1} or if n = m + 1 and ∈ k1 {j, j + 1, . . . , j + k − 1}. The vertex 0 is called the root. The graph G = (V, E) determines the adjacency matrix (operator) on 2 (V ) of the graph G with kernel ∆(v, w) given by  1 if (v, w) ∈ E ∆(v, w) := . (2.1) 0 else That is, for φ ∈ 2 (V ), (∆φ)(v) :=





∆(v, w)φ(w) =

w∈V

φ(w) ,

v∈V .

(2.2)

w∈V :(v,w)∈E

Because of the two conditions (i) and (ii) above on the graph G, the adjacency 2 matrix ∆ is a bounded, self-adjoint operator on  the ¯Hilbert space (V ) 2with respect to the standard scalar product φ, ψ := v∈V φ(v)ψ(v) for φ, ψ ∈ (V ). With some abuse of terminology, ∆ is also called the (discrete) Laplace operator or Laplacian. The total energy, H := −∆ + q, of a quantum mechanical particle on the graph G is described here by the kinetic energy being equal to the negative of the adjacency matrix plus a potential energy term given in terms of a bounded function q : V → R. We identify q with the multiplication operator on 2 (V ) by this function q and call H a Schr¨odinger operator. H is then also a bounded, self-adjoint operator on 2 (V ). λ ∈ C is in the resolvent set of H, if the so-called resolvent, Gλ := (H − λ)−1 , of H exists and if Gλ is a bounded operator on 2 (V ). The complement, σ(H), of the resolvent set in C is called the spectrum of H. Since H is bounded and self-adjoint, σ(H) is a closed, bounded subset of R. By the Spectral Theorem (cf. [18, Theorem VII.6]), there exists a family of orthogonal projections, PΩ , on 2 (V ) indexed by the Borel-measurable sets Ω ⊆ R so that  H= t dPt (2.3) R

with Pt := P(−∞,t] = 1(−∞,t] (H), and 1Ω being the indicator function of Ω. The integral on the right-hand side of (2.3) is meant as a Lebesgue-Stieltjes integral so that  φ, Hψ =

R

t dφ, Pt ψ ,

φ, ψ ∈ 2 (V ) .

By setting µφ,ψ (Ω) := φ, PΩ ψ we define a (complex) Borel measure, µφ,ψ , on R, called a spectral measure (of H).

204

R. Froese, D. Hasler and W. Spitzer

Let λ be in the upper half-plane H := {x + iy : x, y ∈ R, y > 0} (more generally, λ in the resolvent set of H). Then, the kernel of the resolvent of H (for v, w ∈ V we set 1v := 1{v} , µv,w := µ1v ,1w ),  dµv,w (t) −1 −1 Gλ (v, w) = (H − λ) (v, w) = 1v , (H − λ) 1w  = , (2.4) R t−λ is called the Green function; the last identity in (2.4) follows from the Spectral Theorem (cf. [18]). In other words, the Green function, Gλ (v, w), is the Borel transform of the spectral measure, µv,w . Note that (by definition) Gλ (·, w) is the unique function φ ∈ 2 (V ) satisfying (H − λ)φ = 1w ,

w∈V .

(2.5)

We Lebesgue-decompose (cf. [18, Theorem I.14]) the probability measure µv := µv,v with respect to the Lebesgue measure on R into its unique absolutely continuous measure, µac,v , and singular measure, µs,v , and write µv = µac,v ⊕ µs,v .

(2.6)

λ ∈ σ(H) is said to be in the absolutely continuous (ac henceforth) or singular spectrum of H, if for a vertex v ∈ V , λ ∈ supp(µac,v ), respectively if λ ∈ supp(µs,v ). We are here only interested in the ac spectrum of H, σac (H). We use a sufficient criterion (see [16, Theorem 4.1], [19, Theorem 2.1]) for λ ∈ σ(H) to be in σac (H), namely that there exists an interval (c, d) # λ and a vertex v ∈ V so that   lim sup sup Gλ+iε (v, v) ≤ C , (2.7) ε↓0

λ∈(c,d)

for some constant C; in fact, (c, d) ∩ σ(H) is then in σac (H). This follows from Stone’s formula, which says that for c, d ∈ R, c < d, and for all φ ∈ 2 (V ),  d 1 lim π Im φ, Gλ+iε φ dλ ε↓0

c

=

1 lim 2πi ε↓0



d

  φ, (H − λ − iε)−1 − (H − λ + iε)−1 φ dλ

c

= 12 φ, (P[c,d] + P(c,d) )φ .

(2.8)

Consequently, if f ∈ Lq ([c, d]) with q > 1 and 1/q + 1/p = 1, then with φ = 1v ,  d   d  p 1/p   1 f (λ) dµv (λ) ≤ f q lim sup  π Im(Gλ+iε (v, v)) ε↓0

c

c

≤ Cf q . Therefore, by duality, dµv (λ) = g(λ)dλ for some g ∈ Lp ([c, d]).

Absolutely Continuous Spectrum

205

A random potential is a measurable function q from a measure space (A, A) into RV , where RV is equipped with the Borel product σ-algebra. In the simplest case there is a single probability measure ν on R which in turn defines a probability measure, P, on (A, A) by requiring the following conditions: (i) P[ω ∈ A : q(ω)(v) ∈ Ω] = ν(Ω) for all Borel set Ω ⊆ R and for all v ∈ V (q is said to be identically distributed);   (N $N (ii) P ω ∈ A : i=1 q(ω)(vi ) ∈ Ωi ) = i=1 P[ω ∈ A : q(ω)(vi ) ∈ Ωi ] for all vi = vj if i = j, for all N ∈ N, and all Borel sets Ωi ⊆ R (q is said to be independently distributed). We will, without loss of generality, always assume that the mean of ν is zero and, to simplify matters, that ν is compactly supported. The random Schr¨ odinger operator H := −∆ + q on 2 (V ) with iid random potential (that is, q satisfying conditions (i) and (ii) above) is called the Anderson Hamiltonian (or model) on the graph G. For Im(λ) > 0, the random Green function, Gλ (v, v), (the dependence on ω ∈ A is tacitly suppressed) is a random variable on H but simply referred to as the Green function. Since the potential is random so is the spectrum of H = −∆ + q. However, Kirsch and Martinelli [15] proved under some (ergodicity) conditions on the graph (V, E) – which are basically1 satisfied for Nd0 and Tk – that the set σac (H) is P-almost surely equal to one specific set. Most of the time, the probability measure, P, is not mentioned explicitly. Let ρλ,v be the probability distribution of Gλ (v, v), that is, ρλ,v (A) := Prob[ Gλ (v, v) ∈ A] for a Borel subset A ⊆ H. In order to prove ac spectrum of H we show, loosely speaking, that the support of ρλ,v does not leak out to the boundary of H as Im(λ) ↓ 0 but that the support stays inside H. More precisely, for a suitably chosen weight function2 w on H (later denoted by cd), a suitably chosen interval (c, d), and some p > 1 we shall prove that (see [10, Lemma 1])  lim sup sup w(z)p dρλ+iε,v (z) < ∞ . (2.9) ε↓0

λ∈(c,d)

H

Let us scale the potential q by the so-called disorder parameter a ≥ 0 and define Ha := −∆ + a q. A version of the extended states conjecture on a graph G can now be formulated as the property whether for a probability measure ν on R and random potential q defined through ν (obeying the above conditions) and for small coupling a > 0, the ac spectrum of Ha is P-almost surely non-empty, possibly equal to σ(−∆). It is widely believed that this conjecture is true on Nd0 for d ≥ 3 but it is well known not to be true in dimension one. We present a proof of the extended states conjecture on the binary tree in Section 5.

we wanted ergodicity to be satisfied we should switch from the rooted graphs Nd0 and Tk to Zd , respectively the unrooted tree. But as much as the ac spectrum is concerned there is no difference and we stick with the rooted graphs. 2 w satisfies Im(z) ≤ Cw(z) for z near the boundary of H with some constant C, see [11, (5)]. 1 If

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3. One-dimensional graph, N0 We recall here the standard method of transfer matrices and prove some simple geometric properties. This is applied to reproving some known results about decaying potentials. Our goal is to bound the diagonal Green function, Gλ (v, v), for λ ∈ H as Im(λ) ↓ 0 as in (2.7). For the sake of simplicity, let us take v = 0. Let φ = (φ0 , φ1 , . . .) with φn := Gλ (0, n). By recalling (2.5), φ satisfies (−∆ + q − λ)φ = 10 .

(3.1)

This is equivalent to the system of equations −φ1 + (q0 − λ)φ0 − 1 = 0 , −φn+1 + (qn − λ)φn − φn−1 = 0 , Let

 An :=

qn − λ −1 1 0

n ≥ 1.

(3.2)

 ,

n ∈ N0 .

(3.3)

An is called a transfer matrix. Clearly, An ∈ SL(2, C), that is, det(An ) = 1. φ satisfies (3.2) if and only if for all n ≥ 0,     φn+1 φ0 = An An−1 · · · A0 . (3.4) φn 1 There is a unique choice of φ0 ∈ C, namely Gλ (0, 0), so that φn , computed from (3.4), yields a vector φ ∈ 2 (N0 ). An equivalent formulation of (3.4) is     φ0 −1 −1 −1 φn+1 = A0 A1 · · · An . (3.5) 1 φn Here we compute φ0 from the likewise unknown vector [φn+1 , φn ]T . Nevertheless, there is a big difference between (3.4) and (3.5) when it comes to computing φ0 . As an example let us consider  the case  without a potential, that is, with q = 0. −λ −1 Since λ ∈ H, the matrix Ai = has an eigenvalue µ1 with |µ1 | < 1 and 1 0 Im(µ1 ) > 0, and another eigenvalue µ2 with |µ2 | = 1/|µ1 | > 1 and Im(µ2 ) < 0. For φ ∈ 2 (N0 ) we have to choose φ0 so that [φ0 , 1]T is an eigenvector to µ1 . Therefore, the left-hand side of (3.4), namely the vector [φn+1 , φn ]T is very sensitive to the choice of the input vector [φ0 , 1]T . In contrast, the left-hand side of (3.5) (for large n) is quite insensitive to the choice of the input vector [φn+1 , φn ]T . Here, the large n behavior is dominated by the large eigenvalue µ2 , and [φn+1 , φn ]T must not lie in the eigenspace to the eigenvalue µ1 . It is convenient to rewrite the system of equations (3.5), and define for φ = (φn )n∈N0 the sequence α = (αn )n∈N0 with αn :=

φn , φn−1

φ−1 := 1 .

(3.6)

Absolutely Continuous Spectrum

207

Note that for λ ∈ H, φn = 0: For otherwise, λ ∈ H would be an eigenvalue with eigenfunction φ of the self-adjoint operator H restricted to {n, n + 1, . . .} (with Dirichlet boundary condition at n). Let Φn : H → H be the M¨obius transformation associated with the transfer matrix A−1 n . That is, 1 Φn (z) := − . (3.7) z + λ − qn Then (3.5) is equivalent to φ0 = Φ0 ◦ Φ1 ◦ · · · ◦ Φn (αn+1 ) .

(3.8)

The numbers αn can be interpreted as the Green function of the graph N0 truncated at n. To this end, let Nn := {n, n+1, . . . } and En := {(k, k+1), (k+1, k), k ≥ (t) n}. If ∆n denotes the adjacency matrix for the truncated graph (Nn , En ), then αn = Gλ (n, n) := (−∆n(t) + q − λ)−1 (n, n) , (t)

n ∈ N0 ,

(3.9)

and we have the recursion αn = Φn (αn+1 ) ,

n ∈ N0 .

(3.10)

This can be seen from the above equations but we will re-derive this later, see formula (4.5). We equip the upper half-plane H with the hyperbolic (or Poincar´e) metric d, that is,  |z1 − z2 |2  d(z1 , z2 ) := cosh−1 1 + 12 , z 1 , z2 ∈ H , (3.11) Im(z1 )Im(z2 ) or alternatively with the Riemannian line element (see also (4.10) and (4.11)),  dx2 + dy 2 ds = , z = x + iy ∈ H . (3.12) y Proposition 3.1 ([9], Proposition 2.1). (i) For Im(λ) ≥ 0, Φn is a hyperbolic contraction on (H, d), that is, for z1 , z2 ∈ H, d(Φn (z1 ), Φn (z2 )) ≤ d(z1 , z2 ) . (ii) For Im(λ) > 0, Φn (H) ⊂ {z ∈ H : |z| < 1/Im(λ)}. Furthermore, Φn is a strict hyperbolic contraction. That is, for z1 , z2 ∈ H with max{|z1 |, |z2 |} < C there exists a constant δ < 1, e.g., δ := C/(C + Im(λ)), depending on Im(λ) and C so that d(Φn (z1 ), Φn (z2 )) ≤ δ d(z1 , z2 ) . The basic idea is to factor Φn = ρ ◦ τn into the rotation (around the point i and angle π) ρ : z → −1/z and the translation τn : z → z + λ − qn . ρ is a hyperbolic isometry. If Im(λ) > 0 then τn is a strict hyperbolic contraction in the sense that d(τn (z1 ), τn (z2 )) < d(z1 , z2 ) as can be seen directly from definition (3.11). If Im(λ) = 0, then also τn is an isometry. The properties claimed in (i) and (ii) follow from straightforward calculations.


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R. Froese, D. Hasler and W. Spitzer

If Im(λ) > 0 then Φn shifts the upper half-plane upwards. Even more so (recall that the potential q is bounded) we have Proposition 3.2 ([9], Proposition 2.2). For Im(λ) > 0 there exists a hyperbolic disk B ⊂ H so that Φn−1 ◦ Φn (H) ⊂ B. This allows us to state precisely our claim about the stability of our way to compute the Green function. Theorem 3.3 ([9], Theorem 2.3). Let Im(λ) > 0 and let (γn )n∈N be an arbitrary sequence in H. Then we have lim Φ0 ◦ Φ1 ◦ · · · ◦ Φn (γn ) = φ0 = Gλ (0, 0) .

n→∞

(3.13)

Proof. Set wn := Φ0 ◦Φ1 ◦· · ·◦Φn(γn ). Let B be a disk as in Proposition 3.2, and let β := Φn−1 ◦Φn (γn ). Then β ∈ B. The same is true for β
:= Φn−1 ◦Φn ◦Φn+1 (γn+1 ). All further images Φk (β) and Φk (β
) stay in B and the conditions from Proposition 3.1(ii) are fulfilled. Hence we have d(wn+1 , wn ) = d(Φ0 ◦ · · · ◦ Φn−2 (β
), Φ0 ◦ · · · ◦ Φn−2 (β)) ≤ δ d(Φ1 ◦ · · · ◦ Φn−2 (β
), Φ1 ◦ · · · ◦ Φn−2 (β)) ≤ δ n−1 d(β
, β) = C δn . (wn )n∈N is therefore a Cauchy sequence and converges to some w ∈ H. Let (γn
)n∈N be another sequence in H. Then we have analogously d(Φ0 ◦ · · · ◦ Φn−1 ◦ Φn (γn ), Φ0 ◦ · · · ◦ Φn−1 ◦ Φn (γn
)) ≤ C δ n−1 d(Φn−1 ◦ Φn (γn ), Φn−1 ◦ Φn (γn
)) ≤ C δ n−1 . Therefore also Φ0 ◦ · · · ◦ Φn−1 ◦ Φn (γn
) converges to w as n → ∞. Because of (3.8), w = φ0 = Gλ (0, 0).
Proposition 3.4 ([9], Lemma 4.5). Let K be a compact subset of C whose elements have non-negative imaginary parts. For every λ ∈ K, let (zn (λ))n∈N be a sequence in H. Suppose that there exist constants C1 , C2 so that    d Φn+1 (zn+1 (λ)), zn (λ) ≤ C1 (3.14) n≥1

and

  d z1 (λ), i ≤ C2 for all λ ∈ K. Then there exists a constant C3 so that for all λ ∈ K   d Gλ (0, 0), i ≤ C3 .

(3.15) (3.16)

Potentials for which we can find such sequences (zn (λ))n∈N yield ac spectrum for λ ∈ Re(K), and pure ac spectrum for λ ∈ int(Re(K)), the interior of the real part of K.

Absolutely Continuous Spectrum

209

 Proof. Because of Theorem 3.3 there exists an n ∈ N so that d Gλ (0, 0), Φ0 ◦ · · · ◦ Φn (zn ) ≤ 1. Then using the triangle inequality for the Poincar´e metric d and the contraction property of Φn we get (suppressing the dependence of zn on λ),       d Gλ (0, 0), i ≤ d Gλ (0, 0), Φ0 ◦ · · · ◦ Φn (zn ) + d Φ0 ◦ · · · ◦ Φn (zn ), i   ≤ d Φ0 ◦ · · · ◦ Φn−1 (Φn (zn )), Φ0 ◦ · · · ◦ Φn−1 (zn−1 )   + d Φ0 ◦ · · · ◦ Φn−1 (zn−1 ), i + 1     ≤ d Φn (zn ), zn−1 + d Φ0 ◦ · · · ◦ Φn−1 (zn−1 ), i + 1 ≤ ... ≤

n−1 

    d Φk+1 (zk+1 ), zk + d Φ0 (z1 ), i + 1

k=1

  ≤ C1 + d Φ0 (z1 ), i + 1 := C3 .




Examples. 1 (i) Zero potential: Here, Φn (z) = − z+λ . For λ ∈ H, let z+ (λ) ∈ H be the fixed point of Φn , that is, 1 z+ (λ) = − . (3.17) z+ (λ) + λ

Using Theorem 3.3 with γn = z+ (λ) we get φ0 = Gλ (0, 0) = z+ (λ). We have  z+ (λ) = −λ/2 + i 1 − λ2 /4 . (3.18)  z− (λ) := −λ/2−i 1 − λ2 /4 is the second solution to the fixed point equation (3.17), but it lies in the lower half-plane. z± (λ) are also the two eigenvalues of the transfer matrix. z+(λ) and z− (λ) are the stable respectively unstable eigenvalue of this matrix. For λ ∈ R, z+ (λ) ∈ H if and only if |λ| < 2. Therefore, σ(−∆) = σac (−∆) = [−2,  2]. (ii) Short-range potential q, that is, n |qn | < ∞: We choose the constant sequence (zn )n∈N with zn := z+ (λ) for n ∈ N. Then we have  1 d(Φn (zn ), zn ) = d zn + λ − qn , − zn     = d λ/2 + i 1 − λ2 /4 − qn , λ/2 + i 1 − λ2 /4 ≤ C |qn | . By Proposition 3.4, [−2, 2] ⊆ σac (−∆ + q), and on (−2, 2) the spectrum is purely ac.  (iii) A Mourre estimate: Suppose that n≥1 |qn+1 − qn | < ∞. Choose now zn for  n ≥ k to be the fixed point of the map Φn . Then zn = −(λ − qn )/2 + i 1 − (λ − qn )2 /4. zn ∈ H if |λ − q∞ | < 2 with q∞ := limn→∞ qn and k large enough. For 1 ≤ n < k choose arbitrary points in H. Then we have d(Φn+1 (zn+1 ), zn ) = d(zn+1 , zn ) ≤ C |qn+1 − qn | .

210

R. Froese, D. Hasler and W. Spitzer By Proposition 3.4, [−2 + q∞ , 2 + q∞ ] ⊆ σac (−∆ + q). Note, for instance, that by this Mourre estimate, a monotone potential decaying to zero always has pure ac spectrum inside (−2, 2).

By allowing the potential to be random, the decay conditions on the potential can be weakened to guarantee ac spectrum. In one dimension, the 1 -condition can then be replaced by an 2 -condition. Theorem 3.5 ([12], Theorem 1). Let q = (qn )n∈N0 be a family of centered, independent, real-valued random variables with corresponding probability measures νn and  all with support in some compact set K. Suppose that E[ n≥0 |qn |2 ] < ∞, where E 1 is the expectation with respect to the product measure, ν = n≥0 νn . Then almost surely, [−2, 2] is part of the ac spectrum of H, and H is purely ac on (−2, 2). Remarks 3.6. (i) Deylon-Simon-Souillard [7] have proved Theorem 3.5 in 1985 even without assuming compact support of the probability measure. Furthermore, they proved that if C −1 nρ ≤ E[|qn |2 ] ≤ Cnρ for some constant C and ρ < 1/2, then the spectrum of H = −∆ + q is pure point (almost surely) with exponentially localized eigenfunctions. (ii) In [12], we have extended Theorem 3.5 to matrix-valued potentials, and applied to (random) Schr¨odinger operators on a strip. (iii) On the two-dimensional lattice N20 , Bourgain [4] proved σac (∆+q) = σ(∆) for centered Bernoulli and Gaussian distributed, independent random potentials with supn∈N20 E[qn2 ]1/2 |n|ρ < ∞ for ρ > 1/2. In [5], Bourgain improves this result to ρ > 1/3.  Proof of Theorem 3.5. For λ ∈ (−2, 2) let zλ := −λ/2 + i 1 − λ2 /4 be the (truncated) Green function of the Laplace operator −∆, see (3.17). Let us introduce the weight function |z − zλ |2 cd : H → (0, ∞) , z → . (3.19) Im(z) By Proposition 3.2 there is a disk B ⊂ H so that z0,n := Φ0 ◦ · · · ◦ Φn (zλ ) ∈ B for all n ≥ 2 and potentials q with values in a compact set K. Moreover, by Theorem 3.3, Gλ (0, 0) = limn→∞ z0,n . Hence, by the continuity of the function cd, we have limn→∞ cd2 (z0,n ) = cd2 (Gλ (0, 0)). Since cd2 is bounded on the disk B we conclude that E(cd2 (Gλ (0, 0))) = limn→∞ E(cd2 (zn )). It remains to show that this limit is bounded. To this end, we define the rate of expansion, µ(z, qn ) :=

cd2 (Φn (z)) + 1 . cd2 (z) + 1

(3.20)

Noticing that cd(Φn (z)) = |z − zλ − qn |2 /Im(z + λ) and using |qn | ≤ C to bound cubic and quartic terms of q in terms of quadratic ones, we obtain that µ(z, qn ) ≤ A0 (z) + A1 (z)qn + A2 (z)qn2

(3.21)

Absolutely Continuous Spectrum

211

with rational functions Ai (z). The functions A1 and A2 are bounded and A0 ≤ 1. Let us set z
,n := Φ
◦ Φ
+1 ◦ · · · ◦ Φn (zλ ). Note that z
,n = Φ
(z1,n ). By the recursion relation (3.10), E[cd2 (z0,n )] + 1   2  = cd (z1,n ) + 1 dν0 (q0 ) · · · dνn (qn ) K n+1



 cd2 [Φ0 (z1,n )] + 1  2 cd (z1,n ) + 1 dν0 (q0 ) · · · dνn (qn ) 2 cd (z1,n ) + 1 K n+1      2  2 ≤ 1 + A1 (z1,n )q0 + C0 q0 dν0 (q0 ) cd (z1,n ) + 1 dν1 (q1 ) · · · dνn (qn ) K Kn     2  2 cd (z1,n ) + 1 dν1 (q2 ) · · · dνn (qn ) = 1 + C0 E[q0 ] =

Kn

≤

n  

1+



C0 E[qi2 ]

i=0

  ∞ 2 ≤ exp C0 E[qi ] < ∞ .




i=0

4. General graphs We generalize the approach of the previous section to calculating the Green function via transfer matrices (or rather M¨obius transformations) to general graphs G = (V, E), that is, to all graphs that obey the conditions (i) and (ii) of Section 2. Let us choose a point in V which we denote by 0. If dist(v, w) is the graphical distance between the two lattice points v and w then we define the nth sphere, Sn := {v ∈ V : dist(v, 0) = n} . (4.1) 2 2 Clearly, V = n≥0 Sn and 2 (V ) = n≥0 (Sn ). We decompose the adjacency matrix, ∆, of G into the block matrix form   D0 E0T 0 ··· ··· ···  E0 D1 E1T 0 ··· ···   T ∆= 0 , (4.2) E D E 0 ··· 1 2 2   .. .. .. .. .. .. . . . . . . 

where Dn is the adjacency matrix of G restricted to Sn . En : 2 (Sn ) → 2 (Sn+1 ) is the map with kernel  1 if v ∈ Sn , w ∈ Sn+1 , (v, w) ∈ E En (v, w) = . 0 else 2 The potential q = n≥0 qn is diagonal; qn equals the restriction of q to the sphere Sn which is now considered a |Sn |-dimensional diagonal matrix. H = −∆ + q is

212

R. Froese, D. Hasler and W. Spitzer

then of the block matrix form  −D0 + q0 −E0T  −E0 −D1 + q1  H= 0 −E1  .. .. . .

0 −E1T −D2 + q2 .. .

··· 0 −E2T .. .

··· ··· 0 .. .

 ··· ···  . ···  .. .

(4.3)

Let Pn : 

2 (V ) → 2 (Sn ) be the orthogonal projection of 2 (V ) onto 2 (Sn ), and Pn,∞ := k≥n Pk . Then we define the truncated Hamiltonian Hn := Pn,∞ H Pn,∞

(4.4)

and the truncated Green function Gλ (n, n) := Pn (Hn − λ)−1 Pn , (t)

n ∈ N0 .

(t)

(4.5) (t)

Gλ (n, n) is a dn × dn -dimensional matrix with dn = |Sn |. By definition, Gλ (0, 0) equals the Green function, Gλ (0, 0). Furthermore (assuming as usual λ ∈ H), (t)

Gλ (n, n) ∈ SHdn ,

(4.6)

where SHd := {Z = X + iY : X, Y ∈ Mat(d, R), X = X T , Y > 0} is the so-called Siegel half-space. Clearly, SH1 = H. (t) The matrices Gλ (n, n) generalize the numbers αn ∈ H from (3.6). More precisely, let Φn : SHdn+1 × Mat(dn , R) × H → SHdn be defined as Φn (Z, qn , λ) := −(EnT ZEn + Dn − qn + λ)−1 . Then in analogy with (3.9) we have  (t)  (t) Gλ (n, n) = Φn Gλ (n + 1, n + 1), qn , λ .

(4.7)

(4.8)

The proof is simply based upon Schur’s (or Feschbach’s) formula  −1   A BT (A − B T C −1 B)−1 (B T C −1 B − A)−1 B T C −1 = , (4.9) B C C −1 B(B T C −1 B − A)−1 (C − BA−1 B T )−1 by setting A := −Dn + qn − λ, B := (En , 0, . . .) and C := Hn+1 − λ. On SHn , we do not use the standard Riemann metric but a so-called Finsler metric. To this end, let W ∈ Mat(n, C) be an element of the tangent space at Z = X + iY ∈ SHn . Then we set FZ (W ) := Y −1/2 W Y −1/2  ,

(4.10)

where  ·  is the operator norm (rather than the Hilbert-Schmidt norm). [If n = 1 then the length of the tangent vector is |W |/Y as in (3.12).] The Finsler metric on SHn is defined as (thereby suppressing the dimension n)  1   ˙ dt , Z1 , Z2 ∈ SHn , d(Z1 , Z2 ) := inf FZ(t) Z(t) (4.11) Z(t)

0

whereby Z(t) runs through all differentiable paths Z : [0, 1] → SHn with Z(0) = Z1 , Z(1) = Z2 .

Absolutely Continuous Spectrum

213

The Propositions 3.1, 3.2, and 3.4 can be extended to general graphs, see [9, Proposition 3.3, Lemma 3.5, Lemma 4.5]. For instance, for a fixed potential q and fixed λ ∈ H, the transformation Φn is a contraction from (SHdn+1 , d) into (SHdn , d). Theorem 3.3 generalizes as follows. Theorem 4.1 ([9], Theorem 3.6). Let us assume that the matrices Ei in (4.2) all have kernel {0}. Let Im(λ) > 0 and let Zi ∈ SHdi with di = |Si | be an arbitrary sequence. Then for a bounded potential q we have lim Φ0 ◦ · · · ◦ Φn (Zn+1 , qn , λ) = φ0 = Gλ (0, 0) .

n→∞

(4.12)

5. Trees Let us consider for simplicity the (rooted) binary tree, T2 . The recursion relation (4.8) is very simple since diagonal matrices are mapped into diagonal matrices. Hence, the truncated Green functions (or rather matrices) are diagonal by Theorem 4.1. Let qn = diag(qn,1 , . . . , qn,2n ) be the diagonal matrix with diagonal real-valued entries qn,1 , . . . , qn,2n and Z = diag(z1 , . . . , z2n+1 ) a diagonal matrix in SH2n+1 , that is, with zi ∈ H. Then     (5.1) Φn Z, qn , λ = diag Ψ(z1 , z2 , qn,1 , λ), Ψ(z3 , z4 , qn,2 , λ), . . . with the map Ψ : H2 × R ×H → H defined as Ψ(z1 , z2 , q, λ) := −1/(z1 +z2 + λ−q). Now put q = 0 and consider the fixed point equation 1 Ψ(z, z, 0, λ) = − = z. (5.2) 2z + λ  The two solutions are obviously −λ/4 ± λ2 /16 −√1/2. For λ ∈ R, they have non-zero imaginary component if and only if |λ| < 2 2. We choose  zλ := −λ/4 + i 1/2 − λ2 /16 (5.3) for the solution in H. Furthermore, for n ∈ N0 let Zn := diag(zλ , . . . , zλ ) ∈ SH2n . Then     Φn Zn+1 , 0, λ = diag Ψ(zλ , zλ , 0, λ), . . . = Zn , n ∈ N0 . Theorem 4.1 then shows that Gλ (0, 0) = zλ for the rooted binary tree. Hence, √ √ σ(∆) = σac (∆) = [−2 2, 2 2] . Let us consider the Anderson model, Ha = −∆ + a q, on this tree with iid random potential q, which is determined by a probability measure, ν. For simplicity, we assume that ν has compact support. √ Theorem 5.1 ([10], Theorem 1). For every |λ| < 2 2 there is an a0 > 0 so that for all 0 ≤ a ≤ a0 almost surely σac (Ha ) ∩ (−λ, λ) = (−λ, λ) ,

σs (Ha ) ∩ (−λ, λ) = ∅ .

(5.4)

214

R. Froese, D. Hasler and W. Spitzer

This has been proved first by Klein [16] in 1998. The statement σac (Ha ) ∩ (−λ, λ) = ∅ has been proved by Aizenman, Sims, and Warzel [2, 3] in 2005. The following proof is shorter than our first one presented in [10] since we now work directly with the Green function instead of the sum of Green functions, which simplifies the analysis of the functions µ2,p and µ3,p considerably. √ Sketch of Proof. Let λ ∈ H with |Re(λ)| < 2 2. The truncated Green function, (t) Gλ,a (n, n), is an SH2n -valued random variable with range inside the diagonal ma12n trices. In fact, its probability distribution equals i=1 ρa , where, for short, ρa is the probability distribution of Gλ,a (0, 0) := (Ha − λ)−1 (0, 0). Using the recursion relation (4.8) we see that ρa equals the image measure (ρa × ρa × νa ) ◦ Ψ−1 with the function Ψ as in (5.1). Now we define a moment of ρa that we need to control as Im(λ) ↓ 0, that is, we are seeking a uniform bound of Mp (ρa ) below as Im(λ) ↓ 0. As in (3.19) but with zλ from (5.3), let us introduce the weight function cd(z) :=

|z − zλ |2 , Im(z)

Then we define for some p > 1 Mp(ρa ) :=

z ∈ H.

(5.5)

cdp (z) dρa (z) .

(5.6)

 H

Applying the recursion relation we get    Mp (ρa ) = cdp Ψ(z1 , z2 , q, λ) dρa (z1 )dρa (z2 )dνa (q) (5.7) H2 ×R     cdp Ψ(z1 , z2 , q, λ)  1 p p 1 = p p 1 1 2 cd (z1 ) + 2 cd (z2 ) dρa (z1 )dρa (z2 )dνa (q) . 2 H ×R 2 cd (z1 ) + 2 cd (z2 )    =:µ2,p (z1 ,z2 ,q,λ)

√ For zi ∈ H and yi := Im(zi ), let ui := (zi − zλ )/ yi ∈ C. Then cd(zi ) = |ui |2 .    Using u := (u1 , u2 ) and v := y1 /(y1 + y2 ), y2 /(y1 + y2 ) we obtain  2 |z1 + z2 − 2zλ |2 |z1 − zλ + z2 − zλ |2 < 12 = 12 u, v . y1 + y2 + Im(λ) y 1 + y2 By the Cauchy-Schwarz inequality and the strict convexity of x → xp for p > 1 we see that   2 p 1   2 u, v µ2,p (z1 , z2 , 0, λ) ≤ 1 ≤ 1. (5.8) |u1 |2p + 12 |u2 |2p 2 For Im(λ) = 0, the function µ2,p (z1 , z2 , 0, λ) = 1 if and only if u = sv for some s ∈ C and if√|u1 | √ = |u2 |. The function (z1 , z2 , λ) → µ2,p (z1 , z2 , 0, λ) is continuous 2 on C × (−2 2, 2 2) except at u = 0. This implies that in order to have equality, z1 and z2 have to be of the form z1 = x1 + iy, z2 = x2 + iy with |x1 + λ/4| = |x2 + λ/4|. Obviously, we cannot expect that µ2,p (z1 , z2 , q, λ) ≤ 1 − µ < 1 for cd(Ψ(z1 , z2 , 0, λ)) =

1 2

Absolutely Continuous Spectrum

215

z1 , z2 in a neighborhood of the boundary of H2 with a constant µ and q in an interval I # 0. For # the sake of the argument, let us suppose that the average µ ¯2,p (z1 , z2 , λ) := R µ2,p (z1 , z2 , q, λ) dνa (q) ≤ 1 −µ < 1 for z1 , z2 in a neighborhood of the boundary of H2 for small enough disorder, a. Then choose some compact set B ⊂ H2 with B # (zλ , zλ ), and split the integration into an integral over B and its complement in H2 . On the first set, the integrand is bounded and on the second set we use the contraction property of µ ¯2,p . That is, Mp (ρa )  =

µ2,p (z1 , z2 , q, λ)

(B×R)∪(H2 \B)×R



≤ C + (1 − µ)

H2 \B

1 2

1

2 cd

p

 (z1 ) + 12 cdp (z2 ) dρa (z1 )dρa (z2 )dνa (q)

 cdp (z1 ) + 12 cdp (z2 ) dρa (z1 )dρa (z2 )

= C + (1 − µ)Mp (ρa ) ,

(5.9)

where C is a finite constant. This implies Mp (ρa ) < C/µ < ∞. Our assumption that µ ¯2,p (z1 , z2 , λ) ≤ 1 − µ < 1 is not quite true. But this averaging was essential in a similar situation in the proof of Theorem 6.1, see [11]. In order to obtain an estimate of a corresponding function µ3,p (z, q, λ) ≤ 3 1 − µ < 1 for z = (z1 , z2 , z3 ) in a neighborhood of the boundary √ √of H , for q = (q1 , q2 ) in a small square with center at 0, and for all λ ∈ (−2 2, 2 2) we use the recursion relation one more time. Before we define this function µ3,p we extend µ2,p to an upper semi-continuous function onto the boundary of H2 (in terms of the z variables) via a radial compactification of C2 (in terms of the u variables). To this end, let r > 0, (ω1 , ω2 ) ∈ C2 so that 1 = rω1 , u1

1 = rω1 , u1

|ω1 |2 + |ω2 |2 = 1 .

(5.10)

Then, µ2,p (z1 , z2 , q, λ) (5.11)  p 2 2 |ω ω | q 1 1 1 2 2 2 |(ω2 , ω1 ), v| − q r Re(ω2 , ω1 ), v + 2 |ω12 (z1 −zλ )|2 +|ω22 (z2 −zλ )|2 = . 1 |ω |2p + 12 |ω2 |2p 2 1 Now we define for (k1 , k2 ) ∈ ∂H2 and any sequence (z1 , z2 )n in H2 that converges to (k1 , k2 ), µ2,p (k1 , k2 , q, λ) :=

lim sup (z1 ,z2 )n →(k1 ,k2 )

µ2,p (z1 , z2 , q, λ) .

(5.12)

As a next step we define the before-mentioned function µ3,p . First, let H # zi = zλ and q = (q1 , q2 ), then    cdp Ψ(zσ1 , Ψ(zσ2 , zσ3 , q2 , λ), q1 , λ) , (5.13) µ3,p (z, q, λ) := cdp (z1 ) + cdp (z2 ) + cdp (z3 ) σ

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R. Froese, D. Hasler and W. Spitzer

where σ ∈ Σ := {(1, 2, 3), (2, 3, 1), (3, 1, 2)} runs over the cyclic permutations of (1, 2, 3). The function µ3,p can be expressed in terms of µ2,p and the auxiliary function cdp (zj ) nj (z) := p , j = 1, 2, 3 . cd (z1 ) + cdp (z2 ) + cdp (z3 ) Namely,    µ3,p (z, q, λ) = µ2,p zσ1 , Ψ(zσ2 , zσ3 , q2 , λ), q1 , λ (5.14) σ∈Σ

×



1 n (z) 2 σ1

 + 14 µ2,p (zσ2 , zσ3 , q2 , λ)(nσ2 (z) + nσ2 (z) .

Then, like for µ2,p above, we extend µ3,p to an upper semi-continuous function onto the boundary of ∂H3 by taking a lim sup. Now, µ3,p (z, 0, λ) ≤ 1 − 2µ < 1 for some µ > 0. By the compactness of the boundary of H3 and the upper semicontinuity of µ3,p we finally get the pointwise estimate √ √µ3,p (z, q, λ) ≤ 1 − µ for z near the boundary of H3 , small q, and λ ∈ (−2 2, −2 2).
Remark 5.2. This proof is now much easier to generalize to higher branched trees, Tk , with k ≥ 3, which was first accomplished by Halasan in her thesis [13]. In that case, Ψ(z  1 , . . . , zk , 0, λ) := −1/(z1 + · · · + zk + λ) with fixed point zλ√:= −λ/(2k) + i 1/k − λ2 /(4k 2 ). Using u = (u1 , . . . , uk ) with uj := (zj − zλ )/ yj  and v = (v1 , . . . , vk ) with vj := yj /(y1 + · · · + yk ) we see that |z1 + · · · + zk − kzλ |2 < y1 + · · · + yk + Im(λ) Therefore, by the same arguments as above and with p > 1, cd(Ψ(z1 , . . . , zk , 0, λ)) =

1 k

1 k

  u, v2 .

cdp (Ψ(z1 , . . . , zk , 0, λ)) p p 1 1 k cd (z1 ) + · · · + k cd (zk )   p  1  2 k u, v ≤ 1 ≤ 1, |u1 |2p + · · · + k1 |uk |2p k

µ2,p (z1 , . . . , zk , 0, λ) :=

(5.15)

with equality if u = sv and |u1 | = · · · = |uk |. The functions µ3,p and nj have to be changed accordingly. In our first paper [9], we attempted to construct a “large” set of deterministic potentials on a (rooted) binary tree that yield ac spectrum. Since almost always spherically symmetric potentials cause localization we considered potentials that oscillate very rapidly within each sphere. The basic example is the following potential, q0 : Take vertices v = w in the nth sphere and u ∈ Sn−1 so that (u, v) ∈ E and (u, w) ∈ E. For some δ ∈ R, let q0 (v) := δ and q0 (w) := −δ. Then continue this for every sphere Sn except for n = 0, where we may define q0 (0) arbitrarily. λ ∈ R is in the interior of the ac spectrum of −∆ + q0 if and only if the polynomial p(z) := z 3 + 2λz 2 + (2 + λ2 − δ 2 )z + 2λ has two non-real, complex-conjugate roots and one real root.

Absolutely Continuous Spectrum

217

An interesting extension arises when the value δ is allowed to depend on the radius, n. In other words, let δ0 > 0 be fixed and let δ1 , δ2 be real-valued functions on N. Then for vertices v = w in the nth sphere as above, we set q(v) := δ0 + δ1 (n) and q(w) := −δ0 + δ2 (n). Proposition 5.3 ([9], Proposition 4.1). Let q and q0 be the above potentials and let λ ∈ σ(−∆ + q0 ). Then for δ1 ∞ + δ2 ∞ small enough depending on δ0 , the Green function of −∆ + q, Gλ (0, 0), is bounded. However, these potentials (and some modifications thereof) are still a set of measure zero. An an explicit construction of a “large” set (that is, of positive measure) remains an open problem. In percolation models, one is usually interested in the occurrence of infinite clusters. A more sophisticated question is whether the spectrum of the adjacency matrix (of the remaining graph) has an ac component. Let us start with the (rooted) binary tree T2 = (V, E), and let q ∈ [0, 1). At every vertex v ∈ V , say v ∈ Sn for some n ∈ N0 , we delete one and only one (forward) edge (v, v
) ∈ E or (v, v

) ∈ E with v
, v

∈ Sn+1 with probability q. With probability 1 − q we keep both (forward) edges (v, v
), (v, v

) in the set of edges. This defines a probability measure, νq , on Ω := {0, 1}E , which is characterized by (we write ωuv := ω((u, v)))   (i) νq {ω ∈ Ω : ωvv = ωvv = 1} = 1 − q for all v ∈ V ;  (ii) νq {ω ∈ Ω : ωvv = 0, ωvv = 1} = νq {ω ∈ Ω : ωvv = 1, ωvv = 0} = q/2 for all v ∈ V ; (iii) for all u, v ∈ V with u = v the random variables (ωuu , ωuu ) and (ωvv , ωvv ) are independent. For every ω ∈ Ω, we define the adjacency matrix of the remaining random graph,  ∆ω : 2 (V ) → 2 (V ) , (∆ω f )(v) := ∆ω (u, v)f (u) , f ∈ 2 (V ) (5.16) u∈V

with matrix kernel



if ωuv = 1 , u, v ∈ V . (5.17) otherwise √ Theorem 5.4. For every 0 ≤ λ < 2 2 there exists a q0 > 0 such that for all 0 ≤ q ≤ q0 , [−λ, λ] ⊆ σac (∆ω ) νq -almost surely. Furthermore, the spectrum is purely ac on (−λ, λ) νq -almost surely. ∆ω (u, v) :=

1 0

This particular model was suggested to one of us by Shannon Starr to whom we are grateful. Before we enter into some details of the proof let us start with some definitions. For v ∈ Sn ⊂ V , let Gv = (Vv , Ev ) be the binary graph T2 = (V, E) truncated at v, that is, the largest connected subgraph of T2 that contains v but no u ∈ Sk with k < n (or simply the binary tree with root v); this truncation is different from the one in Section 4. For ω ∈ Ω and v ∈ V we define the truncated

218

R. Froese, D. Hasler and W. Spitzer

adjacency matrix, ∆ω(v) : 2 (Vv ) → 2 (Vv ) ,

(∆(v) ω f )(u) :=



∆ω (r, u)f (r) ,

u ∈ Vv , f ∈ 2 (Vv ) .

r∈Vv

(5.18)

Furthermore, for λ ∈ H, we define the two Green functions G(ω, λ) := (∆ω − λ)−1 (0, 0) , (v)

G

(ω, λ) :=

(∆(v) ω

− λ)

−1

(v, v)

(5.19) (5.20)

as the kernels of the respective resolvents. We have G(ω, λ) = G(0) (ω, λ). The recursion formula for G(v) (ω, λ) is 



G(v) (ω, λ) = −(G(v ) (ω, λ) + G(v ) (ω, λ) + λ)−1 .

(5.21)

ρλ,q := νq ◦ G(v) (·, λ)−1

(5.22)

Finally, let (v)

be the Green probability distribution defined as the image of the measure νq under (v) the map ω → G(v) (ω, λ) from Ω to H. By translation-invariance, the measure ρλ,q does, in fact, not depend on v, and we shortly write ρq by also suppressing the spectral parameter λ. Sketch of proof of Theorem 5.4. Using the weight function cd from (5.5) with the same zλ and p > 1 we define the moment  Mp (ρq ) := cdp (z) dρq (z) . (5.23) H

Applying the recursion relation (5.21) and the symmetry between the variables z1 and z2 below we have    Mp(ρq ) = µ2,p,q (z1 , z2 , λ) 12 cdp (z1 ) + 12 cdp (z2 ) dρq (z1 )dρq (z2 ) (5.24) H2

with µ2,p,q (z1 , z2 , λ) := [q cdp (−1/(z1 + λ)) + (1 − q) cdp (−1/(z1 + z2 + λ)−1 )]. Then, as in (5.7), we apply once more the recursion relation and write the result in the form Mp (ρq ) =  H3

(5.25)  1 µ (z , z , z , λ) cdp (z1 ) + cdp (z2 ) + cdp (z3 ) dρq (z1 )dρq (z2 )dρq (z3 ) . 3 3,p,q 1 2 3 

The function µ3,p,q (z1 , z2 , z3 , λ) is expanded as a function of q so that µ3,p,q = (1 − q)2 µ3,p + qR ,

(5.26)

where µ3,p is the function in (5.14) with q1 = q2 = 0 and |R| ≤ CK on H3 \ K for a compact set K. For q small enough we achieve that (1 − q)2 µ3,p + qR ≤ (1 − µ/2) outside such a compact set K with µ > 0. Hence, Mp (ρq ) ≤ C/(1 − µ/2).


Absolutely Continuous Spectrum

219

Remarks 5.5. (i) We do not know the full spectrum of the adjacency matrix, ∆ω , nor do we have information on the remaining (point) spectrum. (ii) In this percolation model, there is always an infinite cluster even when q = 1. This is in contrast to the genuine bond-percolation tree model, where an edge is deleted with probability q independently of other edges. Here, the percolation threshold for the existence of an infinite cluster is qc = 1/2, see [17]. This model seems harder to analyze, at least from the standpoint of our method. The reason is that the point spectrum is dense in the full spectrum of the random percolation graph since almost surely there are arbitrarily large subtrees √ disconnected from the random graph for which the spectrum √ lies inside (−2 2, 2 2). Thus there is no interval of pure ac spectrum if it happens to exist at all. Besides, we are not aware of a conjectured value for a critical (quantum percolation) value qqp up to which the adjacency matrix has an ac component; qqp ≤ 1/2 since an infinite cluster is required to exist.

6. Strongly correlated random potential on a tree There is a large gap between the known results for the tree and the open problem on Zd for d ≥ 3. Therefore it seems worthwhile to address some of the problems that would come up on Zd in simpler toy models. In order to see a strong effect of correlations we consider a transversely 2-periodic random potential. The potential is defined by choosing two values q = (q1 , q2 ) of the potential at random, independently for each sphere in the tree. These two values are then repeated periodically across the sphere and hence the potential is strongly correlated. Such a two-periodic potentials can exhibit either dense point spectrum or absolutely continuous spectrum depending on the correlations of q1 and q2 . We will prove that if the values of q1 and q2 are sufficiently uncorrelated (see assumption (6.3) below) then there will be some ac spectrum, as is the case for the iid Anderson model. However, since in some sense this model is so close to being one-dimensional, the proof has some features not appearing in the tree model of Section 5. This time, the proof follows from an estimate of an average over potential values q of functions µ(z, q), similar in both models, that measure the # contraction of a relevant map of the plane. We seek an estimate of the form µ(z, q) dνa (q) < 1 for z near the boundary of H at infinity. In the proof of Theorem 5.1 we have used the independence of the potentials across the sphere in proving that µ(z, 0) is already less than one. Then small values of q in the integral are handled by semi-continuity. In the present situation, µ(z, 0) = 1 and perturbations in q send it in both directions. Thus we must use cancellations in the integral over q in an essential way. Our method extends to the case where the joint distributions are not identical, as long as they are all centered and satisfy certain uniform bounds. This is significant since in this case we lose the self-similarity that has been used in previous proofs.

220

R. Froese, D. Hasler and W. Spitzer

We make the following assumptions about the measure ν. First, it has compact support, and for simplicity, ν is supported in {q = (q1 , q2 ) : |q1 | ≤ 1, |q2 | ≤ 1} . Then, the measure is centered on zero:  (q1 + q2 ) dν(q) = 0 . R2 # Let cij := R2 qi qj dν(q). Then finally,

(6.1)

(6.2)

2c12 < 1/2 . (6.3) c11 + c22 The first inequality in (6.3) simply says that q is not identically zero. The second is a bound on the correlation. Completely correlated potentials (that is, the onedimensional case where the spectrum is localized) would correspond to δ = 1. We have proved the following theorem. c := c11 + c22 > 0

and

δ :=

Theorem 6.1 ([11], Theorem 2). Let ν(0) be a probability measure of bounded support for the potential at the root, let ν be a probability measure on R2 satisfying (6.1), (6.2) and (6.3) and let Ha be the random discrete Schr¨ odinger operator on the binary tree corresponding to the transversely√two-periodic potential defined by the scaled measure νa . There exists λ0 ∈ (0, 2 2) such that for sufficiently small a the spectral measure for Ha corresponding to δ0 has purely ac spectrum in (−λ0 , λ0 ). Remark 6.2. When the random variables q1 and q2 are independent, that is, when δ = 0, our proof shows that λ0 can be chosen to be 2. The determination of the maximum λ0 remains an open problem.

7. Loop tree models There are several interesting ways to add loops to a tree which are sometimes called decorated trees. Here we present three possibilities of adding new edges that connect vertices inside the same sphere. In our first attempt we connect each vertex (n, 2i ) inside each sphere Sn with (n, 2i + 1) and (n, 2i − 1) modulo 2n . That amounts to adding to the adjacency matrix of the tree the adjacency matrices of the nearest neighbor chains {0, 1, . . . , 2n − 1} with periodic boundary conditions. We call this the regular loop tree model. Every vertex other than the root has five neighbors. In the next subsection we present the derivation of the fixed point equation that determines the spectrum at the root. Finding the spectrum of this new adjacency matrix turns out to be difficult and remains an open problem. In a second attempt we modify these new connections to mean-field connections. This new mean-field Laplacian can be solved explicitly so that we can take on the next step and add a random potential. Here we limit ourselves to a special case, namely to a two-periodic Bernoulli random potential that we have studied in

Absolutely Continuous Spectrum

221

the previous section. We present the model and the main result in Subsection 7.2. Proving ac spectrum for the Anderson model (that is, with iid random potential) on this mean-field tree model is still an open problem. The third loop tree model was suggested to us by Laszlo Erd¨os. In its simplest version, one adds to each sphere of the tree a single loop (of weight γ) that connects two arbitrarily chosen sites within a sphere. It would be interesting to prove the (in)stability of the ac spectrum for small γ > 0. 7.1. Regular loop tree model Each vertex v in the nth sphere of the binary tree, T2 = (V, E), is of the form v = (n, j) with 0 ≤ j ≤ 2n − 1. We now also call v, w ∈ Sn nearest neighbors if w = (n, j ± 1 mod 2n ). The newly added edges are denoted by E rlt . In order to compare with the usual adjacency matrix of the tree we introduce a parameter γ that puts the weight γ on the new connections inside a sphere. The new adjacency matrix, ∆γ , is now defined by the kernel   1 if (v, w) ∈ E ∆γ (v, w) := γ if (v, w) ∈ E rlt . (7.1)  0 else Furthermore, let γD := ∆γ − ∆, and let Dn be D restricted to Sn . For n ∈ N0 , N := 2n , and λ ∈ H we consider the generalized M¨ obius transformations Φn : SH2N → SHN , Φn (Z) := −(EnT ZEn + γDn + λ)−1 between the respective Siegel half-spaces. When γ = 0 then diagonal matrices are no longer mapped to diagonal matrices. An invariant subset of matrices that is preserved under this flow is the set of circulant (or Toeplitz) matrices. Recall that an N × N matrix Z is called circulant if Zi,j = z(j−i)modN . That is,   z0 z1 z2 ··· zN −1  zN−1 z0 z1 ··· zN −2     zN−2 zN−1 z0 ··· zN −3  Z = .  .. .. .. ..  ..  . . . . .  z1

···

z2

zN −1

z0

Circulant matrices are characterized by the condition that they commute with the shift operator. Therefore we can diagonalize circulant matrices by the finite Fourier transform. The finite Fourier transform, Un ∈ Mat(N, C), is defined as (Un )j,k := N −1/2 e2πijk/N ,

j, k = 0, 1, . . . , N − 1 , N = 2n .

(7.2)

Here are some simple properties. Lemma 7.1. 1. Let Z be an N × N circulant matrix with first row z = [z0 , z1 , . . . , zN −1 ]. For N−1 (n) j = 0, 1, . . . , N − 1, let fj :=
=0 z
e2πi
k/N . Then (Un∗ ZUn )j,k = δjk fj

(n)

,

j, k = 0, 1, . . . , N − 1 .

(7.3)

222

R. Froese, D. Hasler and W. Spitzer In particular, for the spherical Laplacian Dn we have   (Un∗ Dn Un )j,k = 2 δj,k cos 2πj , j, k = 0, 1, . . . , N − 1 . N

(7.4)

2. For j = 0, 1, . . . , N − 1, = 0, 1, . . . , 2N − 1 we have  n+1  (Un∗ EnT Un+1 )j,k = 2−1/2 1 + e2πik/2 (δj,k + δj+2n ,k ) .

(7.5)

3. For j, k = 0, 1, . . . , N − 1 we have   Un∗ (EnT ZEn + γDn + λ)−1 Un

(7.6)

j,k

= δj,k

2 cos2



2πk 2n+2



(n+1) fk/2

2

+ 2 sin



1 2πk 2n+2



(n+1)

fk/2+2n + 2γ cos

 2πk  2n

. +λ

Proof. This is all quite easy but nevertheless. . . (Un∗ ZUn )j,k = N −1

N −1 

e−2πi(jm−
k)/N z(
−m)modN

m,
=0

=

N −1 

z
e2πi
/N N −1

(Un∗ EnT Un+1 )j,k =



(n)

.

∗ T Uj
E
,
 U
 k


=0,1,...,N −1,
 =0,1,...,2N −1

= 2−n−1/2 =2

e−2πim(j−k)/N = δj,k fj

m=0


=0

In a similar vein we obtain

N −1 

 −1/2



   e−2πij
/N δ2
,
 + δ2
+1,
 e−2πi
k/2N


,


   δj,k + δj+N,k + 2−1/2 e2πik/2N δj,k + δj+N,k .

The third claim follows from the first two by noticing that Un∗ (EnT ZEn + γDn + ∗ λ)−1 Un = (Un∗ EnT Un+1 Un+1 ZUn+1 (Un∗ En Un+1 )∗ + γ Un∗ Dn Un + λ)−1 . This shows that  ∗ T  Un E Zn+1 EUn j,k   n+1  (n+1)  = 12 δj,k 1 + e2πij/2 fj 1 + e−2πij/2N   (n+1)   + 1 − e2πij/2N fj+N 1 − e−2πij/2N   (n+1)   (n+1) = δj,k 1 + cos( 2πj + 1 − cos( 2πj 2N ) fj 2N ) fj+N  (n+1) (n+1) = 2 δj,k cos2 ( 2πj ) fj + sin2 ( 2πj )fj+N .
4N 4N (7.6) implies that if Z ∈ SH2N is circulant with Fourier transformation f (n+1) then Φn (Z) ∈ SHN is circulant with Fourier transformation f (n) , and so that 1 (n) fk = − . (7.7)  πk  (n+1)  πk  (n+1)   2 2 cos2 2N fk/2 + 2 sin 2N fk/2+N + 2γ cos 2πk +λ N

Absolutely Continuous Spectrum (n)

πk Letting n → ∞ and setting f ( 2N ) := fk

f (θ) = −

2 cos2

θ 4

f ( θ2 ) + 2 sin2

223

we obtain the fixed point equation, 1 θ 4

f ( θ2 + π) + 2γ cos(θ) + λ

,

(7.8)

for functions f : [0, 2π] → H. [For γ = 0 and Im(λ) > 0 the only solution to (7.8) is the constant function with value zλ from (5.3).] The truncated Green functions Z = Zn ∈ SHN are further restricted by the condition that Zn has to be symmetric (not Hermitean). This implies that the first row, z = [z0 , z1 , . . . , z2n −1 ] of Zn is symmetric with respect to the middle co-ordinate, 2n−1 . That is, z = [z0 , z1 , . . . , z2n−1 −1 , z2n−1 , z2n−1 −1 , . . . , z1 ] . (n+1)

(7.9)

(n+1)

Therefore, f2n −k = f2n +k and consequently, f (π − θ) = f (π + θ). The only place where we are able to evaluate the solution of (7.8) explicitly is for θ ∈ {0, 2π}, where we find that Gλ (0, 0) = f (0) = f (2π) = 2γ+λ 4 √+  i 2 . For f (0) to be in H we get the condition that |2γ + λ| < 2 2. 8 − (2γ + λ) 4 √ √ Therefore, [−2γ − 2 2, −2γ + 2 2] is in the ac spectrum of ∆γ . On the other hand, let φn ∈ 2 (V ) with φn (v) := 2−n/2 for v ∈ Sn and zero otherwise. Then, the variational energy, φn , ∆γ√φn  = 2γ. So for large γ, this energy is outside the √ interval [−2γ − 2 2, −2γ + 2 2] and thus, unlike for γ = 0, f (0) does not alone determine the full spectrum. 7.2. Mean-field loop model Now we add a weighted complete graph to every sphere in the binary tree. Since the weights are chosen to make the total added weights the same in each sphere, this is a sort of mean-field model. Pick a number γ > 0. Each added edge (dotted line in the figure below) in the nth sphere Sn is given the weight γ2−n . That is, we define the adjacency matrix, ∆mf γ , through the kernel  if (v, w) ∈ E  1 −n γ2 if v, w ∈ Sn . ∆mf (v, w) := (7.10) γ  0 else We call the new (weighted) graph the mean-field binary tree. The spectrum of √ √ the mean-field adjacency matrix, ∆mf , is the union of two intervals [−2 2 + γ, 2 2+ γ √ √ γ] ∪ [−2 2, 2 2] and is purely ac. This can be seen by using a Haar basis [11]. For simplicity, we considered a random potential that is transversely twoperiodic and defined by the product of two independent Bernoulli measures for q1 and q2 , dν(q1 , q2 ) =

  1 δ(q1 − 1) + δ(q1 + 1) δ(q2 − 1) + δ(q2 + 1) . 4

(7.11)

224

R. Froese, D. Hasler and W. Spitzer

S0

S1

S2

S3

...

Figure 1. Rooted binary tree with mean-field edges insides spheres and transversely 2-periodic potential. Then we have the following theorem. Theorem 7.2 ([11], Theorem 9). Let ν(0) be a probability measure of bounded support for the potential at the root and ν be the product of Bernoulli measures defined above and let Ha,γ := −∆mf odinger operator on γ + a q be the random discrete Schr¨ the mean-field binary tree corresponding to the transversely two-periodic potential √ defined by the scaled distribution νa and weight γ. There exist 0 < λ0 , λ1 < 2 2 such that for sufficiently small a the spectral measure for Ha corresponding to δ0 has purely ac spectrum in {λ : |λ| ≤ λ0 , |λ − γ| ≤ λ1 }. In this theorem, the constant λ0 has the same value √ as in Theorem 6.1, while λ1 can be taken to be any positive number less than 2 2. Acknowledgment Wolgang Spitzer is indepted to Florian Sobieczky for organizing the wonderful Alp-workshop in St. Kathrein. We are also grateful to the referee for many useful comments.

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References [1] M. Aizenman and S. Molchanov, Localization at large disorder and at extreme energies: An elementary derivation, Commun. Math. Phys. 157 (1993), 245–278. [2] M. Aizenman, R. Sims, and S. Warzel, Stability of the Absolutely Continuous Spectrum of Random Schr¨ odinger Operators on Tree Graphs, Prob. Theor. Rel. Fields 136, no. 3 (2006), 363–394. [3] M. Aizenman, R. Sims and S. Warzel, Absolutely Continuous Spectra of Quantum Tree Graphs with Weak Disorder, Commun. Math. Phys. 264 (2006), 371–389. [4] J. Bourgain, On random Schr¨ odinger operators on Z2 , Discrete Contin. Dyn. Syst. 8, no. 1 (2002), 1–15. [5] J. Bourgain, Random lattice Schr¨ odinger operators with decaying potential: some higher dimensional phenomena, V.D. Milman and G. Schechtman (Eds.) LNM 1807, 70–98, 2003. [6] H. Cycon, R. Froese, W. Kirsch, and B. Simon, Schr¨ odinger operators with application to quantum mechanics and global geometry, Springer-Verlag, 1987. [7] F. Delyon, B. Simon, and B. Souillard, From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincar´e Phys. Th´eor. 42, no. 3 (1985), 283–309. [8] J. Fr¨ ohlich and T. Spencer, Absence of diffusion in the Anderson tight binding model for large disorder or low energy, Commun. Math. Phys. 88 (1983), 151–184. [9] R. Froese, D. Hasler, and W. Spitzer, Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schr¨ odinger operators on graphs, Journ. Funct. Anal. 230 (2006), 184–221. [10] R. Froese, D. Hasler, and W. Spitzer, Absolutely continuous spectrum for the Anderson model on a tree: geometric proof of Klein’s theorem, Commun. Math. Phys. 269 (2007), 239–257. [11] R. Froese, D. Hasler, and W. Spitzer, Absolutely continuous spectrum for random potentials on a tree with strong transverse correlations and large weighted loops, Rev. Math. Phys. 21 (2009), 1–25. [12] R. Froese, D. Hasler, and W. Spitzer, On the ac spectrum of one-dimensional random Schr¨ odinger operators with matrix-valued potentials arXiv:0912.0294, 13 pp, to appear in Mathematical Physics, Analysis and Geometry. [13] F. Halasan, Absolutely continuous spectrum for the Anderson model on trees, Ph.D. thesis at the University of British Columbia, Department of Mathematics, 2009, https://circle.ubc.ca/handle/2429/18857, 63pp. [14] W. Kirsch, An Invitation to Random Schr¨ odinger operators, Soc. Math. France 2008, Panoramas & Synth`esis, no 25, 1–119. [15] W. Kirsch and F. Martinelli, On the ergodic properties of the spectrum of general random operators, Journ. Reine und Angew. Math. 334 (1982), 141–156. [16] A. Klein, Extended States in the Anderson Model on the Bethe Lattice, Advances in Math. 133 (1998), 163–184. [17] R. Lyons, Random walks and percolation on trees, Ann. Probab. 18 (1990), 931–958. [18] M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, Revised and Enlarged Edition, Academic Press, 1980.

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[19] B. Simon, Lp Norms of the Borel Transform and the Decomposition of Measures, Proceedings AMS 123, no. 12 (1995), 3749–3755. [20] P. Stollmann, Caught by Disorder: Bound States in Random Media, Birkh¨ auser, 2001. Richard Froese University of British Columbia Department of Mathematics Vancouver, British Columbia, Canada e-mail: [email protected] David Hasler College of William & Mary Department of Mathematics Williamsburg, Virginia, USA e-mail: [email protected] Wolfgang Spitzer FernUniversit¨ at Hagen Fakult¨ at f¨ ur Mathematik und Informatik Hagen, Germany e-mail: [email protected]

Progress in Probability, Vol. 64, 227–234 c 2011 Springer Basel AG


Some Spectral and Geometric Aspects of Countable Groups Alexander Bendikov, Barbara Bobikau and Christophe Pittet Abstract. We discuss the relationship between the isospectral profile and the spectral distribution of a Laplace operator on a countable group. In the case of locally finite countable groups, we emphasize the relevance of the metric associated to a natural Markov operator: it is an ultra-metric whose balls are optimal sets for the isospectral profile. Mathematics Subject Classification (2000). Primary: 60B15, 20F65; Secondary: 58C40. Keywords. Probability measures on graphs and groups, geometric group theory, spectral theory and eigenvalue problems for Laplace operators.

1. Introduction We review the relationship of the spectral distribution and the isospectral profile of a Laplace operator on a countable group. We investigate properties of metrics associated to Markov operators on countable locally finite groups, and emphasize their relevance in studying the isospectral profiles and the spectral distributions of Laplace operators on these groups.

2. Countable groups Let G be a countable group with the counting measure as the Haar measure. Let µ be an irreducible symmetric probability measure on G. In other words, the kernel defined as K(x, y) = µ({x−1 y}) is Markovian, symmetric, and irreducible. If x, y ∈ G are such that µ({x−1 y}) > 0, we attach an edge of length (or weight) 1/µ({x−1 y}) with end points x and y. The resulting graph Γ(G, S) is the Cayley graph of G with respect to the symmetric generating set S = support(µ). The restriction of the edge path metric on Γ(G, S) to G is the weighted word metric dK defined by µ. For example, if S = support(µ) is finite and µ is the

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homogeneous measure on S then the weighted word metric defined by S is the usual word metric defined by S, multiplied by the cardinality |S| of S.

3. Locally finite groups and ultra-metrics Let G be a locally finite group, that is if F ⊂ G is any finite subset of G then the smallest subgroup of G containing F is finite. If we assume that G is countable, then there exists an increasing chain of finite subgroups {e} = G0 ⊂ G1 ⊂ · · · ⊂ Gn ⊂ · · · , such that 9 G= Gn . n≥0

Conversely, any group which is equal to such a union is obviously countable and locally finite. For example let G be the set of functions f : N → Z/2Z with finite support.  Then G = n≥0 Gn where Gn ∼ = (Z/2Z)n is the subgroup of such functions whose support is included in N ∩ [0, n] (if n = 0, the support N ∩ [0, 0] is empty hence the function is identically 0 and G0 is indeed the trivial group). Another example is the group G of permutations of N whose support is finite. Then G = n≥0 Gn where Gn ∼ = Sn+1 is the subgroup of such permutations whose support is included in N ∩ [1, n + 1].  A natural class of measures associated to a strictly increasing union G = Gn is obtained by choosing strictly positive numbers c0 , c1 , . . . , cn , . . . such n≥0 ∞ that n=0 cn = 1. If mn denotes the normalized Haar measure on Gn , then the series ∞  µ= cn mn n=0

defines an irreducible symmetric probability measure on G. Lemma 3.1. Let G and µ be as above and let S = support(µ). Let q ∈ G \ {e}. Any edge path in Γ(G, S) between e and q which is geodesic (i.e., whose length equals the distance between e and q) travels along the unique edge joining e with q. Proof. Let n be minimal such that q ∈ Gn . Notice that n ≥ 1. Let c : [0, 1] → Γ(G, S) be a shortest edge path with constant speed and with c(0) = e and c(1) = q. Let d ∈ N and 0 = t0 < · · · < td = 1, such that the set of points xk = c(tk ), 0 ≤ k ≤ d, is the intersection Im(c) ∩ G of the image of the edge path c with G. There exists 1 ≤ k0 ≤ d such that −1 x−1 k0 −1 xk0 ∈ G \ Gn−1 . Indeed, if xk−1 xk ∈ Gn−1 for all 1 ≤ k ≤ d, then q = xd =

d  k=1

x−1 k−1 xk

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would also belongs to Gn−1 . But this would contradict the minimality of n. Hence, as x−1 k0 −1 xk0 ∈ G \ Gn−1 , and as q ∈ Gn , we deduce that µ({x−1 k0 −1 xk0 }) ≤ µ({q}). The length of the restriction of c to [tk0 −1 , tk0 ] is equal to 1/K(xk0 −1 , xk0 ) = 1/µ({x−1 k0 −1 xk0 }). The edge between e and q has length 1/µ({q}). This implies that the geodesic edge path c has only one edge (i.e., tk0 −1 = 0 and tk0 = 1). As there is exactly one edge in Γ(G, S) between any two points of G, the proof is finished.
Proposition 3.2. Let G and µ be as above and let S = support(µ). 1. The distance dK (p, q) between any two distinct points p, q ∈ G is realized in the Cayley graph Γ(G, S) by a unique geodesic which is the edge of length 1/K(p, q) with end points p and q. 2. The set of values taken by the distance dK is the set {rn : n ∈ N} ∪ {0}, where  −1  cn  , rn =  |Gn | k≥n

and the ball of radius rn with center the identity is the subgroup Gn . 3. The metric dK is an ultra-metric, hence for any radius r ≥ 0, G is partitioned by its balls of radius r. Proof. As dK is left-invariant, in order to prove the first statement, it is enough to prove it when p = e. The distance dK (e, q) is equal to the length of an edge path c : [0, 1] → Γ(G, S) such that c(0) = e and c(1) = q. Lemma 3.1 shows that (the image of) c is the unique edge of Γ(G, S) between e and q and its length is 1/K(e, q). To prove the second statement, notice that G is the disjoint union 9 G = G0 ∪ Gn \ Gn−1 . n∈N

For any q ∈ Gn \ Gn−1 , we have µ({q}) =

 cn . |Gn |

k≥n

Hence according to the first statement,



−1  cn  . dK (e, q) = rn =  |Gn | k≥n

This proves that the ball of radius rn with center the identity is the subgroup Gn . It also proves that the distances to the identity are {rn : n ∈ N} ∪ {0}. As dK is left-invariant this finishes the proof of the second statement.

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The third statement is an immediate consequence of left-invariance and the fact that the balls with center the identity are subgroups. Indeed, for q ∈ G, let us denote qK = dK (e, q). If pqK = rn , that is pq ∈ Gn \ Gn−1 then either p or q belongs to G \ Gn−1 , that is pqK ≤ max{pK , qK }. As dK is left-invariant, the above inequality shows that dK defines an ultra-metric. Hence two balls of the same radius which meet have to coincide.


4. The spectral distribution and the isospectral profile of a Laplace operator on a countable group Let G be a countable group with the Haar measure equals the counting measure. Let µ be a symmetric irreducible probability measure on G. The right convolution with µ defines a self-adjoint operator Rµ : L2 (G) → L2 (G) whose norm is bounded by µ(G) = 1. Hence the Laplacian ∆µ = 1 − Rµ is a positive self-adjoint operator on L2 (G) of norm bounded by 2. Let  ∆µ = λdEλ [0,∞)

be a spectral resolution of ∆µ . As Rµ commutes with the left-regular representation λ : G → U (L2 (G)) (λ(g)f )(x) = f (g −1 x), the orthogonal projector Eλ = Eλ∗ = Eλ2 commutes with λ(g) for any g ∈ G. Hence Eλ belongs to the von Neumann algebra B(L2 (G))G of bounded operators on L2 (G) which commute with λ. Let us denote Nµ (λ) = trG (Eλ ) = (Eλ δe , δe ) the von Neumann trace of Eλ (here δx denotes the characteristic function of the point x ∈ G and the scalar product is chosen such that the set {δx : x ∈ G} is an orthonormal Hilbert basis). The group G is non-amenable if and only there exists a neighborhood U of zero such that the spectral density function Nµ (λ) is identically equal to zero on U . This follows from [5]. Let Ω be a finite subset of G, and let λ1 (Ω) denotes the first eigenvalue for the Dirichlet problem on Ω. That is λ1 (Ω) =

inf

(∆µ (f ), f )/f 2.

∅ =support(f )⊂Ω

The isospectral profile associated to (G, µ) is the function Λµ (v) = inf λ1 (Ω) |Ω|≤v

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where v ∈ [1, ∞). The isospectral profile Λµ is obviously a positive decreasing right-continuous function. It satisfies lim Λµ (v) = 0

v→∞

if and only if G is amenable. If G is amenable, one can use a Nash type inequality to deduce that limv→∞ Λµ (v) = 0. Let us show that if the isospectral profile goes to zero then G is amenable. Lemma 4.1. (See [3, Proposition 4.2].) Let Ω ⊂ G be a finite subset of G such that 0 ≤ λ1 (Ω) < 1. Let λ = λ1 (Ω). Let α ≥ 1 such that α ≥ − log(1 − λ)/λ = 1 + λ/2 + λ2 /3 + · · · . Then for any integer time t ∈ N, the return probability p2t (e, e) at time 2t for the random walk defined by µ satisfies p2t (e, e) ≥

exp(−2αtλ1 (Ω)) . |Ω|

Let us choose α = 2 in Lemma 4.1. We obtain for all t ∈ N, and all finite subset Ω of G such that λ1 (Ω) ≤ 1/2, p2t (e, e) ≥ exp(−4tλ1 (Ω) − log(|Ω|)). In other words, if we define for each t ∈ N the positive number R(2t) by the equality p2t (e, e) = exp(−2tR(2t)), we obtain R(2t) ≤ 2λ1 (Ω) + log(|Ω|)/2t. Hence, if limv→∞ Λµ (v) = 0, we deduce that limt→∞ R(2t) = 0. Hence G is amenable [5].

5. The relationship between the spectral distribution and the isospectral profile In the case the countable group G is amenable, the (generalized) inverse of the isospectral profile Λ−1 µ (λ) = inf{v ≥ 1 : Λµ (v) ≤ λ} takes only finite values for λ ∈ (0, ∞) because limv→∞ Λµ (v) = 0. Following [4], we say that two positive functions f, g are dilatationally equivalent near zero, respectively near infinity, and we write f (λ) ∼ = g(λ), respectively f (v) ∼ = g(v), if there exist strictly positive numbers a, b, such that near zero f (aλ) ≤ g(λ) ≤ f (bλ), respectively such that for near infinity, f (av) ≤ g(v) ≤ f (bv).

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In the case the above inequalities hold with the constants a, b as external factors, namely if af (λ) ≤ g(λ) ≤ bf (λ), respectively af (v) ≤ g(v) ≤ bf (v), hold (here b ≥ 1), we say that f and g are factor-equivalent near zero, respectively near infinity, and we write f (λ) ) g(λ), respectively f (v) ) g(v). Remark 5.1. The relations ∼ = and ) are not comparable, i.e., no one implies the other, rather they are dual to each other when taking inverses. Question 5.2. Let (G, µ) be a countable group endowed with a symmetric irreducible probability measure. Does the dilatational equivalence near zero 1 Nµ (λ) ∼ = −1 Λµ (λ) hold true? Remark 5.3. In the case G is non-amenable, if λ is small enough, then N (λ) = 0 and Λ−1 µ (λ) is infinite (provided we agree that inf(∅) = ∞). 5.1. Finitely generated groups and measures with finite second moment If the group G is generated by a finite symmetric set S we say that µ has finite second moment with respect to the word metric defined by S if  µ({g})g2S < ∞. g∈G

Notice that this condition on µ does not depend on the choice of the finite symmetric generating set S. In this subsection we assume that G is finitely generated and we only consider probability measures which are symmetric irreducible and have finite second moment. In this setting, if µ and ν are two measures on G, then we have the dilatational equivalence near zero Nµ (λ) ∼ = Nν (λ), and the factor-equivalence near infinity Λµ (v) ) Λν (v). See [2]. If G has polynomial growth, on the one hand it follows from [4] and [2] that Nµ (λ) ∼ = λd/2 , where d ∈ N ∪ {0} is the growth degree of G. On the other hand, the following factor-equivalence near infinity 1 Λµ (v) ) 2/d v

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holds true (balls with respect to the word metric are asymptotically optimal sets for the isospectral profile) [3]. Hence, near zero 1 . λd/2 Hence we have the dilatational equivalence near zero 1 Nµ (λ) ∼ . = −1 Λµ (λ) ∼ Λ−1 µ (λ) =

If G is amenable and if the isospectral profile pre-composed with the exponential function is doubling, that is if there exists  > 0 such that for all v ≥ 0, Λµ (exp(2v)) ≥ Λµ (exp(v)), then the dilatational equivalence near zero Nµ (λ) ∼ =

1 Λ−1 µ (λ)

holds true [2]. For example, if G is polycyclic with exponential growth, then Λµ (v) )

1 , log(v)2

(the balls with respect to the word metric are not asymptotically optimal for the isospectral problem) and 1 Nµ (λ) ∼ = exp(− 1/2 ). λ 5.2. Countable locally finite groups and series of Haar measures Suppose (G, µ) isas in Section 3, that is G is countable, locally finite and µ is ∞ of the form µ = n=0 cn mn , where mn is the normalized Haar measure on Gn , ∞ cn > 0 and n=0 cn = 1. Then the balls Gn with respect to the metric dK defined by K(x, y) = µ({x−1 y}) are optimal for the isospectral profile [1]. It means that λ1 (Gn ) =

inf

|Ω|≤|Gn |

λ1 (Ω).

If the decay of the measure µ is not too fast, namely, if there exists  > 0, such that for all n ∈ N,  ck ≥ cn , k>n

then the dilatational equivalence near zero Nµ (λ) ∼ =

1 Λ−1 µ (λ)

holds true [1]. For example if G is the group of permutations of N with finite  support, we have G = n≥0 Gn with Gn = Sn+1 . If α is a strictly positive real, and if there exists a, b > 0 such that for all n ∈ N, a/nα+1 ≤ cn ≤ b/nα+1 ,

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then near infinity

 Λµ (v) )

and near zero

log log(v) log(v)

α ,

  1 1 Nµ (λ) ∼ . = exp − 1/α log λ λ

Acknowledgment A. Bendikov and B Bobikau were supported by the Polish Goverment Scientific Research Fund, Grant N N201 371736. We are grateful to Daniel Lenz, Florian Sobieczky, and Wolfgang Woess, for inviting us to the workshop “Spectral and probabilistic properties of random walks on random graphs” funded by the FWF (Austrian Science Fund), within the framework of the project P18703 “Random walks on random subgraphs of transitive graphs”, and to the workshop “Boundaries” founded by TU Graz and the Marie-Curie series of events “GROUPS: European training courses and conferences in group theory”.

References [1] Alexander Bendikov, Barbara Bobikau, and Christophe Pittet, Spectral properties of a class of random walks on locally finite groups, preprint. [2] Alexander Bendikov, Christophe Pittet, and Roman Sauer, Spectral distribution and L2 -isoperimetric profile of Laplace operators on groups, arXiv:0901.0271 (2009), 1–22. [3] Thierry Coulhon, Alexander Grigor’yan, and Christophe Pittet, A geometric approach to on-diagonal heat kernel lower bounds on groups, Ann. Inst. Fourier (Grenoble) 51 (2001), no. 6, 1763–1827 (English, with English and French summaries). [4] Mikhail Gromov and Mikhail Shubin, von Neumann spectra near zero, Geom. Funct. Anal 1 (1991), no. 4, 375–404. [5] Harry Kesten, Full Banach mean values on countable groups, Math. Scand. 7 (1959), 146–156. Alexander Bendikov and Barbara Bobikau Institute of Mathematics, Wroclaw University e-mail: [email protected] [email protected] Christophe Pittet CMI Universit´e d’Aix-Marseille I e-mail: [email protected]

Progress in Probability, Vol. 64, 235–258 c 2011 Springer Basel AG


Percolation Hamiltonians Peter M¨ uller and Peter Stollmann Abstract. There has been quite some activity and progress concerning spectral asymptotics of random operators that are defined on percolation subgraphs of different types of graphs. In this short survey we record some of these results and explain the necessary background coming from different areas in mathematics: graph theory, group theory, probability theory and random operators. Mathematics Subject Classification (2000). Primary 05C25; Secondary 82B43. Keywords. Random graphs, random operators, percolation, phase transitions.

1. Preliminaries Here we record basic notions, mostly to fix notation. Since this survey is meant to be readable by experts from different communities, this will lead to the effect that many readers might find parts of the material in this section pretty trivial – never mind. 1.1. Graphs A graph is a pair G = (V, E) consisting of a countable set of vertices V together with a set E of edges. Since we consider undirected graphs without loops, edges can and will be regarded as subsets e = {x, y} ⊆ V . In this case we say that e is an edge between x and y, respectively adjacent to x and y. Sometimes we write x ∼ y to indicate that {x, y} ∈ E. The degree, the number of edges adjacent to x, is denoted by degG := deg : V → N0 , deg(x) := #{y ∈ V | x ∼ y}. A graph with constant degree equal to k is called a k-regular graph. A path is a finite family γ := (e1 , e2 , . . . , en ) of consecutive edges, i.e., such that ek ∩ ek+1 = ∅; the set of points visited by γ is denoted by γ ∗ := e1 ∪ · · · ∪ en . This gives a natural notion of clusters or connected components as well as a natural distance in the following way. If x is a vertex, then Cx , the cluster containing x, is the set of all vertices y, for which there is a path γ joining x and y, i.e., so

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that x, y ∈ γ ∗ . The length of a shortest path joining x and y is called the distance dist(x, y). With the convention inf ∅ := ∞ it is defined on all of V , its restriction to any cluster induces a metric. A subgraph G
= (V
, E
) of G is given by a subset V
⊆ V and a subset
E ⊆ E. The subgraph G
= (V
, E
) induced by V
has the edge set E
= {e ∈ E | e ⊆ V
}. A one-to-one mapping Φ : V → V is called an automorphism of the graph G = (V, E) if {x, y} ∈ E if and only if {Φ(x), Φ(y)} ∈ E. The set of all automorphisms Aut(G) is a group, when endowed with the composition of automorphisms as group operation. An action of a group Γ on G is a group homomorphism j : Γ → Aut(G), and we write γx := (j(γ))(x) for γ ∈ Γ, x ∈ V . An action is called free, if γx = x only happens for the neutral element γ = e of Γ. A group action is called transitive, if the orbit Γx := {γx | γ ∈ Γ} of x equals V for some (and hence every) vertex x ∈ V . Note that in this case G looks the same everywhere. Example. A prototypical example is given by the d-dimensional integer lattice graph Ld with vertex set Zd and edge set given by all unordered pairs of vertices with Euclidean distance one. Clearly, the additive group Zd acts transitively and freely on Ld by translations. For any group action, due to the group structure of Γ, it is clear that two orbits Γx = Γy must be disjoint. If there are only a finite number of different orbits under the action of Γ, the action is called quasi-transitive, in which case there are only finitely many different ways in what the graph can look like locally. For quasi-transitive actions, there are finite minimal subsets F of V so that 9 Γx = V. (1.1) x∈F

These are called fundamental domains. 1.2. The adjacency operator and Laplacians The adjacency operator of a given graph G = (V, E) acts on the Hilbert space

2 (V ) of complex-valued, square-summable functions on V and is given by  A := AG : 2 (V ) → 2 (V ), Af (x) := f (y) for f ∈ 2 (V ), x ∈ V. y∼x

We will assume throughout that the degree deg is a bounded function on V , and so A is a bounded linear operator. The (combinatorial or graph) Laplacian is defined as  ∆ := ∆G : 2 (V ) → 2 (V ), ∆f (x) := [f (x) − f (y)] for f ∈ 2 (V ), x ∈ V y∼x

so that ∆G = DG − AG , where D := DG denotes the bounded multiplication operator with deg. Signs are a notorious issue here: note that (contrary to the convention in most of the second author’s papers) there is no minus sign in front of the triangle.

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237

For a subgraph G
= (V
, E
) of a given graph, certain variants of ∆G are often considered: The Neumann Laplacian is just ∆N G := ∆G , meaning that the ambient larger graph plays no role at all. The Dirichlet Laplacian ∆D G (the notation agrees with that of [36, 5, 7, 51]) penalizes boundary vertices of G
in G, that is vertices with a lower degree in G
than in G: N 2
2
∆D G := 2(DG − DG ) + ∆G = 2DG − DG − AG : (V ) → (V ).

A third variant is called pseudo-Dirichlet Laplacian in [36, 51]; here we use the notation from [5, 7], where it is named adjacency Laplacian: N 2
2
∆A G := DG − DG + ∆G = DG − AG : (V ) → (V ).

The motivation and origin for the terminology of the different boundary conditions are discussed in [36] – together with some basic properties of these operators. Most importantly, they are ordered in the sense of quadratic forms A D 0  ∆N G  ∆G  ∆G  2DG  2degG ∞ Id

(1.2)

on 2 (V
). Here, Id stands for the identity operator. We recall that for bounded operators on a Hilbert space H, the partial ordering A  B means ψ, (B − A)ψ  0 for all ψ ∈ H, where the brackets denote the scalar product on H. Thus the specX trum of each Laplacian ∆X G , X ∈ {N, A, D}, is confined according to spec(∆G ) ⊆   0, 2 degG ∞ . The names Dirichlet and Neumann are chosen in reminiscence of the different boundary conditions of Laplacians on open subsets of Euclidean space. In fact one can easily check that for disjoint subgraphs G1 , G2 ⊂ G, N N D D D ∆N G1 ⊕ ∆G2  ∆G1 ∪G2  ∆G1 ∪G2  ∆G1 ⊕ ∆G2 .

The adjacency Laplacian does not possess such a monotonicity. On bipartite graphs, such as the lattice graph Ld , the different Laplacians are related to each other by a special unitary transformation on 2 (V ). We recall that a graph is bipartite if its vertex set can be decomposed into two disjoint subsets V± so that no edge joins two vertices within the same subset. Define a unitary involution U = U ∗ = U −1 on 2 (V ) by (U f )(x) := ±f (x) for x ∈ V± . Clearly, we have U ∗ DU = D and U ∗ AU = −A. The latter holds because of       A(U f ) (x) = (U f )(y) = ∓f (y) = − U (Af ) (x) y∼x

y∼x

for every x ∈ V± . In particular, for any subgraph G
of a k-regular bipartite graph G we get ∗ A ∆A G = 2k Id −U ∆G U ∗ D ∆N G = 2k Id −U ∆G U

∆D G

= 2k Id −U

∗

(1.3)

∆N G U.

Consequently, spectral properties of the different Laplacians at zero – the smallest possible spectral value as allowed by (1.2) – can be translated into spectral properties (of another Laplacian) at 2k.

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1.3. Amenable groups and their Cayley graphs Here we record several basic notions and results that will be used later on; we largely follow [5]. Let Γ be a finitely generated group and S ⊂ Γ a symmetric (i.e., S −1 ⊆ S) finite set of generators that does not contain the identity element e of Γ. The Cayley graph G = G(Γ, S) has Γ as a vertex set and an edge connecting x, y ∈ Γ provided xy −1 ∈ S. By symmetry of S we get an undirected graph in this fashion, and G is |S|-regular. Moreover, it is clear that Γ acts transitively and freely on G by left multiplication. Examples. (1) The d-dimensional integer lattice graph Ld is the Cayley graph of the group Zd (written additively, of course) with the set of generators S = {ej , −ej | j = 1, . . . , d} with ej the unit vector in direction j. (2) Changing the set of generators to S
:= S ∪ {±ej ± ek | 1  j < k  d} gives additional diagonal edges; see Figure 1 for an illustration in d = 2.

Figure 1. Two Cayley graphs of Zd . (3) The Cayley graph of the free group with n ∈ N \ {1} generators g1 , . . . , gn can be formed with S = {g1 , . . . , gn , g1−1 , . . . , gn−1 }; it is a 2n-regular rooted infinite tree. More generally, a (κ+ 1)-regular rooted infinite tree, κ ∈ N\ {1}, is also called Bethe lattice Bκ , honouring Bethe [11] who introduced them as a popular model of statistical physics. Every vertex other than the root e in Bκ possesses one edge leading “towards” the root and κ “outgoing” edges, see Figure 2 for an illustration for n = 2, respectively κ = 3. Due to fundamental theorems of Bass [10], Gromov [26] and van den Dries and Wilkie [59], the volume, i.e., the number of elements, of the ball B(n) consisting of all those vertices that are at distance at most n from the identity e, / 0 V (n) := |B(n)| := # x ∈ Γ | distG(Γ,S) (x, e)  n , (1.4) has an asymptotic behaviour that obeys one of the following alternatives:

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Figure 2. Bethe lattice B3 , the Cayley graph of the free group with n = 2 generators a, b. Theorem 1.1. Let G = G(Γ, S) be the Cayley graph of a finitely generated group. Then exactly one of the following is true: (a) G has polynomial growth, i.e., V (n) ∼ nd for some d ∈ N. (b) G has superpolynomial growth, i.e., for all d ∈ N and b ∈ R there are only finitely many n ∈ N so that V (n)  bnd . The growth behavior, in particular the exponent d, is independent of the chosen set S of generators. There is another issue of importance to us, amenability. A definition in line with our subject matter here goes as follows: Definition 1.2. A discrete group Γ is called amenable, if there is a Følner sequence, i.e., a sequence (Fn )n∈N of finite subsets which exhausts Γ with the property that for every finite F ⊂ Γ: |(F · Fn )+Fn | →0 |Fn |

for

n → ∞,

where A+B := (A \ B) ∪ (B \ A) denotes the symmetric difference of two sets A and B. There is quite a number of different equivalent characterisations of amenability. The notion goes back to John von Neumann [63]. In its original form he required the existence of a mean on ∞ (Γ), i.e., a positive, normed, Γ-invariant functional. Remarks 1.3. (1) The defining property of a Følner sequence is that the volume of the boundary of Fn becomes small with respect to the volume of Fn itself as n → ∞. Boundary as a topological term is of no use here; instead, thinking of the associated Cayley graph, F · Fn can be thought of as a neighborhood around Fn (at least for F containing the identity) and so |(F · Fn )+Fn | represents

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the volume of a boundary layer around Fn . Thinking of F as the ball B(r) makes this picture quite suggestive. (2) Discrete groups of subexponential growth are amenable. (3) The lamplighter groups (see below) are amenable but not of subexponential growth. Consequently, growth does not determine amenability. (4) The standard example of a nonamenable group is the free group on two generators. Let us end this subsection with the example we already referred to above: Example. Fix m ∈ N, m  2. The wreath product Zm  Z is the set Zm  Z := {(ϕ, x) | ϕ : Z → Zm , supp ϕ finite, x ∈ Z}, (ϕ1 , x1 ) ∗ (ϕ2 , x2 ) := (ϕ1 + ϕ2 (· − x1 ), x1 + x2 ) and is called the lamplighter group. It is amenable, see [7].

2. Spectral asymptotics of percolation graphs This section contains the heart of the matter of the present survey. After introducing percolation, we begin discussing the relevant properties of the random operators associated with percolation subgraphs. The central notion is the integrated density of states, a real-valued function. We then explain a number of results on the asymptotic behaviour of this function and how methods from analysis, geometry of groups, graph theory and probability are used to derive these results. 2.1. Percolation Percolation is a probabilistic concept with a wide range of applications, usually related to some notion of conductivity or connectedness. Its importance in (statistical) physics lies in the fact that, despite its simplicity, percolation yet exposes a phase transition. The mathematical origin of percolation can be traced back to a question of Broadbent that was taken up in two fundamental papers by Broadbent and Hammersley in 1957 [15, 28]. Percolation theory still has an impressive list of easy-to-state open problems to offer, some with well-established numerical data and conjectures based on physical reasoning. We refer to [25, 31] for standard references concerning the mathematics, as well as Kesten’s recent article in the Notices of the AMS [32]. Mathematically speaking, and presented in accordance with our subject matter here, percolation theory deals with random subgraphs of a given graph G = (V, E) that is assumed to be infinite and connected. A good and important example is the d-dimensional lattice graph Ld , the particular case d = 1 being very special, however. There are two different but related random procedures to delete edges and vertices from G, called site percolation and bond percolation. In both cases, everything will depend upon one parameter p ∈ [0, 1] that gives the probability of keeping vertices or edges, respectively.

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Let us start to describe site percolation. We consider the infinite product :  Ω := Ωsite := {0, 1}V , Pp := p · δ1 + (1 − p) · δ0 , x∈V

as probability space with elementary events ω := (ωx )x∈V , ωx ∈ {0, 1}, and a product Bernoulli measure Pp that formalizes the following random procedure. Independently for all vertices (also called sites in this context) of V , we delete the vertex x from the graph with probability 1 − p, along with all edges adjacent to x. This corresponds to the event ωx = 0, and we call the site x closed. On the other hand, we keep the vertex x and its adjacent edges in the graph with probability p. This corresponds to the event ωx = 1, in which case we speak of an open site. Every possible realisation or configuration is given by exactly one element ω = (ωx )x∈V ∈ Ω, and the measure Pp above governs the statistics according to the rule we just mentioned. Note that we omit the superscript in the notation of the product measure. The graph we just described is illustrated in Figure 3 and formally defined by Gω = (Vω , Eω ), where Vω := {x ∈ V | ωx = 1},

Eω := {e ∈ E | e ⊆ Vω },

i.e., the subgraph of G induced by Vω . Note that for p = 0 the graph Gω is empty with probability 1 and for p = 1 we get Gω = G with probability 1. The second variant, bond percolation, works quite similarly: : Ω := Ωbond := {0, 1}E , Pp := (p · δ1 + (1 − p) · δ0 ), x∈E

leading to the subgraph Gω = (Vω , Eω ) with Vω := V,

Eω := {e ∈ E | ωe = 1}.

Figure 3. Part of a realisation Gω for bond percolation on L2 for p = 12 .

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It amounts to deleting edges (also called bonds in this context) with probability 1 − p, independently of each other. The choice Vω = V is merely a convention. Other authors keep only those vertices that are adjacent to some edge. In both site and bond percolation, the issue is the connectedness of the soobtained random subgraphs. Note that the realisations Gω themselves do not depend upon p, while assertions concerning the probability of certain events or the stochastic expectation of random variables constructed from the subgraphs surely do. A typical question is whether the cluster Cx that contains vertex x ∈ V is finite in the subgraph Gω for Pp -almost all ω ∈ Ω or whether it is infinite with non-zero probability. In the latter case one says that percolation occurs. Let us assume from now on that G is quasi-transitive, so that the above question will have an answer that is independent of x. The percolation threshold or critical probability is then defined as  / 0 pH := sup p ∈ [0, 1] Pp [|Cx | = ∞] = 0 . It is independent of x since, globally, G looks the same everywhere, cf. (1.1), and Pp is a product measure consisting of identical factors. A related critical value is given by  / 0 pT := sup p ∈ [0, 1] Ep [|Cx |] < ∞ , and it is clear that pT  pH . Here, Ep stands for the expectation on the probability space (Ω, Pp ). The equality of these two critical values is often dubbed sharpness of the phase transition, and we write pc := pH = pT in this case for the critical probability. Clearly, sharpness of the transition is a desirable property, as both pH and pT represent two equally reasonable ways to distinguish a phase with Pp almost surely only finite clusters, the subcritical or non-percolating phase, from a phase where there exists an infinite cluster with probability one, the supercritical or percolating phase. Apart from that, sharpness of the phase transition has been used as an important ingredient in the proof of Kesten’s classical result that pc = 1 for bond percolation on the 2-dimensional integer lattice L2 . Together with 2 estimates known for p < pT , it gives that the expectation of the cluster size decays exponentially, i.e., Pp {|Cx | = n}  e−αp n , n ∈ N, with some constant αp > 0 for all p < pc . This fact is also heavily used in some proofs of Lifshits tails for percolation subgraphs, see below. Fundamental papers that settle sharpness of the phase transition for lattices and certain quasi-transitive percolation models are [2, 47, 48]. Recent results valid for all quasi-transitive graphs can be found in [6] together with a discussion of the generality of earlier literature. Theorem 2.1 ([6], Theorem 2, Theorem 3). For every quasi-transitive graph pT = pH =: pc , and for every p < pc there exists a constant αp > 0 so that Pp {|Cx |  n}  e−αp n for all x ∈ V, n ∈ N.

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It is expected that sharpness of the phase transition also holds for percolation on more general well-behaved graphs even without quasi-transitivity. The celebrated Penrose tiling gives rise to such a graph without quasi-transitivity but some form of aperiodic order. A result analogous to Theorem 2.1 was proven for the Penrose tiling in [30]. The general case of graphs with aperiodic order has not yet been settled. We refer to [49] for partial results in this direction. 2.2. The integrated density of states The study of the random family (∆Gω )ω∈Ω of Laplacians on percolation graphs was proposed by de Gennes [19, 20] and often runs under the header quantum percolation in physics. In this paper we focus on the integrated density of states (IDS), also called spectral distribution function, of this family of operators. In general, the IDS is the distribution function of a (not necessarily finite) measure on R that is meant to describe the density of spectral values of a given selfadjoint operator. In the cases of interest to us here, the underlying Hilbert space is

2 (V ), with V being the countable vertex set of some graph. In this situation the IDS is even the distribution function of a probability measure on R, as we shall see. Before giving the rigorous definition that applies in this setting, let us first start with a discussion at a heuristic level. For elliptic operators acting on functions on some infinite configuration space V with a periodic geometric structure, one typically does not have eigenvalues, but rather continuous spectrum. However, the restrictions of these operators to compact subsets K of configuration space V (more precisely to 2 (K), actually) come with discrete spectrum. Therefore, one can count eigenvalues, including their multiplicities. The idea of the IDS is to calculate the number of eigenvalues per unit volume for an increasing sequence Kn of compact subsets and take the limit. For this procedure to make sense, the operator has to be homogenous, at least on a statistical level. Two situations are typical: Firstly, a periodic operator, quite often the Laplacian of a periodic geometry. And, secondly, an ergodic (statistically homogenous) random family of operators, in which case the above mentioned limit will exist with probability one. Let H be a self-adjoint operator in 2 (V ). An intuitive ansatz for the definition of the IDS might be N : R → [0, 1],    tr 1Fn 1]−∞,E] (H) x∈Fn δx , 1]−∞,E] (H)δx  E → N (E) := lim = lim , n→∞ n→∞ |Fn | |Fn | (2.1) where (Fn )n∈N is an appropriate sequence of finite sets exhausting V . Before we go on, let us add some remarks on our notation in (2.1). In general, we write 1A for the indicator function of some set A. Above, 1Fn is to be interpreted as the multiplication operator corresponding to the indicator function 1Fn . In view of the functional calculus for self-adjoint operators we write 1B (H) for the spectral projection of H associated to some Borel set B ⊆ R. Finally, tr stands for the trace on 2 (V ) and δx ∈ 2 (V ) for the canonical basis vector that is one at vertex x and zero everywhere else.

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As was already mentioned, a certain homogeneity property is necessary in order for the limit in (2.1) to exist. A careful choice of the exhausting sequence is necessary, too. For amenable groups tempered Følner sequences will do the job, as is ensured by a general ergodic theorem of Lindenstrauss [43]. We refer to [39, 49, 50] for more details in the present context and sum up the main points in the following definition and the subsequent results. Definition 2.2. Let G be a graph and let Γ be an infinite group that acts quasitransitively on G. We fix a fundamental domain F. For E ∈ R we define   1 Nper (E) := tr 1F 1]−∞,E] (∆G ) (2.2) |F | to be the IDS of the full graph. Secondly, the expression /  0 1 (p) Ep tr 1F 1]−∞,E] (∆X NX (E) := NX (E) := Gω ) |F |

(2.3)

is the IDS of the Laplacians on random percolation subgraphs, where X ∈ {N,A,D} stands for one of the possible boundary conditions discussed in Subsection 1.2. Remarks 2.3. (1) We could have chosen a more general probability measure than Pp , as long as it is invariant under Γ. (2) Usually, we will omit the superscript p and write simply NX for the quantity in (2.3). (1) (3) Note that Nper = NX for any X ∈ {N, A, D}. (4) Note also that NX is not defined in terms of a single operator ∆X Gω , but X rather using the whole family (∆Gω )ω∈Ω ; see also the subsequent result for a clarification. The next theorem establishes the connection between the heuristic picture displayed in (2.1) and the preceding definition. The point here is the generality of the group involved. In the more conventional setting of random operators on Euclidean space Rd (with the group action of Zd ), the equation is the celebrated Pastur-Shubin trace formula. Theorem 2.4 ([39], Theorem 2.4). Let G be a graph and let Γ be an infinite group that acts quasi-transitively on G. Then there is a sequence (Fn )n∈N of finite subsets of V so that   1 NX (E) = lim tr 1]−∞,E] (1Fn ∆X (2.4) Gω 1Fn ) , n→∞ |Fn | uniformly in E ∈ R for Pp -a.e. ω ∈ Ω. Remarks 2.5. (1) We refer to [22, 36, 38, 45, 61] for further predecessors of the latter theorem. (2) The inequalities in (1.2) imply ND  NA  NN .

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(3) A comprehensive theory of the IDS in the (more conventional) set-up of random Schr¨ odinger operators can be found in the monographs [17, 53, 58]; see also the surveys [34, 35, 62] and the references therein. Interestingly, the IDS links quite a number of different areas in mathematics: We started with an elementary operator theoretic point of view. If we rephrase the basic existence problem in the way that we regard the counting of eigenvalues as evaluating the trace of the corresponding eigenprojection, we arrive at the question, whether appropriate traces exist on certain operator algebras. Typically, the operators we have in mind are intimately linked to some geometry, so that quantities derived from the IDS play an important role in geometric analysis. An important example is the Novikov-Shubin invariant of order zero, which equals the van Hove exponent in the mathematical physics language and will be discussed in our setting further below; see [52, 27] and the Oberwolfach report [21]. Another well-known principle provides a link to stochastic processes and random walks: The Laplace transform of NN is the return probability of a continuous time random walk on the graph; details geared towards the applications we have in mind can be found in [51]. The original motivation and the name IDS come from physics. The Laplacians we consider show up as energy operators for a quantum-mechanical particle which undergoes a free motion on the vertices of the graph. If v, v
∈ V are connected by an edge, the particle can “hop” directly from v to v
or vice versa. In this way, the spectrum of the Laplacian appears as the set of possible energy values the particle may attain, hence the name IDS for the quantities in Def. 2.2. In the percolation case, the motion is interpreted to be a quantum mechanical motion of a particle in a random environment. Thm. 2.4 is interpreted as the self-averaging of the IDS for a family of random ergodic operators: for P-a.e. realisation ω of the environment, the normalised finite-volume eigenvalue counting function converges to a non-random quantity. In particular, if one had taken an expectation on the r.h.s. of (2.4), one would have ended up with the very same expression in the macroscopic limit. The IDS is one of the simplest, but nonetheless physically important spectral characteristics of the operators we consider. It encodes all thermostatic properties of a corresponding gas of non-interacting particles. As an example we mention a systems of electrons in a solid, where this is a reasonable approximation in many situations. Besides, the IDS enters transport coefficients such as the electric conductivity and determines the ionisation properties of atoms and molecules. For this reason, the IDS (more precisely, its derivative with respect to E, the density of states) is a widely studied quantity in physics. 2.3. The integer lattice In this subsection we are concerned with the asymptotics at spectral edges of the IDS of the family of Laplacians (∆X Gω )ω∈Ω on bond-percolation subgraphs of the d-dimensional integer lattice graph Ld (or bond percolation on Zd , for short).

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The spectral edges of these Laplacians turn out to be 0 and 4d. In fact, standard arguments [36], which are based on ergodicity w.r.t. Zd -translations, yield that even the whole spectrum equals almost surely the one of the Laplacian ∆Ld on the full lattice spec(∆X Gω ) = [0, 4d]

for Pp -almost every ω ∈ Ω,

any p ∈]0, 1] and X ∈ {N, A, D}. Thus, the left-most and right-most inequality in (1.2) are sharp in this case. Since the lattice Ld is bipartite, it follows from (1.3) with k = 2d that the different Laplacians are related to each other by a unitary involution, which implies the symmetries NA (E) = 1 − lim NA (ε) , ε↑4d−E

ND(N ) (E) = 1 − lim NN (D) (ε)

(2.5)

ε↑4d−E

for their integrated densities of states for all E ∈ [0, 4d]. The limits on the righthand sides of (2.5) ensure that the discontinuity points of NX are approached from the correct side. As before we write pc ≡ pc (d) for the unique critical probability of the bondpercolation transition in Zd . We recall from [25] that pc = 1 for d = 1, otherwise pc ∈]0, 1[. Let us first think about what to expect. At least for small p, the random graph Gω is decomposed into relatively small pieces, due to Theorem 2.1 above. This means that there cannot be many small eigenvalues as the size of the components limits the existence of low lying eigenvalues. Consequently, the eigenvalue-counting function for small E must be small. It turns out that the IDS vanishes even exponentially fast. This striking behaviour is called Lifshits tail, to honour Lifshits’ fundamental contributions to solid state physics of disordered systems [40, 41, 42]. In fact, Lifshits tails continue to show up in the percolating phase for the adjacency and the Dirichlet Laplacian at the lower spectral edge. This follows from a large-deviation principle. Theorem 2.6 ([51], Theorem 2.5). Assume d ∈ N and p ∈]0, 1[. Then the integrated density of states NX of the Laplacians (∆X Gω )ω∈Ω on bond-percolation graphs in Zd exhibits a Lifshits tail at the lower spectral edge ln | ln NX (E)| d =− E↓0 ln E 2 and at the upper spectral edge ln | ln[1 − NX (E)]| d lim =− E↑4d ln(4d − E) 2 lim

for

X ∈ {A, D}

(2.6)

for

X ∈ {N, A} .

(2.7)

Actually, slightly stronger statements without logarithms are proven in [51], see the next lemma. Together with the symmetries (2.5), these bounds will imply the above theorem.

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Lemma 2.7 ([51], Lemma 3.1). For every d ∈ N and every p ∈]0, 1[ there exist constants εD , αu , αl ∈]0, ∞[ such that exp{−αl E −d/2 }  ND (E)  NA (E)  exp{−αu E −d/2 }

(2.8)

holds for all E ∈]0, εD [. Remarks 2.8. (1) In the non-percolating phase, p ∈]0, pc [, the content of Theorem 2.6 has already been known from [36], where it is proved by a different method. The method of [36], however, does not seem to extend to the critical point or the percolating phase, p ∈]pc , 1[. (2) The Lifshits asymptotics of Theorem 2.6 are determined by those parts of the percolation graphs which contain large, fully-connected cubes. This also explains why the spatial dimension enters the Lifshits exponent d/2. (3) We expect that (2.6) can be refined in the adjacency case X = A as to obtain the constant ln NA (E) lim =: −c∗ (d, p) . (2.9) E↓0 E −d/2 An analogous statement is known from Theorem 1.3 in [12] for the case of site-percolation graphs. Moreover, it is demonstrated in [4] that the bondand the site-percolation cases have similar large-deviation properties. The second main result of this subsection complements Theorem 2.6 in the non-percolating phase. Theorem 2.9 ([36], Theorem 1.14). Assume d ∈ N and p ∈]0, pc [. Then the integrated density of states of the Neumann Laplacians (∆N Gω )ω∈Ω on bond-percolation d graphs in Z exhibits a Lifshits tail with exponent 1/2 at the lower spectral edge lim

E↓0

ln | ln[NN (E) − NN (0)]| 1 =− , ln E 2

(2.10)

while that of the Dirichlet Laplacians (∆D Gω )ω∈Ω exhibits one at the upper spectral edge − ln | ln[ND (4d) − ND (E)]| 1 lim =− , (2.11) E↑4d ln(4d − E) 2 − where ND (4d) := limE↑4d ND (E) = 1 − NN (0).

Remarks 2.10. (1) This theorem also follows from sandwich bounds analogous to those in Lemma 2.7. We do not state them here but refer to Lemmas 2.7 and 2.9 in [36] for details. Using interlacing techniques, [56] establishes a better control on the constants in these bounds. For example, it was found that for all sufficiently small energies E NN (E) − NN (0)  AE exp{−α+ E −1/2 }

(2.12)

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√ with α+ := 4/[3 3χ4p ], where χp stands for the expected number of vertices in the cluster containing the origin and where the constant A > 0 can also be made explicit. (2) The constant NN (0) appearing in Theorem 2.9 is given by   tr
2 (Λ) 1[0,∞[ −∆N Gω ,Λ NN (0) = lim = ρ(p) + (1 − p)2d (2.13) |Λ| Λ↑Zd and equals the mean number density ρ(p) of clusters with at least two and at most finitely many vertices, see, e.g., Chapter 4 in [25], plus the number density of isolated vertices. This follows from the fact that the operator 1[0,∞[ (−∆N Gω ,Λ ) is nothing but the projector onto the null space of the restricN 2 tion ∆N Gω ,Λ of ∆Gω to (Λ). The dimensionality of this null space equals the number of finite clusters and isolated vertices of Gω in Λ, see Remark 1.5(iii) in [36]. (3) The Lifshits tail for NN at the lower spectral edge – and hence the one for ND at the upper spectral edge – is determined by the linear clusters of bond-percolation graphs. This explains why the associated Lifshits exponent −1/2 is not affected by the spatial dimension d. Technically, this relies on a Cheeger inequality [18] for the second-lowest Neumann eigenvalue of a connected graph, see also Prop. 2.2 in [36]. The third main result of this subsection is the counterpart of Theorem 2.9 in the percolating phase. Theorem 2.11 ([51], Theorem 2.7). Assume d ∈ N \ {1} and p ∈]pc , 1[. Then the integrated density of states of the Neumann Laplacians (∆N Gω )ω∈Ω on bondpercolation graphs in Zd exhibits a van Hove asymptotic at the lower spectral edge ln[NN (E) − NN (0)] d lim = , (2.14) E↓0 ln E 2 while that of the Dirichlet Laplacian ∆D Gω exhibits one at the upper spectral edge lim

E↑4d

− ln[ND (4d) − ND (E)] d = . ln(4d − E) 2

(2.15)

Similar to the two theorems above, Theorem 2.11 also follows from upper and lower bounds and the symmetries (2.5). Lemma 2.12 ([51], Lemma 4.1). Assume d ∈ N \ {1} and p ∈]pc , 1[. Then there exist constants εN , Cu , Cl ∈]0, ∞[ such that Cl E d/2  NN (E) − NN (0)  Cu E d/2

(2.16)

holds for all E ∈]0, εN [. Remarks 2.13. (1) Lemma 2.12 relies mainly on recent random-walk estimates [46, 8, 29] for the long-time decay of the heat kernel of ∆N Gω on the infinite cluster.

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(2) There is also an additional Lifshits-tail behaviour with exponent 1/2 due to finite clusters as in Theorem 2.9, but it is hidden under the dominating van Hove asymptotic of Theorem 2.11. Loosely speaking, Theorem 2.11 is true because the percolating cluster looks like the full regular lattice on very large length scales (bigger than the correlation length) for p > pc . On smaller scales its structure is more like that of a jagged fractal. The Neumann Laplacian does not care about these small-scale holes, however. All that is needed for the van Hove asymptotic to be true is the existence of a suitable d-dimensional, infinite grid. The adjacency and Dirichlet Laplacians though do care about those small-scale holes, as we infer from Theorem 2.6. (3) In the physics literature the terminology van Hove “singularity” is also used for this kind of asymptotic. This refers to the fact that for odd dimensions d derivatives seize to exist for high enough order. The above three theorems cover all cases for p and X except the behaviour at the critical point p = pc of NN at the lower spectral edge, respectively that of ND at the upper spectral edge. In dimension d = 2 upper and lower power-law bounds have been obtained in [57]. However, the exponents differ so that the asymptotics is still an open problem; see also Remark 2.17 (3) below for further properties at criticality. 2.4. The regular infinite tree (Bethe lattice) In this subsection we report results from [55] on the asymptotics at spectral edges for the IDS of the family of Laplacians (∆X Gω )ω∈Ω on bond-percolation subgraphs of the (κ+1)-regular rooted infinite tree, a.k.a. Bethe lattice Bκ , where κ ∈ N\{1}. Percolation on regular trees is well studied, see, e.g., [54], and it turns out that the bond-percolation transition occurs sharply at the unique critical probability pc = κ−1 . Here, sharpness of the phase transition is implied by, e.g., Theorem 2.1, but it can also be verified by explicit computations. In contrast to percolation on the hypercubic lattice Ld , where the infinite cluster of the percolating phase is unique, there exist infinitely many percolating clusters simultaneously for p > pc on Bκ . The results on spectral asymptotics of the IDS are analogous in spirit to the ones of the previous subsection, but restricted to the non-percolating phase. However, as the Bethe lattice Bκ exhibits an exponential volume growth of the ball B(n) of radius n about its root V (n) = |B(n)| = 1 + (κ + 1)

n  ν=1

κν−1 = 1 + (κn − 1)

κ+1 , κ−1

cf. Figure 2, there will be natural differences. The next lemma determines the spectral edges of the operators under consideration. As a consequence of the exponential growth of the graph, and in contrast to the preceding subsection, the spectrum of the Laplacian on the Bethe lattice does not start at zero, neither does it extend up to twice the degree 2(κ + 1).

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Lemma 2.14. Let κ ∈ N \ {1} and let ∆Bκ be the Laplacian on the (full) Bethe lattice Bκ . Then √ spec(∆Bκ ) = [Eκ− , Eκ+ ], where Eκ± := ( κ ± 1)2 . Moreover, for P-almost every realisation Gω of bond-percolation subgraphs of Bκ we have + spec(∆N Gω ) ⊆ [0, Eκ ],

− + spec(∆A Gω ) = [Eκ , Eκ ],

− spec(∆D Gω ) ⊆ [Eκ , 2(κ + 1)].

Remarks 2.15. (1) We believe that equality (and not only “⊆”) holds for the statements involving the Neumann and the Dirichlet Laplacians, too. (2) Since the Bethe lattice is bipartite the above lemma reflects the symmetries (1.3). (3) Almost-sure constancy of the spectra (i.e., independence of ω) is again a consequence of ergodicity of the operators, see, e.g., [1] for a definition of the ergodic group action. The ergodic group action on the Bethe lattice, which was referred to in the last remark above, is even transitive so that the IDS NX of the family (∆X Gω )ω∈Ω can be defined as in Definition 2.2 with the fundamental cell F consisting of just the root. Clearly, NX will then obey the symmetry relations NA (E) = 1 − ND(N ) (E) = 1 −

lim ε↑2(κ+1)−E

lim ε↑2(κ+1)−E

NA (ε) , NN (D) (ε)

(2.17)

for all E ∈ [0, 2(κ + 1)]. Our first result concerns the asymptotic of NN at the lower edge, resp. of ND at the upper edge. Since these two spectral edges are unaffected by the exponential volume growth, it comes as no surprise that we find the same type of Lifshits tail as in the Zd -case. Theorem 2.16 ([55]). Assume κ ∈ N\{1} and p ∈]0, pc [. Then the integrated density of states of the Neumann Laplacians (∆N Gω )ω∈Ω on bond-percolation graphs in Bκ exhibits a Lifshits tail with exponent 1/2 at the lower spectral edge lim

E↓0

ln | ln[NN (E) − NN (0)]| 1 =− , ln E 2

(2.18)

while that of the Dirichlet Laplacian ∆D Gω exhibits one at the upper spectral edge lim

E↑2(κ+1)

− ln | ln[ND (2(κ + 1)) − ND (E)]| 1 =− , ln(2(κ + 1) − E) 2

− where ND (2(κ + 1)) := limE↑2(κ+1) ND (E) = 1 − NN (0).

(2.19)

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Remarks 2.17. (1) These asymptotics are again determined by the linear clusters of bond-percolation graphs, cf. Remark 2.10 (3). The interpretation of the reference value NN (0) in terms of the cluster plus isolated vertex density is analogous to Remark 2.10 (2). (2) In contrast to this Lifshits-tail behaviour in the subcritical phase, one expects NN (E) − NN (0) to obey a power-law for small E at the critical point pc , caused by the finite critical clusters. This is not yet fully confirmed, but upper and lower algebraic bounds (with different exponents) follow from the random-walk estimates in [57]. (3) It should be noted that the power-law behaviour at pc mentioned in the previous remark is not the one referred to by the famous Alexander-Orbach conjecture [3]. The latter concerns the E 4/3 -behaviour as E → 0 of NN (E) on the incipient infinite percolation cluster. For the case of the Bethe lattice this asymptotic was proven in [9]. (Here no subtraction of NN (0) is necessary. Instead, one kind of conditions on the event that the origin belongs to an infinite cluster, see, e.g., [14] for details of the definition.) The Alexander-Orbach conjecture says that the E 4/3 -asymptotic should also hold for percolation in Zd for every d  2. Extensive numerical simulations indicate that this is not true in d = 2 [24]. We refer to [16] for a comprehensive discussion and further references from a Physics perspective. In order to reveal the characteristics of the Bethe lattice we now turn to the spectral edges Eκ± . Theorem 2.18 ([55]). Assume κ ∈ N \ {1} and p ∈]0, pc [. Then the integrated density of states of (∆X Gω )ω∈Ω on bond-percolation graphs in Bκ exhibits a doubleexponential tail with exponent 1/2 at the lower spectral edge   ln ln | ln NX (E)| 1 lim− =− for X = A, D (2.20) − 2 ln(E − E ) E↓Eκ κ and one at the upper spectral edge     ln ln  ln 1 − NX (E)  1 lim+ =− + 2 ln(Eκ − E) E↑Eκ

for X = N, A.

(2.21)

Remarks 2.19. (1) The extremely fast decaying asymptotic of (2.20) – and similarly that of (2.21) – is determined by the lowest eigenvalues E ∼ Eκ− + R−2 of those clusters in the percolation graph which are large fully connected balls of radius R. − −1/2 Their volume is exponentially large in the radius, V (R) ∼ eR ∼ e(E−Eκ ) , and their probabilistic occurrence is exponentially small in the volume. (2) One would expect Theorem 2.18 to be valid beyond the non-percolating phase. However, the region p  pc is still unexplored. (3) A double-exponential tail as in (2.20) will also be found in Theorem 2.24 (3) below. This concerns the lower spectral edge of the IDS for percolation on

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P. M¨ uller and P. Stollmann the Cayley graph of the lamplighter group, which is amenable. These doubleexponential tails in two concrete situations should also be compared to the less precise last statement of Theorem 2.21 below, which, however, holds for superpolynomially growing Cayley graphs of arbitrary, finitely generated, infinite, amenable groups.

2.5. Equality and non-equality of Lifshits and van Hove exponents on amenable Cayley graphs . . . is almost the title of a paper by Antunovi´c and Veseli´c [7]. Here we record their main results. In our definition of the IDS in Subsection 2.2 above, two entirely different cases were treated. Let us first consider the deterministic case of the Laplacian on the full graph, denoted by Nper . In our case of a quasi-transitive graph the geometry looks pretty regular; just like in the case of a lattice, the local geometry has the same local structure everywhere. Specializing to Cayley graphs this allows one to relate the asymptotic of Nper near 0 to the volume growth V (n) defined in (1.4). The latter is the same for the different Cayley graphs of the same group, see Theorem 1.1 above. Theorem 2.20. Let Γ be an infinite, finitely generated, amenable group, G = G(Γ, S) a Cayley graph of Γ and Nper the associated IDS. If G has polynomial growth of order d, then ln Nper (E) d lim = . (2.22) E↓0 ln E 2 If G has superpolynomial growth, then ln Nper (E) lim = ∞. E↓0 ln E Proofs can be found in [60, 44]. Note that the limit appearing in (2.22) is exactly the zero-order Novikov-Shubin invariant, where zero-order refers to the fact that we deal with the Laplacian on 0-forms, i.e., functions. Next we turn to the asymptotic of the IDS NX of the corresponding percolation subgraphs. Again, Lifshits tails are found. Theorem 2.21 ([7], Theorem 6). Let G = G(Γ, S) be the Cayley graph of an infinite, finitely generated, amenable group. Let NX be the IDS for the Laplacians (∆X Gω )ω∈Ω of percolation subgraphs of G with boundary condition X ∈ {A, D} in the subcritical phase, i.e., for p < pc . Then there is a constant ap > 0 so that for all E > 0 small enough    ap ˜ 1 − 12 √ ND (E)  NA (E)  exp − V E −1 , 2 2 2|S| where V˜ (t) := V (,t-), the volume V (n) is given by (1.4) and ,t- denotes the integer part of t ∈ R. If G has polynomial growth of order d, then there are constants − α+ D , αD > 0 so that for E > 0 small enough   −d −d 2 2 exp −α−  N . E (E)  N (E)  exp −α+ D A D DE

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If G has superpolynomial growth, then lim

E↓0

ln | ln ND (E)| ln | ln NA (E)| = lim = ∞. E↓0 | ln E| | ln E|

(2.23)

Theorem 2.18 and Theorem 2.24 (3) provide much more detailed information as compared to (2.23), but only in two specific situations: the non-amenable free group with n  2 generators and the amenable lamplighter group. The equality that is mentioned in the title of this subsection is now an easy consequence. Corollary 2.22. In the situation of the preceding theorem the van Hove exponent and Lifshits exponents for X ∈ {A, D} coincide, i.e., lim

E↓0

ln | ln ND (E)| ln | ln NA (E)| ln Nper (E) = lim = lim . E↓0 E↓0 | ln E| | ln E| ln E

Note that the asymptotic proved for ND and NA in the case of polynomially growing Cayley graphs is actually more precise than the double-log-limit that appears in the preceding corollary. For Cayley graphs with superpolynomial growth, a lower estimate is missing. However, for the lamplighter groups a more precise statement can be proven, see Theorem 2.24 below. The results of the previous section for the lattice case indicate that one should expect a different behaviour for the IDS NN of the Neumann Laplacian at the lower spectral edge: it should be dominated by the linear clusters for p < pc . This is indeed true. Theorem 2.23 ([7], Theorem 14). In the situation of the previous theorem there − exist constants α+ N , αN > 0 so that for all E > 0 small enough   − 12 − 12 exp −α− E  N (E) − N (0)  exp −α+ . N N N NE The dimension d is replaced by 1 in these estimates, since linear clusters are effectively one-dimensional and independent of the volume growth of G. This latter result remains true for quasi-transitive graphs with bounded vertex degree. As already announced, here are the more detailed estimates for the lamplighter group. Theorem 2.24 ([7], Theorems 11 and 12). Let G be a Cayley graph of the lamplighter group Zm  Z. + (1) There are constants a+ 1 , a2 > 0 so that for all E > 0 small enough  + − 12 . Nper (E)  a+ 1 exp −a2 E

(2) For every r > enough

1 2

− there are constants a− 1,r , a2−r > 0 so that for all E > 0 small

 − −r  2 Nper(E)  a− . 1,r exp −a2,r E

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(3) For every p < pc there are constants b1 , b2 , c1 , c2 > 0 so that for all E > 0 small enough     −1 −1 exp −c1 ec2 E 2  ND (E)  NA (E)  exp −b1 eb2 E 2 . 2.6. Outlook: some further models To conclude, we briefly mention two other percolation graph models for which the Neumann Laplacian exhibits a Lifshits-tail behaviour with Lifshits exponent 1 at the lower spectral edge E = 0 in the non-percolating phase. As in the cases 2 we discussed above, see Theorem 2.9 for the integer lattice, Theorem 2.16 for the Bethe lattice and Theorem 2.23 for amenable Cayley graphs, these Lifshits tails will also be caused by the dominant contribution of linear clusters. For this reason they occur quite universally, as long as the cluster-size distribution of percolation follows an exponential decay – no matter how complicated the “full” graph G may look like. This structure will not be seen by the linear clusters of percolation! The first class of models [49, 50] consists of graphs G which are embedded into Rd (or, more generally, into a suitable locally compact, complete metric space) with some form of aperiodic order. The celebrated Penrose tiling in R2 constitutes a prime example. But one can consider rather general graphs whose vertices form a uniformly discrete set in Rd and whose edges do not extend over arbitrarily long distances. Amazingly, the main point that needs to be dealt with to establish Lifshits tails for such models concerns the definition of the IDS. In contrast to the definition in (2.3), one cannot expect to benefit from a quasi-transitive group action on G with a finite fundamental cell in this aperiodic situation. The way out is to consider the hull of the graph G, that is the set of all Rd -translates of G, closed in a suitable topology which renders the hull a compact dynamical system. As such it carries at least one Rd -ergodic probability measure µ, and the expectation in (2.3) will be replaced by a two-stage expectation: one with respect to µ over all graphs G
in the hull of G, and inside of it, for each graph G
, the expectation (G ) Ep over all realisations of percolation subgraphs of G
. The interested reader is referred to [37, 50] for more details. The second model, Erd˝ os-R´enyi random graphs [23, 13], has a combinatorial background. There we consider bond percolation on the complete graph Kn over n vertices with bond probability p := c/n. The n-independent parameter c > 0 corresponds to twice the expected number density of bonds, if n is large. This is sometimes referred to as the (very) sparse case. For c ∈]0, 1[, the fraction of vertices belonging to tree clusters tends to 1 as n → ∞, and the limiting clustersize distribution decays exponentially. In this model the IDS is defined by  (K )  NN (E) := lim Ec/nn δ1 , 1]−∞,E] (∆N Gω )δ1  , n→∞

and it exhibits a Lifshits tail at the lower spectral edge E = 0 with exponent

1 2

[33].

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Acknowledgment Many thanks to the organisers of the Alp-Workshop at St. Kathrein for the kind invitation and the splendid hospitality extended to us there.

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Progress in Probability, Vol. 64, 259–275 c 2011 Springer Basel AG


Survey of Scalings for the Largest Connected Component in Inhomogeneous Random Graphs Tatyana S. Turova Abstract. We review some recent results on the exact asymptotics of the components in the inhomogeneous random graph models of rank 1. We discuss the relevance of these results to the analysis of random walk on random graphs. Mathematics Subject Classification (2000). Primary 60C05; Secondary 05C80. Keywords. Inhomogeneous random graphs, phase transitions.

1. Introduction 1.1. Phase transitions in Inhomogeneous random graphs A wide variety of random graph models strongly attracts interest of nearly all the natural science disciplines in the last decade. The watershed paper by Bollob´ as, Janson and Riordan [3] published in 2007 provided a unified approach to many of these models (see the reference list in [3]). Since then the theory of inhomogeneous random graphs, whose core and basis is in the paper [3], is a rapidly developing area of probability. The authors of [3] themselves extended the theory to an even more general class of models [4], and provided a powerful criterion to classify the limiting behaviour of different graph models [14]. Among the directions of the research prompted by publication [3] is the analysis of one particular (simple but versatile for the applications) class of inhomogeneous models, the so-called rank 1 model. In the last 2 years a number of new results were obtained about this model, complementing the ones known from [3]. One of the most intriguing features of the random graphs is the phase transition. Let us explain this briefly. Let Gn denote some random graph on the vertices {1, . . . , n}, and let C1 (Gn ) denote the size of the largest connected component of Gn . Hence, C1 (Gn ) is also random. The question of interest is: For Research was supported by the Swedish Natural Science Research Council.

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which (deterministic) monotone increasing function K(n) does there exist a limit in distribution C1 (Gn ) lim ? n→∞ K(n) When this limit is non-trivial (i.e., identical neither to zero nor to infinity) the function K(n) defines a scaling for C1 (Gn ). Consider for example, the Erd¨ osR´enyi model Gn,p [9], where any two vertices i and j in {1, . . . , n} are connected independently with a probability pij (n) = c/n. The following convergence results, along with even more precise ones, were derived already in [9]: if

c<1 if

then c>1

C1 (Gn,c/n ) P 1 → ; log n c − 1 + | log c|

(1.1)

C1 (Gn,c/n ) P → β, n

(1.2)

then

where β is the largest positive root of β = 1 − e−cβ ;

(1.3)

C1 (Gn,c/n ) d → γ, (1.4) n2/3 where γ is some non-degenerated positive random variable. (We shall cite below a more exact statement for the latter case due to Aldous [1].) Summarizing, we see that the largest component in the Gn,p model with p = c/n exhibits three scalings: log n, n2/3 and n. We shall collect here the recent results which are counterparts to (1.1), (1.2) and (1.4) for the inhomogeneous graphs, where the probability of an edge pij (n) depends on the vertices i and j. We will conclude with a fairly complete picture of the possible scalings for the largest component in a special class of inhomogeneous models, namely the rank 1 model. As one may predict, the spectra of scalings is much richer in the inhomogeneous case. We refer to [3] and the references therein on the results related to the scalings of some other macro-characteristics of the graph, e.g., the diameter or the distances between the vertices. In particular, the scalings for the distance in rank 1 models were studied in [11]. if c = 1

then

1.2. Inhomogeneous random graphs We begin by recalling some basic definitions and assumptions from [3]. Let S be a separable metric space and µ be a Borel probability measure on S. For each n (n) (n) let (x1 , . . . , xn ) be a deterministic or a random sequence of points in S. Call (n)

V = (S, µ, (x1 , . . . , xn(n) )n≥1 ) (n)

(n )

a vertex space. No relationship is assumed between xi and xi , but to simplify (n) (n) notation we shall from now on write (x1 , . . . , xn ) = (x1 , . . . , xn ). Assume that

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for any µ-continuity set A ⊆ S #{i : xi ∈ A} P → µ(A). (1.5) n Definition 1.1. Given the sequence x1 , . . . , xn , we let GV (n, κ) be the random graph on {1, . . . , n}, such that any two vertices i and j are connected by an edge independently of the others vertices with probability pij (n) = min{κ(xi , xj )/n, 1},

(1.6)

where κ is a symmetric nonnegative measurable kernel defined on S × S. Furthermore, we assume that the kernel κ is graphical on V, which means that (i) κ is continuous a.e. on S × S; (ii) κ ∈ L1 (S × S, µ × µ); (iii)   1  V 1 Ee G (n, κ) → κ(x, y)dµ(x)dµ(y), n 2 S2 where e(G) denotes the number of edges in a graph G. We shall also consider some generalization of the kernels (hence, probabilities (1.6)) allowing dependence on n. In this case we denote the model GV (n, κn ). Recall another definition from [3]. Definition 1.2. A sequence of kernels (κn ) on S × S is graphical on V with limit κ if, for a.e. (y, z) ∈ S 2 , yn → y and zn → z imply that κn (yn , zn ) → κ(y, z), κ satisfies conditions (i) and (ii) of Definition 1.1, and   1  V 1 Ee G (n, κn ) → κ(x, y)dµ(x)dµ(y). n 2 S2 A simple but yet a non-trivial example of the inhomogeneous graph model is when κ(x, y) = ψ(x)ψ(y), x, y ∈ S (1.7) for some nonnegative function ψ. This is one of the most studied and well-understood special cases of the inhomogeneous graphs, which is called rank 1 model (see Section 16.4 in [3]). This case has proved to be versatile for the applications. One may interpret ψ(x) as the “activity” of a vertex of type x. One particularly often seen choice of ψ is ψ(x) = x on S = {1, 2, . . .}. Here “type x” can represent the degree of a node as in [8] or the size of a macro-vertex as in [27] (many more examples one can find in [3]). Remark 1.3. Without loss of generality one can consider the rank 1 model with the kernel given by κ(x, y) = cxy, x, y ∈ S ⊆ R+. (1.8) Indeed, by rescaling the space S and, correspondingly, measure µ, one can reduce model (1.7) to the one with kernel (1.8).

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Example. (Gn,p model.) The Erd¨ os-R´enyi model Gn,p [9] with p = c/n can be presented also as the rank 1 model GV (n, κ) with kernel (1.8), if we simply set S = {1} to be a one point space. Example. (Rank 1 model Gn,X on i.i.d. vertices.) Let X denote a positive random variable with distribution µ. Let x1 , . . . , xn be independent copies of X. Assume, EX < ∞. Then we can define a model GV (n, κ) with S = R+ and κ(x, y) = xy, x, y ∈ R+. This means by (1.6) that the probability of edge between i and j is equal to xi xj pij = (1.9) n Denote the corresponding model Gn,X . This will be our principle example here. We will see that C1 (Gn,X ) admits (for different parameters of distribution of X) not only the former scalings as for the Gn,p model, but also scalings in a form nα for all 1/2 < α ≤ 2/3 and 0 < α < 1/2.

2. Branching processes associated with the inhomogeneous graph 2.1. The breadth-first walk for inhomogeneous random graph and branching processes It was observed in [16] that a random graph can be naturally related to a certain branching process underlying the algorithm of revealing the connected components in the graph. The algorithm is defined as follows. Take any vertex 1 ≤ i ≤ n to be the root. Find all the vertices connected to this vertex i in the graph GV (n, κ), and then mark i as “saturated”. Then for each non-saturated revealed vertex j we find all the vertices connected to it but which have not been revealed previously, and then mark j as “saturated”. We continue this process until we end up with a tree of saturated vertices. The set of vertices in the tree constructed according to this algorithm forms a connected component of the graph. It is clear at least intuitively, that the described algorithm can be approximated by a multi-type branching process with the type space S. Consider first a simple case when S is at most countable. Then for any vertex i in the graph GV (n, κ) the number of the neighbouring vertices of type x has (for all large n) the Binomial distribution  κ(xi , x) Bin # {j ∈ {1, . . . , n} \ {i} : xj = x} , , n which by the assumption (1.5) converges to the Poisson distribution Po(κ(xi x)µ(x))

(2.1)

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(here Po(m) denotes the Poisson distribution with expectation m.) Therefore in this case it is natural to approximate the algorithm of finding a connected component by a branching process with Po(κ(xi x)µ(x)) number of the offspring of type x. The branching processes approach for the random graphs was (probably for the first time) used in [16] and then developed also in [13], in particular for the supercritical phase. Already in [16] it was shown that the constant β in (1.2) equals the survival probability of the associated branching process. This idea was extended in [23] where a multi-type branching process was introduced to study connectivity in certain inhomogeneous random graphs. A solid part of theory of branching processes relevant to the study of inhomogeneous graphs was elaborated in [3]. In particular, it was proved in [3] that the size of the connected component (scaled by n) is also given by the survival probability of the associated branching process. Observe that this is mostly relevant to the supercritical phase, where the survival probability is positive (see Section 3). More recent study of the subcritical case in [24] and [25] continues approach of [3] (see Section 4). It was proved in [24] and [25] that in the subcritical case, i.e., when the survival probability is zero, the size of the connected component (scaled by log n) can be derived from another characteristic of the branching process, namely the distribution of total progeny. Observe that the methods of investigating the off-critical phases are rather different from the ones in the critical regime despite the fact they are based as well on the branching process approach through the breadth-first walk. Aldous [1] developed branching process approach in combination with martingale technique for the study of critical graphs [1]. Briefly, studying the critical phase involves consideration of a sequence of components, whose distributions are dependent. Therefore in the inhomogeneous case an essential part of the algorithm is the definition of the roots for the consecutively revealed components. In particular, size-biased choice is very important for the theory of multiplicative coalescent developed in [1] and [2]. The independent work of Martin-L¨of [18] is very similar in a spirit to [1] (although written in the context of critical epidemics). This approach received a very recent development in [5], [6] and [26] (see Section 5) for the models that we consider here, but also for some different graph models in [19]. 2.2. Branching processes results Define the multi-type Galton-Watson process Bκ (x) as follows. The type space of Bκ (x) is S, and initially there is a single particle of type x ∈ S. Then at any step, a particle of type x ∈ S is replaced in the next generation by a set of particles whose types are distributed as Poisson process on S with intensity measure κ(x, y)dµ(y). (This is a generalization of (2.1).) 2.2.1. Survival probability. Let ρκ (x) denote the survival probability of process Bκ (x), i.e., the probability that a particle of type x produces an infinite population.

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First we state a general result on ρκ (x) which was proved in [3]. Define  Tκ f (x) = κ(x, y)f (y)dµ(y), S

and Tκ  = sup{Tκ f 2 : f ≥ 0, f 2 ≤ 1}. Theorem 2.1 ([3], Theorem 6.1). Suppose that κ is the kernel on (S, µ), that κ ∈ L1 , and  κ(x, y)dµ(y) < ∞

(2.2)

S

for every x ∈ S. Then ρκ (x) is the maximal solution to f (x) = 1 − exp{−Tκf (x)}.

(2.3)

Furthermore: (i) If Tκ  ≤ 1 then ρκ (x) = 0 for every x, and (2.3) has only the zero solution. (ii) If 1 < Tκ  ≤ ∞ then ρκ (x) > 0 on a set of a positive measure. If, in addition, κ is irreducible, i.e., if A ⊆ S and κ = 0 a.e. on A × (S \ A) implies µ(A) or µ(S \ A) = 0, then ρκ (x) > 0 for a.e. x, and ρκ (x) is the only non-zero solution of (2.3). Notice that f = 0 is always a solution to (2.3) independent of value Tκ . Hence, the parameters when Tκ  = 1 are critical for the introduced branching process. Remark 2.2. If κ(x, y) = ψ(x)ψ(y), x, y ∈ S, then operator Tκ has rank 1 (therefore the name rank 1 model). In this case   1/2  2 Tκ  = κ (x, y)dµ(x)dµ(y) = ψ(x)2 dµ(x). (2.4) S

S

S

(For further details consult [3].) 2.2.2. Total progeny. Let us consider now another characteristic of the process Bκ (x) which is also important for the study of random graphs. Denote X (x) the size of the total progeny of Bκ (x), and let    rκ = sup {z ≥ 1 : E z X (x) dµ(x) < ∞}. (2.5) S

One should notice the direct relation of rκ to the tail of distribution of the total progeny X (x). If the tail of distribution of X (x) decays exponentially, then rκ defines the constant in the exponent. In particular, we will discuss the conditions for the exponential decay of the tail, i.e., whether rκ = 1 or rκ > 1. In the case of a single-type branching process the exact result on the relation between rκ and the distribution of the total progeny was proved in [21].

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inf κ(x, y) > 0.

(2.6)

Theorem 2.3 ([25]). Let x,y∈S

Then rκ is the supremum value of all z ≥ 1 for which equation f (x) = 1 − ze−Tκ f (x)

(2.7)

has µ-a.s. finite solution f ≤ 0. Observe that when z = 1 equation (2.7) is exactly the same as equation (2.3). Notice however, that here we are looking for the negative solutions, which are unbounded, unlike the positive ones. Next we state some sufficient conditions for rκ > 1. Let Tκ have a finite Hilbert-Schmidt norm, i.e.,   1/2 2 Tκ HS := κL2 (S×S) = κ (x, y)dµ(x)dµ(y) < ∞. (2.8) S

Define



S

1/2 κ (x, y)dµ(y) , 2

Ψ(x) =

(2.9)

S

and assume that for some positive constant a > 0  eaΨ(x) dµ(x) < ∞.

(2.10)

S

Theorem 2.4 ([25]). I. Let κ satisfy (2.2). Then rκ = 1

if

Tκ  ≥ 1.

(2.11)

rκ > 1

if

Tκ  < 1

(2.12)

II. Let κ satisfy (2.10). Then

and at least one of the following conditions is satisfied (C1) supx,y∈S κ(x, y) < ∞, or (C2) Tκ HS < 1, or (C3) S ⊆ R and κ(x, y) is non-decreasing in both arguments, and such that for some constant c1 > 0 κ(x, y) ≤ c1 Tκ [1](x)Tκ [1](y),

(2.13)

for all x, y ∈ S. Observe that for all kernels Tκ  ≤ Tκ HS , where the equality holds only in the rank 1 case (1.7). Hence, Theorem 2.4 yields immediately the following necessary and sufficient conditions for the rank 1 case.

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Corollary 2.5. Let κ(x, y) = ψ(x)ψ(y), x, y ∈ R+ , and distribution µ on R+ satisfy  ∞ eaψ(x) dµ(x) < ∞ (2.14) 0

for some positive constant a > 0. Then rκ > 1

if and only if Tκ  < 1,

where

 Tκ  =

∞

ψ(x)2 dµ(x).

(2.15)

(2.16)

0

Otherwise, rκ = 1. Notice that under the assumptions of this corollary, i.e., in the rank 1 case, (2.14) is equivalent to (2.10), and (2.16) holds by (2.4). The statement of this corollary was first derived in [24] where the simple form of the rank 1 case was exploited. In the rank 1 case it is possible to compute rκ in a rather closed form as we state below. Theorem 2.6 ([24]). Assume, the conditions of Corollary 2.5 are satisfied. Let X be a random variable with distribution µ on S, and define a positive constant M = Eψ(X). If Tκ  < 1 (i.e., Eψ 2 (X) < 1) there exists a unique y > 1 which satisfies y=

1 Eψ(X) exp {M (y − 1)ψ(X)} , M Eψ 2 (X) exp {M (y − 1)ψ(X)}

(2.17)

1 . Eψ 2 (X) exp {M (y − 1)ψ(X)}

(2.18)

and then rκ =

In the following sections we will show how the quantities ρκ and rκ define the size of the largest components in the random graphs.

3. Supercritical phase: Tκ > 1 Let ρκ (x) be the maximum solution to (2.3), and define  ρκ = ρκ (x)dµ(x). S

Theorem 3.1 ([3], Theorem 3.1). Let (κn ) be a graphical sequence of kernels on a generalized vertex space V with limit κ. (i) If Tκ  ≤ 1, then C1 (GV (n, κn )) = oP (n), while if Tκ  > 1, then C1 (GV (n, κn )) = Θ(n) whp.

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(ii) For any ε > 0, whp we have 1 C1 (GV (n, κn )) ≤ ρκ + ε. n (iii) If κ is quasi-irreducible, then 1 P C1 (GV (n, κn )) → ρκ . n In all cases ρκ > 0 if and only if Tκ  > 1. This theorem gives us the exact scaling (n) for the largest component in the supercritical phase, i.e., when Tκ  > 1. Also, it tells us that when Tκ  ≤ 1 the largest connected component is o(n) with probability tending to one as n → ∞. Although [3] provides no more exact information on the scalings of C1 in the inhomogeneous case, the detailed analysis of the phase transition at criticality from above (i.e., when Tκ  ↓ 1) indicates what are the other characteristics besides Tκ  which come into play when the phase transitions (i.e., different scalings) are concerned. Consider now a rank 1 model GV (n, κ) with S = R+ , where x1 , . . . , xn are independent copies of X, and κ(x, y) = cxy. In this case Tκ  = cEX by (2.4). Assume that X has a density 1 f (x) = x−τ , x ≥ 1. τ −1 Define ρ(c) = ρκ as given by Theorem 3.1 for this model. 2

(3.1)

Corollary 3.2 (Section 16.4 in [3]). Let τ > 3 and set c0 := 1/EX 2 . (i) If 3 < τ < 4 then ρ(c0 + ε) ∼ a1 ε1/(τ −3) as ε ↓ 0, (ii) if τ = 4 then ρ(c0 + ε) ∼ a2 ε/ ln(1/ε) as ε ↓ 0, (iii) if τ > 4 then ρ(c0 + ε) ∼ a3 ε as ε ↓ 0, where ai are some positive constants. This result indicates that if EX 3 = ∞ (3 < τ < 4) there might be different scalings for C1 depending on the tail of the distribution of X, parameter τ in the above example. This we will account indeed in the critical and subcritical phases.

4. Subcritical case: Tκ < 1 4.1. Rank 1 model: sufficient conditions for the log n scaling We consider here the rank 1 case of the inhomogeneous random graph GV (n, κ) with κ(x, y) = ψ(x)ψ(y), x, y ∈ S, where S ⊆ R+ is finite or countable. We shall also assume a number of additional conditions.

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Assumption 1. There is a positive a such that  eaψ(x) dµ(x) < ∞.

(4.1)

S

Assumption 2. For any ε > 0 and q > 0  #{i : xi = x} P − µ(x) ≤ εeqψ(x) µ(x), n as n → ∞.

) for all x ∈ S

→1

(4.2)

Remark 4.1. If S is finite Assumption 1 trivially holds, while convergence (4.2) follows by (1.5). Remark 4.2.If S is infinite then convergence (4.2) holds (under condition (1.5)) when, e.g., x e−qψ(x) < ∞ for any q > 0 and E #{i : xi = x} ≤ (1 + o(1))nµ(x), where o(1) → 0 as n → ∞ uniformly in x. Remark 4.3. If a positive random variable X has some finite exponential moment then the corresponding model Gn,X satisfies both Assumption 1 and Assumption 2. Remark 4.4. Roughly speaking, condition (4.2) together with (4.1) rules out whp the presence of vertices of too large type in GV (n, κ). More precisely, in order to have vertices of (large) type x in the graph GV (n, κ) condition (4.2) requires  −1 n ≥ (εeqψ(x) + 1)µ(x) . Theorem 4.5 ([24]). Under Assumptions 1 and 2 we have   C1 GV (n, κ) P 1 → log n log rκ as n → ∞.

(4.3)

Observe that the statement of Theorem 4.5 complements Theorem 3.1 under the conditions of Theorem 4.5, since rκ > 1 implies ρκ = 0, whereas ρκ > 0 implies rκ = 1. Notice, however, that when Tκ  = 1 then both rκ = 1 and ρκ = 0, and none of the Theorems 3.1 or 4.3 provides a substantial information. One may think that a statement similar to (4.3) should remain true without the rank 1 assumption as well. Taking into account Remark 4.3 we immediately derive from Theorem 4.5 and Corollary 2.5 the following result on the rank 1 model Gn,X . Corollary 4.6. Assume, that X is a positive random variable, such that EeaX < ∞ for some positive a. Then C1 (Gn,X ) P 1 → , log n log rκ

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where rκ is defined by Theorem 2.6, and in particular,  > 1, if EX 2 < 1, rκ = 1, if EX 2 ≥ 1. Notice that Theorems 4.5 and 2.6 immediately yield √ (1.1). Indeed, in the case of a homogeneous model Gn,c/n we have ψ(X) ≡ c = M in Theorem 2.6, trivially implying y = 1/c, which together with (2.18) and (4.3) gives (1.1). 4.2. Sufficient conditions for the nα scaling, α < 1/2 Assume now that we do not have the assumption of the exponentially decaying tail (4.1), but assume instead P{X ≥ x} = O(x−(τ −1) ), τ > 3.

(4.4)

Then the scaling of the largest component proved to be the same as for the maximal degree of the graph, which under condition (4.4) might be as large as n1/(τ −1) , as the following theorem tells us. Theorem 4.7 ([15], Corollary 4.4). Assume that (4.4) holds. Denote ∆(Gn,X ) the largest degree in the graph Gn,X . Then   ∆(Gn,X ) 1/(τ −1) C1 (Gn,X ) = + o n . (4.5) P 1 − EX 2 Remark 4.8. Notice that Theorem 4.7 was proved in [15] for a slight modification of the model Gn,X . Namely, it is assumed in [15] that the probability of an edge between i and j is given by xi xj pij = , n + xi xj which only asymptotically equivalent to (1.9). However, it is proved in [14] that under the condition EX 2 < ∞ (which holds here) these models are asymptotically equivalent, and Theorem 4.7 follows as well.

5. Critical case: Tκ = 1 5.1. Model Gn.c/n We begin with the result due to Aldous [1] for the Erd¨ os-R´enyi random graph Gn.c/n . Let a c = 1 + 1/3 , n i.e., a (critical) window of order n−1/3 is allowed around the critical value c = 1. Let (W (s), s ≥ 0) be the standard Brownian motion, and 1 W a (s) = W (s) + as − s2 . (5.1) 2 Denote γ1 , γ2 , . . . the ordered lengths of the excursions of B(s) = W a (s) − min W a (s
),  0≤s ≤s

s ≥ 0.

(5.2)

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T.S. Turova

To formulate the convergence result define l 2 to be the set of infinite sequences  2 2 x = (x1 , x 2 , . . .) with x1 ≥ x2 ≥ · · · ≥ 0 and i xi < ∞, and give l metric d(x, y) = (xi − yi )2 . Theorem 5.1 (Aldous [1]). Let C1 (n), C2 (n), . . . denote the ordered sizes of the connected components in the graph Gn,p with 1 a  p= 1 + 1/3 . n n Then the convergence in distribution n−2/3 (C1 (n), C2 (n), . . .) → (γ1 , γ2 , . . .) d

(5.3)

2

holds with respect to the l topology. In particular, it follows from (5.3) that C1 (n) d → γ1 , (5.4) n2/3 i.e., the largest connected component in the critical random graph has scaling n2/3 , which was also known from [9]. Convergence (5.4) was independently proved in [18]. A closed formula for the limiting distribution of C1 /n2/3 based on a combinatorial analysis is derived in [22], seemingly independent of earlier published in [18] formula based on the branching process approach. 5.2. Rank 1 case on i.i.d. with finite 3d moment The first result on the scaling for the connected components of some inhomogeneous random graph (formulated however, different from the models which we consider here) was also proved by Aldous [1]. Only recently this approach was extended for the study of the inhomogeneous random graph model. Here we consider the following case of rank 1 model. Let X denote a random variable with values in R+ and the distribution given by µ. Assume that x1 , . . . , xn are i.i.d. as random variable X. Consider now GV (n, κn ) with  a  κn (x, y) = 1 + 1/3 xy → xy =: κ(x, y), n where a is any fixed real constant. Recall that by Definition 1.1 this means that given the sequence x1 , . . . , xn , we let GV (n, κn ) be the random graph on {1, . . . , n}, such that any two vertices i and j are connected by an edge independently of the others and with the probability ;x x  a  < i j pij (n) = min 1 + 1/3 , 1 . (5.5) n n Let us denote here Gn (X, a) = GV (n, κn ). Recall that by (2.4) we have Tκ  = EX 2 .

(5.6)

Hence, our assumption on criticality (i.e., that Tκ  = 1) is equivalent here to EX 2 = 1.

(5.7)

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It turns out that under the assumption EX 3 < ∞

(5.8)

the model Gn (X, a) falls into the same universality class as Gn,c/n . Let " γi denote the ordered lengths of the excursions of the process
"a (s) − min W
"a (s
), W 

s≥0

0≤s ≤s

(recall (5.2)), where " a = a(EX 3 )−2/3 . The following result (inspired very much by Aldous’ Theorem 5.1) was proved independently in [5] and [26]. Theorem 5.2 ([5] and [26]). Let C1 (n), C2 (n), . . . denote the ordered sizes of the connected components in Gn (X, a) with C1 (n) being the largest one. Then if EX 2 = 1 and EX 3 < ∞ we have n−2/3 (C1 (n), C2 (n), . . .) → d

EX 1/3

(EX 3 )

(" γ1 , " γ2 , . . .) .

The proof of Theorem 5.2 relies essentially on the theory of the multiplicative coalescent developed by Aldous [1]. Theorem 5.2 places all the rank 1 graphs with a finite third moment into the same universality class as the critical homogeneous random graph Gn,p model [1], and the critical random regular graphs (recent result [19]) as long as the scaling n2/3 concerns. This fact was observed by van der Hofstad already in [12], where possible critical scalings when only EX 2 < ∞ were classified. 5.3. Rank 1 case with infinite 3d moment The scalings for the discussed above model Gn,X with EX 3 = ∞ when EX 2 = 1 are not yet found. However, the recent results of Bhamidi, van der Hofstad and van Leeuwaarden [6] on some particular modification of this model indicate possibility to solve this problem. The model studied in [6] is the so-called Poissonian random graph or NorrosReittu model [20]. This is a particular case of GV (n, κn ) model, where the sequence x1 , . . . , xn is deterministic with xi = [1 − F ]−1 (i/n) := inf{s : [1 − F ](s) ≤ i/n} for some distribution function F , and

 ) xi xj . pij (n) = 1 − exp − n k=1 xk

Let us denote the resulting graph GF (n).

(5.9)

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To see that GF (n) is indeed a special case of rank 1 model GV (n, κn ), one first notices that for all y > 0 #{i : xi ≤ y} → F (y), n and thus F corresponds measure µ in (1.5).  Let W denote random variable with n distribution function F . Observe also that k=1 xk /n → EW . Hence, for any fixed xi and xj we have in (5.9) pij (n) = where

κn (xi , xj ) , n

  ) xi xj 1  κn (xi , xj ) := n 1 − exp − n → xi xj =: κ(xi , xj ). x EW k=1 k

One can deduce from here that the limiting kernel is simply κ(x, y) = xy/EW . In this case the norm of the corresponding operator Tκ is Tκ  = EW 2 /EW , and thus the critical phase Tκ  = 1 corresponds to EW 2 = 1. EW

(5.10)

Hence, one may view GF (n) as some approximation of the rank 1 model with i.i.d. sequence xi with distribution F , and kernel κ(x, y) = xy/EW . Assume that for some τ ∈ (3, 4) 1 − F (x) = cF x−(τ −1) (1 + o(1))

(5.11)

as x → ∞ for some positive constant cF . For any τ ∈ (3, 4) we have EW 3 = ∞, which corresponds the model Gn,X with EX 3 = ∞. Let GF,λ (n) denote a generalization of model GF (n), which is obtained from GF (n) by replacing xi with xi (λ) = (1 + λn(τ −3)/(τ −1) )xi . In particular, GF,0 (n) = GF (n). Theorem 5.3 ([6], Theorem 1.1). Assume that (5.10) holds and τ ∈ (3, 4). Let C1 (n), C2 (n), . . . denote the ordered (decreasing) sizes of the connected components in GF,λ (n). Then for all λ ∈ R we have n−(τ −2)/(τ −1) (C1 (n), C2 (n), . . .) → (γ1 (λ), γ2 (λ), . . .) . d

for some non-degenerate limit (γi (λ))i≥1 . Remark 5.4. The limiting random variables are identified in [6] explicitly in terms of certain hitting times of zero of the scaling limit of the breadth-first walk. Refer to [6] for the details.

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273

We see that indeed as it was indicated already by Corollary 3.2, in the case when the third moment is infinite (τ ∈ (3, 4)) the scaling depends on the tail of the distribution. One may conjecture that a result of this type should hold for the model Gn,X as well, but as the authors of [6] predict, the scaling limits might be different.

6. Spectra of rank 1 graphs The recent progress in the study of inhomogeneous graphs opens up a possibility to extend some of the results obtained for homogeneous random graphs for the inhomogeneous case. As an example we consider here the graph Laplacian of the introduced above model Gn,X . We shall argue that the results of [17] on the Lifshitz tail for the spectra of the Erd¨ os-R´enyi random graph should also hold for some class of inhomogeneous graphs. The following definition is an extension of the one given in [17] for the Erd¨ osR´enyi graphs. Definition 6.1. Let GV (n, κ) denote a random inhomogeneous graph. Given a se(n) quence x1 , . . . , xn and a graph GV (n, κ), let e[i,j] for any pair i, j ∈ {1, . . . , n}, be a random variable which is one if i = j and the edge [i, j] is present in GV (n, κ); otherwise, it is zero. The graph Laplacian ∆(n) of GV (n, κ) is the random linear operator on C n with matrix elements    (n) (n) (n) ∆ij =  e[i,l]  δij − e[i,j] (1 − δij ) l =i n

in the canonical basis of C . Recall that given a sequence x1 , . . . , xn the edges in a graph GV (n, κ) are independent and present with probabilities pij (n) = min{κ(xi , xj )/n, 1}. Hence, (n) (n) given a sequence x1 , . . . , xn the random variables e[i,j] ≡ e[j,i] , i = j, are independent Bernoulli random variables with parameter pij (n). For a given sequence x1 , . . . , xn matrix ∆(n) is a (random) self-adjoint and (n) non-negative, hence it possesses n non-negative eigenvalues λj , j = 1, . . . , n, which are random variables (and functions of x1 , . . . , xn ). Following [17] let us introduce the normalized eigenvalue counting function: ; < 1 (n) σ (n) (E) := E# 1 ≤ j ≤ n : λj ≤ E , E ≥ 0. n When GV (n, κ) is a homogeneous Gn,p model, it is proved in [17] that for any p > 0 the sequence σ (n) weakly converges to some distribution function σp . Then it is proved in [17] that σp for any p < 1 has a Lifshitz tail at the lower edge of the spectrum, E = 0, with a Lifshitz exponent 1/2. The proof relies essentially on the exponential decay of the distributions of clusters in the subcritical regime. By

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Corollary 2.5 and Corollary 4.6 the component size distribution in Gn.X has also exponential tail in the subcritical regime. Therefore it should be possible to derive a similar result for the subcritical Gn,X as we state below. Conjecture. Let Gn,X be a rank 1 model, where EX 2 < 1 and EeaX < ∞ for some positive a. Then σ (n) converges weakly to some distribution σ which satisfies lim

E↓0

ln | ln (σ(E) − σ(0)) | 1 =− . ln E 2

Furthermore, having exact formula (2.18) for the exponent in the distribution of clusters in the subcritical case, one can get even more precise results on the spectra of ∆(n) . However, to verify the result conjectured in [7] remains a big challenge even in the case of homogeneous Erd¨ os-R´enyi graphs. Acknowledgment The author thanks F. Sobieczky for the helpful discussions resulted in Section 6.

References [1] D. Aldous, Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Probab. 25 (1997), no. 2, 812–854. [2] D. Aldous and V. Limic. The entrance boundary of the multiplicative coalescent. Electron. J. Probab. 3 (1998) 1–59. [3] B. Bollob´ as, S. Janson and O. Riordan, The phase transition in inhomogeneous random graphs. Random Structures and Algorithms 31 (2007), 3–122. [4] B. Bollob´ as, S. Janson and O. Riordan, The cut metric, random graphs, and branching processes. arXiv:0901.2091 [math.PR] [5] Bhamidi S., van der Hofstad R. and van Leeuwaarden J., Scaling limits for critical inhomogeneous random graphs with finite third moments, arXiv:0907.4279v1 [math.PR] [6] Bhamidi S., van der Hofstad R. and van Leeuwaarden J., Novel scaling limits for critical inhomogeneous random graphs, arXiv:0909.1472 [math.PR] [7] A.J. Bray and G.J. Rodgers, Diffusion in a sparsely connected space: A model for glassy relaxation. Phys. Rev. B 38 (1988) 11461–11470 [8] T. Britton, M. Deijfen and A. Martin-L¨ of, Generating simple random graphs with prescribed degree distribution. Journal of Statistical Physics, 124 (2006), 1/2, 1377– 1397. [9] P. Erd¨ os and A. R´enyi, On the evolution of random graphs. Publ. Math. Inst. Hung. Acad. Sci. 5 (1960), 17–61. [10] F. Chung and L. Lu, The volume of the giant component of a random graph with given expected degree. SIAM J. Discrete Math. 20 (2006), 395–411 [11] H. van den Esker, R. van der Hofstad, and G. Hooghiemstra, Universality for the distance in finite variance random graphs. J. Stat. Phys. 133 (2008), no. 1, 169–202.

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[12] R. van der Hofstad, Critical behavior in inhomogeneous random graphs. Report 1, 2008/2009 Spring, Institute Mittag-Leffler. [13] S. Janson, T. L  uczak, and A. Ruci´ nski, Random graphs, Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley-Interscience, New York, 2000. [14] S. Janson, Asymptotic equivalence and contiguity of some random graphs. Random Structures Algorithms, 36 (2010) 26–45. [15] S. Janson, The largest component in a subcritical random graph with a power law degree distribution Ann. Appl. Probab. (2008), 1651–1668. [16] R.M. Karp, The transitive closure of a random digraph. Random Structures Algorithms 1 (1990), no. 1, 73–93. [17] O. Khorunzhiy, W. Kirsch, and P. M¨ uller, Lifshitz tails for spectra of random Erd¨ osR´enyi graphs. Ann. Appl. Probab. 6 (2006) 295–309. [18] A. Martin-L¨ of, The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier. J. Appl. Probab. 35 (1998), no. 3, 671–682. [19] A. Nachmias and Y. Peres, Critical percolation on random regular graphs. To appear in Random Structures and Algorithms. [20] I. Norros and H. Reittu. On a conditionally Poissonian graph process. Adv. in Appl. Probab. 38 (2006) 59–75. [21] R. Otter, The multiplicative process. Ann. Math. Statistics 20 (1949), 206–224. [22] B. Pittel, On the largest component of the random graph at a nearcritical stage. J. of Combinatorial Theory, Series B 82 (2001) 237–269. [23] T.S. Turova, Long paths and cycles in the dynamical graphs. Journal of Statistical Physics, 110 (2003), 1/2, 385–417. [24] T.S. Turova, The size of the largest component below phase transition in inhomogeneous random graphs, arXiv:0706.2106v1 [math.PR] [25] T.S. Turova, Asymptotics for the size of the largest component scaled to “log n” in inhomogeneous random graphs, Combinatorics, Probability and Computations 20 (2010) 131–154. [26] T.S. Turova, Diffusion approximation for the components in critical inhomogeneous random graphs of rank 1, arXiv:0907.0897 [math.PR] [27] T.S. Turova and T. Vallier, Merging percolation on Z d and classical random graphs: Phase transition. Random Structures and Algorithms 36, 185–217. Tatyana S. Turova Mathematical Center University of Lund P.O. Box 118 S-221 00 Lund, Sweden e-mail: [email protected]

Progress in Probability, Vol. 64, 277–304 c 2011 Springer Basel AG


Partition Functions of the Ising Model on Some Self-similar Schreier Graphs Daniele D’Angeli, Alfredo Donno and Tatiana Nagnibeda Abstract. We study partition functions and thermodynamic limits for the Ising model on three families of finite graphs converging to infinite self-similar graphs. They are provided by three well-known groups realized as automorphism groups of regular rooted trees: the first Grigorchuk’s group of intermediate growth; the iterated monodromy group of the complex polynomial z 2 −1 known as the “Basilica group”; and the Hanoi Towers group H (3) closely related to the Sierpi´ nski gasket. Mathematics Subject Classification (2000). Primary 82B20; Secondary 05A15. Keywords. Ising model, partition function, self-similar group, Schreier graph.

1. Introduction 1.1. The Ising model The famous Ising model of ferromagnetism was introduced by W. Lenz in 1920, [14], and became the subject of the PhD thesis of his student E. Ising. It consists of discrete variables called spins arranged on the vertices of a finite graph Y . Each spin can take values ±1 and only interacts with its nearest neighbours. Configuration of spins at two adjacent vertices i and j has energy Ji,j > 0 if the spins have opposite values, and −Ji,j if the values are the same. Let | Vert(Y )| = N , and let σ = (σ1 , . . . , σN ) denote the configuration of spins, with σi ∈ {±1}. The total energy of the system in configuration σ is then  E(σ ) = − Ji,j σi σj , i∼j

where we write i ∼ j if the vertices i and j are adjacent in Y . This research has been supported by the Swiss National Science Foundation Grant PP0022 118946.

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The probability of a particular configuration at temperature T is given by 1 P(σ ) = exp(−βE(σ )), Z where β is the “inverse temperature” conventionally defined as β ≡ 1/(kB T ), and kB denotes the Boltzmann constant. As usual in statistical physics, the normalizing constant that makes the distribution above a probability measure is called the partition function:  Z= exp(−βE(σ )).

σ

One can rewrite this formula by using exp(Kσi σj ) = cosh(K) + σi σj sinh(K), so as to get the so-called “high temperature expansion”:    Z= cosh(βJi,j ) (1 + σi σj tanh(βJi,j ) i∼j

+



σ

i∼j

(σi σj )(σl σm ) tanh(βJi,j ) tanh(βJl,m ) + · · · ).

i∼j l∼m

After changing the order of summation, observe that the non-vanishing terms in Z are exactly those with an even number of occurrences of each σi . We can interpret this by saying that non-vanishing terms in this expression are in bijection with closed polygons of Y , i.e., subgraphs in which every vertex has even degree. Consequently we can rewrite Z as      Z =  cosh(βJi,j ) · 2N tanh(βJi,j ) , (1) i∼j

X closed polygon of Y (i,j)∈Edges(X)

where in the RHS we have the generating series of closed polygons of Y with weighted edges, the weight of an edge (i, j) being tanh(βJi,j ). In the case of constant J, the above expression specializes to  | Edges(Y )| N cl Z = cosh(βJ) · 2 Γ (tanh(βJ)) ∞ cl cl n cl with Γ (z) = n=0 An z , where An is the number of closed polygons with n edges in Y . (In particular, the total number of closed polygons is given by Γcl (1).) From the physics viewpoint it is interesting to study the model when the system (i.e., the number of vertices in the graph) grows. One way to express this mathematically is to consider growing sequences of finite graphs converging to an infinite graph. If the limit log(Zn ) lim n→∞ | Vert(Yn )| for a sequence of finite graphs Yn with partition functions Zn exists, it is called the thermodynamic limit. In the thermodynamic limit, at some critical temperature, a phase transition can occur between ordered and disordered phase in the behaviour of the model.

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Existence of a phase transition depends on the graph. In his thesis in 1925, Ising studied the case of one-dimensional Euclidean lattice; computed the partition functions and the thermodynamic limit and showed that there is no phase transition [13]. The Ising model in Zd with d ≥ 2 undergoes a phase transition. This was first established for d = 2 by R. Peierls. At high temperature, T > TC , the clusters of vertices with equal spins grow similarly for two different types of spin, whereas for T < TC the densities of the types of spin are different and the system “chooses” one of them. The infinite graphs that will be studied in this paper all have finite-order R of ramification, i.e., for any connected bounded part X of the graph there exists a set A of at most R vertices such that any infinite self-avoiding path in the graph that begins in X necessarily goes through A. Finite-order of ramification ensures that the critical temperature is T = 0, and there is no phase transition in the Ising model (see [7]). Typically, an infinite lattice is viewed as the limit of an exhaustive sequence of finite subgraphs. This is a simple example of the so-called pointed HausdorffGromov convergence (see Proposition 1.3 below). Another typical case of this convergence is that of covering graph sequences. Ising model on towers of coverings was considered previously by Grigorchuk and Stepin [9] in the case of Cayley graphs of finite quotients of an infinite residually finite group. In this paper we will be studying the Ising model on families of finite graphs coming from the theory of self-similar groups (see Definition 1.2 below), and their infinite limits. Any finitely generated group of automorphisms of a regular rooted tree provides us with a sequence of finite graphs describing the action of the group on the levels of the tree. When the action is self-similar the sequence converges in the above sense to infinite graphs describing the action of the group on the boundary of the tree. The graphs that we study here are determined by group actions, and so their edges are labeled naturally by the generators of the acting group. Different weights on the edges lead to weighted partition functions, with Ji,j depending on the label of the edge (i, j). Schreier graphs of self-similar groups have already served as source of interesting examples for various problems with motivation from physics. They were mainly studied till now from the viewpoint of spectral computations ([1, 11, 10]), providing examples of regular graphs with unusual spectra and spectral measures. Moreover, Schreier graphs form approximating sequences for fractals (Julia sets of rational maps) via the notion of iterated monodromy group and its limit space, introduced by Nekrashevych [17], and can therefore be used in spectral computations on fractals (see [18] for the case of the Basilica group). In a recent work [16] the Basilica Schreier graphs were used to construct uncountably many non-isomorphic graphs of quadratic growth with critical Abelian sand pile model. These and other related results have motivated our interest in the dimers and the Ising model on families of Schreier graphs of self-similar groups. Let us mention also that the Ising model has been considered previously on (exhaustive sequence of finite subgraphs of) Cayley graphs of some finitely generated groups, in particular, the modular

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group [15], and more generally discrete cocompact lattices in the hyperbolic plane [19]. These negatively curved examples are very different from the ones studied in this paper, which are in fact much closer to Euclidean lattices. 1.2. Groups of automorphisms of rooted regular trees Let T be the infinite regular rooted tree of degree q, i.e., the rooted tree in which each vertex has q children. Each vertex of the nth level of the tree can be regarded as a word of length n in the alphabet X = {0, 1, . . . , q − 1}. Moreover, one can identify the set X ω of infinite words in X with the set ∂T of infinite geodesic rays starting at the root of T . Now let G < Aut(T ) be a group acting on T by automorphisms, generated by a finite symmetric set of generators S. Suppose moreover that the action is transitive on each level of the tree. Definition 1.1. The nth Schreier graph Σn of the action of G on T , with respect to the generating set S, is a graph whose vertex set coincides with the set of vertices of the nth level of the tree, and two vertices u, v are adjacent if and only if there exists s ∈ S such that s(u) = v. If this is the case, the edge joining u and v is labeled by s. For any infinite ray ξ ∈ ∂T , the orbital Schreier graph Σξ has vertices G · ξ and edges determined by the action of generators, as above. The vertices of Σn are labeled by words of length n in X and the edges are labeled by elements of S. The Schreier graph is thus a regular graph of degree d = |S| with q n vertices, and it is connected since the action of G is level-transitive. Definition 1.2. A finitely generated group G < Aut(T ) is self-similar if, for all g ∈ G, x ∈ X, there exist h ∈ G, y ∈ X such that g(xw) = yh(w), for all finite words w in the alphabet X. Self-similarity implies that G can be embedded into the wreath product Sym(q)  G, where Sym(q) denotes the symmetric group on q elements, so that any automorphism g ∈ G can be represented as g = τ (g0 , . . . , gq−1 ), where τ ∈ Sym(q) describes the action of g on the first level of T and gi ∈ G, i = 0, . . . , q − 1 is the restriction of g on the full subtree of T rooted at the vertex i of the first level of T (observe that any such subtree is isomorphic to T ). Hence, if x ∈ X and w is a finite word in X, we have g(xw) = τ (x)gx (w). It is not difficult to see that the orbital Schreier graphs of a self-similar group are infinite and that the finite Schreier graphs {Σn }∞ n=1 form a sequence of graph coverings (see [17] and references therein for more information about this interesting class of groups, also known as automata groups).

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Take now an infinite ray ξ ∈ X ω and denote by ξn the nth prefix of the word ξ. Then the sequence of rooted graphs {(Σn , ξn )} converges to the infinite rooted graph (Σξ , ξ) in the space of rooted graphs, in the following sense. Proposition 1.3 ([12], Chapter 3.). Let X be the space of connected graphs having a distinguished vertex called the root; X can be endowed with the following metric: given two rooted graphs (Y1 , v1 ) and (Y2 , v2 ),  ) 1 Dist((Y1 , v1 ), (Y2 , v2 )) := inf ; BY1 (v1 , r) is isomorphic to BY2 (v2 , r) r+1 where BY (v, r) is the ball of radius r in Y centered in v. Under the assumption of uniformly bounded degrees, X endowed with the metric Dist is a compact space. 1.3. Plan of the paper Our aim in this paper is to study the Ising model on the Schreier graphs of three key examples of self-similar groups: – the first Grigorchuk’s group of intermediate (i.e., strictly between polynomial and exponential) growth (see [8] for a detailed account and further references); – the “Basilica”group that can be described as the iterated monodromy group of the complex polynomial z 2 − 1 (see [17] for connections of self-similar groups to complex dynamics); – and the Hanoi Towers group H (3) whose action on the ternary tree models the famous Hanoi Towers game on three pegs, see [10]. It is known [2] that the infinite Schreier graphs associated with these groups (and, more generally, with all groups generated by bounded automata) have finite order of ramification. Hence the Ising model on these graphs exhibits no phase transition. We first compute the partition functions and prove existence of thermodynamic limit for the model where interactions between vertices are constant: in Section 2 we treat the Grigorchuk’s group and the Basilica group, and in Section 3 the Hanoi Towers group H (3) and its close relative the Sierpi´ nski gasket are considered. In Section 4, we study weighted partition functions for all the graphs previously considered, and we find the distribution of the number of occurrences of a fixed weight in a random configuration. The relation between the Schreier graphs of H (3) and the Sierpi´ nski gasket is also discussed from the viewpoint of Fisher’s theorem establishing a correspondence between the Ising model on the Sierpi´ nski gasket and the dimers model on the Schreier graphs of H (3) .

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2. Partition functions and thermodynamic limit for the Grigorchuk’s group and for the Basilica group 2.1. Grigorchuk’s group This group admits the following easy description as a self-similar subgroup of automorphisms of the binary tree. It is generated by the elements a = (id, id),

b = e(a, c), c = e(a, d), d = e(id, b),

where e and  are respectively the trivial and the non-trivial permutations in Sym(2). These recursive formulae allow easily to construct finite Schreier graphs for the action of the group on the binary tree. Here are three first graphs in the sequence, with loops erased. Σ1

•

a

•

•

a

b •

•

a

•

•

a

Σ2

c •

a

b •

•

a

b •

•

c

a

b •

•

Σ3

c

d

In general, the Schreier graph Σn has the same linear shape, with 2n−1 simple edges, all labeled by a, and 2n−1 − 1 cycles of length 2. It is therefore very easy to compute the generating function of closed polygons of Σn , for each n. Theorem 2.1. The generating function of closed polygons for the nth Schreier graph 2 2n−1 −1 of the Grigorchuk group is Γcl . In particular, the number of n (z) = (1 + z ) n−1 2 −1 all closed polygons in Σn is 2 . The partition function of the Ising model is given by n−1

Zn = cosh(βJ)3·2

−2

 2n−1 −1 n · 22 · 1 + tanh2 (βJ)

and the thermodynamic limit exists and satisfies:   log(Zn ) 3 1 = log(cosh(βJ)) + log 2 + log 1 + tanh2 (βJ) . n n→∞ 2 2 2 lim

Proof. Is is clear that a closed polygon in Σn is the union of 2-cycles. So we can easily compute the number Acl k,n of closed polygons with k edges in Σn , for all n k = 0, 1, . . . , 2 − 2. For k odd, one has Acl k,n = 0. For k even, we have to choose k cycles of length 2 to get a closed polygon with k edges, which implies 2  n−1 2 −1 cl Ak,n = . k 2

Ising Model on Schreier Graphs

283

So the generating function of closed polygons for Σn is given by Γcl n (z)

=

2n−1 −1  n−1

2

−1

k

k=0

n−1 z 2k = (1 + z 2 )2 −1 .




2.2. The Basilica group The Basilica group is a self-similar group of automorphisms of the binary tree generated by the elements a = e(b, id),

b = (a, id).

The associated Schreier graphs can be recursively constructed via the following substitutional rules: Rule I

Rule II

a • 1w

• u

⇓

b

Rule III

• v

• 0u

⇓

b a R• 11w b

• 01w

• 0u

a

• 0v

⇓

a

a • 10v

b

• 0v

• 00u

b • 00v

The starting point is the Schreier graph Σ1 of the first level:

Σ1

b

a R• 0

• a 1I

b

The following pictures of graphs Σn for n = 1, 2, 3, 4, 5 with loops erased give an idea of how finite Schreier graphs of the Basilica group look like. See [5] for a comprehensive analysis of finite and infinite Schreier graphs of this group. Note also that {Σn }∞ n=1 is an approximating sequence for the Julia set of the polynomial z 2 − 1, the famous “Basilica” fractal (see [17]). b Σ1

•

b •

b

•

a •

b

b •

a

• b

Σ2

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D. D’Angeli, A. Donno and T. Nagnibeda

•

Σ3

•

b

a

b •

b

• a

b

a

b

• b

•

b

a

b

•

•

a

b

• b

b b •

• b

a

b

•

•

• a

Σ4

•

b

a

b

•

b

•

b

• a

•

a

a •

b

•

b

b •

•

a

b

b

b • • b

b •

• b

a

b •

• b Σ5

a

a a b • b

b • • • a b b b a • • • a a b b • b • • b b b • b • a a • b b •

• b b • • a b • b b a • • • b b • a b • b • b b •

a

b •

a

• b

In general, it follows from the recursive definition of the generators, that each Σn is a cactus, i.e., a union of cycles (in this example all of them are of length power of 2) arranged in a tree-like way. The maximal length of a cycle in Σn is n+1 n 2 2 if n is odd and 2 2 if n is even. Denote by aij the number of cycles of length j labeled by a in Σi and analogously denote by bij the number of cycles of length j labeled by b in Σi . Proposition 2.2. For any n ≥ 4 consider the Schreier graph Σn of the Basilica group.

Ising Model on Schreier Graphs

285

For each k ≥ 1, the number of cycles of length 2k labeled by a is  2n−2k−1 for 1 ≤ k ≤ n−1 n 2 −1 , a2k = for n odd, 2 for k = n−1 2  2n−2k−1 for 1 ≤ k ≤ n2 − 1 n a2k = , for n even 1 for k = n2 and the number of cycles of  n−2k  2 n b2k = 2   1  2n−2k bn2k = 2

length 2k labeled by b is for 1 ≤ k ≤ for k = n−1 2 for k = n+1 2

n−1 2

for 1 ≤ k ≤ for k = n2

n 2

−1 ,

−1

,

for n odd,

for n even.

Proof. The recursive formulae for the generators imply that, for each n ≥ 3, one has an2 = bn−1 and bn2 = an−1 = 2n−2 2 1 n−2(k−1)

n−2(k−1)

and in general an2k = a2 and bn2k = b2 . In particular, for each n ≥ 4, the number of 2-cycles labeled by a is 2n−3 and the number of 2-cycles labeled by b is 2n−2 . More generally, the number of cycles of length 2k is given by an2k = 2n−2k−1 ,

bn2k = 2n−2k ,

where the last equality is true if n − 2k + 2 ≥ 4, i.e., for k ≤ n2 − 1. Finally, for n n+1 odd, there is only one cycle of length 2 2 labeled by b and four cycles of length n−1 2 2 , two of them labeled by a and two labeled by b; for n even, there are three n cycles of length 2 2 , two of them labeled by b and one labeled by a.
Corollary 2.3. For each n ≥ 4, the number of cycles labeled by a in the Schreier graph Σn of the Basilica group is  n−1 2 +2 for n odd, 3 n−1 2 +1 for n even, 3 and the number of b-cycles in Σn is  n

2 +1 3 2n +2 3

for n odd, for n even.

The total number of cycles of length ≥ 2 is 2n−1 + 1 and the total number of edges, without loops, is 3 · 2n−1 . The computations above lead to the following formula for the partition function of the Ising model on the Schreier graphs Σn associated with the action of the Basilica group.

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Theorem 2.4. The partition function of the Ising model on the nth Schreier graph Σn of the Basilica group is n

Zn = 22 · cosh(βJ)3·2

n−1

· Γcl n (tanh(βJ)),

where Γcl n (z) is the generating function of closed polygons for Σn given by 4  3·2n−2k−1  n−1 n+1   2k 2 2 2 2 = 1+z · 1+z · 1+z , n−1 2 −1

Γcl n (z)

k=1

for n ≥ 5 odd and n 2 −1

Γcl n (z)



=

1 + z2

k

3·2n−2k−1  n 3 · 1 + z2 2 ,

k=1 2 cl 2 3 cl 2 4 4 for n ≥ 4 even. Moreover, Γcl 1 = 1 + z , Γ2 = (1 + z ) and Γ3 = (1 + z ) (1 + z ). cl Proof. Recall that Zn = 2|Vert(Σn )| cosh(βJ)|Edges(Σn )| ·Γcl n (tanh(βJ)), where Γn (z) is the generating function of closed polygons in Σn . In our case we have | Edges(Σn )| = 3 · 2n−1 and | Vert(Σn )| = 2n . The formulae for Γcl n (z) with n = 1, 2, 3 can be directly verified. For n ≥ 4, we can use Proposition 2.2. Since the length of each cycle of Σn is even, it is clear cl that the coefficient Acl k,n is zero for every odd k. The coefficient Ak,n is nonzero n−1 n−1 for every even k such that 0 ≤ k ≤ 3 · 2 . In fact, 3 · 2 is the total number of edges of Σn (2n labeled by b and 2n−1 labeled by a). By taking the exact number of cycles of length 2i in Σn , we get the assertion.


Theorem 2.5. The thermodynamic limit limn→∞

log(Zn ) | Vert(Σn )|

exists.

Proof. Since | Edges(Σn )| = 3 · 2n−1 and | Vert(Σn )| = 2n , the limit reduces to (choosing, for example, n even) log(2) +

3 log(Γcl n (z)) log(cosh(βJ)) + lim , n→∞ 2 2n

where z = tanh(βJ) takes values between 0 and 1. Now n  n2 −1 k n−2k−1 log(1 + z 2 ) + 3 log(1 + z 2 2 ) log(Γcl n (z)) k=1 3 · 2 lim = lim n→∞ n→∞ 2n 2n n ∞ 2k  3 log(1 + z ) 3 log(1 + z 2 2 ) = + lim n→∞ 2 4k 2n k=1 ∞

3  log(2) ≤ < ∞, 2 4k k=1

giving the assertion.




Ising Model on Schreier Graphs

287

3. Partition functions and thermodynamic limits for the Hanoi Towers group H (3) and for the Sierpi´ nski gasket 3.1. Hanoi Towers group H (3) The Hanoi Towers group H (3) is generated by three automorphisms of the ternary rooted tree admitting the following self-similar presentation [10]: a = (01)(id, id, a)

b = (02)(id, b, id)

c = (12)(c, id, id),

where (01), (02) and (12) are transpositions in Sym(3). The associated Schreier graphs are self-similar in the sense of [20], that is, each Σn+1 contains three copies of Σn glued together by three edges. These graphs can be recursively constructed via the following substitutional rules [10]: 11u • a

c 1u • Rule I

a

21u• a

=⇒

c

• 01u

b

c

20u• • 0u

• 2u

b

• 02u a

c

b • 00u a

• 10u b

b

• 12u c

• 22u

22u • c 2u • Rule II

12u•

a

• 02u c

b

=⇒

c

b

b

10u• • 0u

a

• 1u

a • 00u b

Rule III 0u • c • 0v

=⇒

• 01u c • 20u a

Rule IV 00u •

1u •

c

b

• 00v

• 1v

=⇒

a

b • 21u c

• 11u

Rule V 11u •

2u •

b

a

• 11v

• 2v

22u • =⇒

a • 22v

288

D. D’Angeli, A. Donno and T. Nagnibeda

The starting point is the Schreier graph Σ1 of the first level. b •

b • Σ1 a c R•

a

c • Σ2

c

c

•

• a I

b

•

b

a

•

c R•

a

c

b

•

• a I Remark 3.1. Observe that, for each n ≥ 1, the graph Σn has three loops, at the vertices 0n , 1n and 2n , labeled by c, b and a, respectively. Moreover, these are the only loops in Σn . The Ising model will be studied on Σn considered without loops. a

•

b

b

c

Let us now proceed to the computation of closed polygons in Σn . Denote by Pn the set of closed polygons in Σn , and by Ln the set of all subgraphs of Σn whose vertices have even degree, except for the left-most vertex and the right-most vertex of Σn which have odd degree. • • •

•

• •

• •

• •

•

•

•

•

• • • Two elements of P2 .

• •

•

•

•

• •

•

• • •

•

• • • Two elements of L2 .

•

• •

•

Each closed polygon in Σn can be obtained in the following way: either it is a union of closed polygons living in the three copies Σn−1 or it contains the three special edges joining the three subgraphs isomorphic to Σn−1 . The subgraphs of the first 3 type can be identified with the elements of the set Pn−1 , whereas the other ones are obtained by joining three elements in Ln−1 , each one belonging to one of the

Ising Model on Schreier Graphs

289

three copies of Σn−1 , so that they can be identified with elements of the set L3n−1 . This gives = 3 Pn = Pn−1 L3n−1 . (2) On the other hand, each element in Ln can be described in the following way: if it contains a path that does not reach the up-most triangle isomorphic to Σn−1 , it can be regarded as an element in L2n−1 × Pn−1 ; if it contains a path which goes through all three copies of Σn−1 , then it is in L3n−1 . This gives  = 3 Ln−1 , Ln = L2n−1 × Pn−1 (3) from which we deduce Proposition 3.2. For each n ≥ 1 the number |Pn | of closed polygons in the Schreier 3n −1 graph Σn of H (3) is 2 2 . We are now ready to compute the generating series for closed polygons and the partition function of the Ising model on Schreier graphs of H (3) . Denote by Γcl n (z) the generating function of the set of subgraphs in Pn and by Υn (z) the generating function of the set of subgraphs in Ln . The equation (2) gives  cl 3 Γcl + z 3 Υ3n−1 (z). (4) n (z) = Γn−1 (z) The factor z 3 in (4) is explained by the fact that each term in Υ3n (z) corresponds to a set of edges that becomes a closed polygon after adding the three special edges connecting the three copies of Σn−1 . We have consequently that the second summand is the generating function for the closed polygons containing the three special edges. Analogously, from (3) we have 2 3 Υn (z) = zΥ2n−1 (z)Γcl n−1 (z) + z Υn−1 (z).

(5)

Theorem 3.3. For each n ≥ 1, the partition function of the Ising model on the Schreier graph Σn of the group H (3) is n

Zn = 23 · cosh(βJ) with n

3 Γcl n (z) = z

where ψ1 (z) =

z+1 z

and ψk (z) =

n 

ψk3

3n+1 −3 2

n−k

k=1 2 ψk−1 (z)

· Γcl n (tanh(βJ)),

(z) · (ψn+1 (z) − 1),

− 3ψk−1 (z) + 4, for each k ≥ 2.

cl Proof. Recall that Zn = 2|Vert(Σn )| cosh(βJ)|Edges(Σn )| ·Γcl n (tanh(βJ)), where Γn (z) is the generating function of closed polygons in Σn . In our case we have | Edges(Σn )| n+1 = 3 2 −3 and | Vert(Σn )| = 3n . We know that the generating functions Γcl n (z) and Υn (z) satisfy equations (4) and (5), and the initial conditions can be easily computed as: 3 Γcl 1 (z) = 1 + z

Υ1 (z) = z 2 + z.

290

D. D’Angeli, A. Donno and T. Nagnibeda

We now show by induction on n that the solutions of the system of equations (4) and (5) are  $n 3n 3n−k Γcl (z) · (ψn+1 (z) − 1) n (z) = z k=1 ψk $ n n 3 −1 3n−k Υn (z) = z (z). k=1 ψk 3 3 2 For n = 1, we get Γcl 1 (z) = z ψ1 (z)(ψ2 (z) − 1) = z + 1 and Υ1 (z) = z ψ1 (z) = 2 z + z and so the claim is true. Now suppose that the assertion is true for n and let us show that it is true for n + 1. One gets:

& Γcl n+1 (z)

=

z

= z3 = z3

3n

n 

n+1

k=1 n 

n+1

k=1 n+1 

'3 n−k ψk3 (z)

· (ψn+1 (z) − 1)

& +z

3

z

3n −1

n 

'3 n−k ψk3 (z)

k=1

ψk3

n−k+1

n−k+1

ψk3

 3  2 (z) ψn+1 (z) − 3ψn+1 (z) + 3ψn+1 (z) (z)(ψn+2 (z) − 1)

k=1

and

&

Υn+1 (z) = z

z

3n −1

+z

n−k ψk3 (z)

z

k=1

& 2

'2 &

n 

z

n 

3n −1

'3

3n

n 

' n−k ψk3 (z)

· (ψn+1 (z) − 1)

k=1

n−k ψk3 (z)

k=1

= z3

n+1

−1

n 

ψk3

n−k+1

(z) · ψn+1 (z)

k=1

= z3

n+1

−1

n+1 

n−k+1

ψk3




(z).

k=1

Theorem 3.4. The thermodynamic limit limn→∞ Proof. Since | Edges(Σn )| = log(2) +

3n+1 −3 2

log(Zn ) | Vert(Σn )|

exists.

and | Vert(Σn )| = 3n , the limit reduces to

3 log(Γcl n (z)) log(cosh(βJ)) + lim , n→∞ 2 3n

where tanh(βJ) takes values between 0 and 1. It is straightforward to show, by (z) induction, that ψk (z) = ϕ2kk−1 , for every k ≥ 1, where ϕk (z) is a polynomial of z

degree 2k−1 in z, such that ϕk (z) = ϕ2k−1 (z) − 3z 2

k−2

k−1

ϕk−1 (z) + 4z 2

. Hence, the

Ising Model on Schreier Graphs limit limn→∞

291

log(Γcl n (z)) 3n

lim

becomes $   n n 3n−k log (z) · ϕn+1 (z) − z 2 k=1 ϕk

n→∞

= lim

n→∞

n  k=1

∞

Let us show that the series limn→∞ log(ϕn+13n(z)−z

2n

)

3n n log(ϕk (z)) log(ϕn+1 (z) − z 2 ) + lim . n→∞ 3k 3n k=1

log(ϕk (z)) 3k

converges absolutely, and that

= 0. It is not difficult to show by induction that 2z 2

k−1

≤ ϕk (z) ≤ 22

k

−1

for each k ≥ 2 and z ∈ [0, 1]. We want to show that, for each k ≥ 2, one has | log(ϕk (z))| ≤ 2k log(2). Note that 1 ≤ ϕ1 (z) ≤ 2 for each 0 ≤ z ≤ 1. Moreover, one can directly verify that ϕ2 (z) has a local minimum at c2 = 1/4 and ϕ
2 (z) < 0 for each z ∈ (0, c2 ). Let us call ck the point where ϕk (z) has the first local minimum. One can prove by induction that ϕ
k+1 (z) < 0 for each z ∈ (0, ck ] and so ck < ck+1 for every k ≥ 2. In particular, ck ≥ 14 for each k ≥ 2. Hence, 2

1−2k

 2k−1 k−1 1 =2 ≤ 2(ck )2 ≤ ϕk (ck ) ≤ ϕk (z) 4 k−1

for each z ∈ [0, 1], since 2z 2

is an increasing function. So ϕk (z) satisfies

−(2k − 1) log(2) ≤ log(ϕk (z)) ≤ (2k − 1) log(2), that gives | log(ϕk (z))| ≤ 2k log(2) for each k ≥ 2. So we can conclude that ∞  | log(ϕk (z))| k=1

3k

log(2)  2k log(2) + < ∞. 3 3k k=2

n

Moreover limn→∞

∞

≤

| log(ϕn+1 (z)−z 2 )| 3n

≤ limn→∞

2n log(2) 3n

= 0.




3.2. The Sierpi´ nski gasket In this section we use the high temperature expansion and counting of closed polygons in order to compute the partition function for the Ising model on a sequence of graphs {Ωn }n≥1 converging to the Sierpi´ nski gasket. The graphs Ωn are close relatives of the Schreier graphs Σn of the group H (3) considered above. More precisely, one can obtain Ωn from Σn by contracting the edges between copies of Σn−1 in Σn . The graphs Ωn are also self-similar in the sense of [20], as can be seen in the picture.

292

D. D’Angeli, A. Donno and T. Nagnibeda •

•

Ωn−1

•

Ω1

•

Ωn

Ωn−1 Ωn−1 • • • • • (3) Similarly to the case of H above, define sets Pn and Ln . The same recursive 3n −1 rules hold, and the total number of closed polygons is again 2 2 , since the initial conditions are the same. Let Γcl n (z) denote the generating function of the subgraphs in Pn and let Υn (z) denote the generating function of the subgraphs in Ln . From relations (2) and (3) we deduce the following formulae:  cl 3 Γcl + Υ3n−1 (z), (6) n (z) = Γn−1 (z) and 3 Υn (z) = Υ2n−1 (z)Γcl n−1 (z) + Υn−1 (z). 2

(7)

3

Note that in (6) and (7) there are no factors z, z , z occurring in (4) and (5), because the special edges connecting elementary triangles have been contracted in Ωn . Theorem 3.5. For each n ≥ 1, the partition function of the Ising model on the nth Sierpi´ nski graph Ωn is Zn = 2

3n +3 2

with Γcl n (z)

=z

3n 2

n

· cosh(βJ)3 · Γcl n (tanh(βJ)), n 

n−k

ψk3

(z) · (ψn+1 (z) − 1),

k=1

where ψ1 (z) = k ≥ 3.

z+1 , z 1/2

ψ2 (z) =

z 2 +1 z

2 and ψk (z) = ψk−1 (z) − 3ψk−1 (z) + 4, for each

Proof. Again we shall use the expression Zn = 2| Vert(Ωn )| cosh(βJ)| Edges(Ωn )| · cl Γcl n (tanh(βJ)), where Γn (z) is the generating function of nclosed polygons in Ωn . In our case we have | Edges(Ωn )| = 3n and | Vert(Ωn )| = 3 2+3 . We know that the generating functions Γcl n (z) and Υn (z) satisfy the equations (6) and (7), with the initial conditions 3 Γcl 1 (z) = 1 + z

Υ1 (z) = z 2 + z.

Let us show by induction on n that the solutions of the system of equations (6) and (7) are  3n $n 3n−k 2 Γcl (z) · (ψn+1 (z) − 1) n (z) = z k=1 ψk n $n 3 3n−k Υn (z) = z 2 ψ (z). k=1 k

Ising Model on Schreier Graphs

293

3

3

3 2 2 For n = 1, we get Γcl 1 (z) = z ψ1 (z)(ψ2 (z) − 1) = z + 1 and Υ1 (z) = z ψ1 (z) = 2 z + z and so the claim is true. Now suppose that the assertion is true for n and let us show that it is true for n + 1. One gets: & '3 & '3 n n   n−k n−k 3n 3n cl 3 3 Γn+1 (z) = z 2 ψk (z) · (ψn+1 (z) − 1) + z 2 ψk (z)

=z

3n+1 2

=z

3n+1 2

k=1 n  k=1 n+1 

k=1 n−k+1

ψk3

n−k+1

ψk3

 3  2 (z) ψn+1 (z) − 3ψn+1 (z) + 3ψn+1 (z) (z) · (ψn+2 (z) − 1)

k=1

and

& Υn+1 (z) =

z

3n 2

=z

'2 & n−k ψk3 (z)

k=1

& +

n 

z

3n+1 2

3n 2

n 

k=1 n 

z '3

3n 2

n 

' n−k ψk3 (z)

· (ψn+1 (z) − 1)

k=1

n−k ψk3 (z)

n−k+1

ψk3

(z) · ψn+1 (z) = z

3n+1 2

k=1

n+1 

n−k+1

ψk3

(z).




k=1

Remark 3.6. The existence of the thermodynamic limit can be shown in exactly the same way as for the Schreier graphs of the Hanoi Towers group. 3.3. Renormalization approach Expressions for the partition function of the Ising model on the Sierpi´ nski gasket are well known to physicists. A renormalization equation for it can be found for example in [7] (see also references therein), and a more detailed analysis is given in [3]. Using renormalization, Burioni et al. [3] give the following recursion for the partition function of the Ising model on the graphs Ωn , n ≥ 1: Zn+1 (y) = Zn (f (y))[c(y)]3

n−1

,

where y = exp(βJ); f (y) is a substitution defined by  8 1/4 y − y4 + 4 y → f (y) = ; y4 + 3 and c(y) =

y4 + 1 4 [(y + 3)3 (y 8 − y 4 + 4)]1/4 ; y3

with Z1 (y) = 2y 3 + 6y −1 .

(8)

294

D. D’Angeli, A. Donno and T. Nagnibeda

A similar computation can be performed in the case of the Schreier graph Σn of the group H (3) , where the self-similarity of the graph allows to compare the partition function Z1 of the first level with the partition function of level 2, where the sum is taken only over the internal spins σ2 , σ3 , σ5 , σ6 , σ8 , σ9 (see figure below). The resulting recurrence is the same as (8), but with  8 1/4 y − 2y 6 + 2y 4 + 2y 2 + 1 y → f (y) = 2(y 4 + 1) and c(y) =

(y 4 − y 2 + 2)(y 2 + 1)3 (8(y 4 + 1)3 (y 8 − 2y 6 + 2y 4 + 2y 2 + 1))1/4 . y6 σ •1 Σ2

σ2 •

• σ9

σ3 • • σ4

• σ8 • σ5

• σ6

• σ7

Remark 3.7. The above recursions for the partition functions for Ωn ’s and Σn ’s can be deduced from our Theorems 3.3 and 3.5 by rewriting the formulae in the variable y = exp(βJ) and substituting z = tanh(βJ) = (y 2 − 1)/(y 2 + 1).

4. Statistics on weighted closed polygons This Section is devoted to the study of weighted generating functions of closed polygons, i.e., we allow the edges of the graph to have different weights tanh(βJi,j ), as in RHS of (1). We also take into account the fact that the graphs we consider are Schreier graphs of some self-similar group G with respect to a certain generating set S, and their edges are therefore labeled by these generators. It is thus natural to allow the situations where the energy between two neighbouring spins takes a finite number of possible values encoded by the generators S. Logarithmic derivatives of the weighted generating function with respect to s ∈ S give us the mean density of s-edges in a random configuration. We can further find the variance and show that the limiting distribution is normal. 4.1. The Schreier graphs of the Grigorchuk’s group Recall from 2.1 that the simple edges in Σn are always labeled by a. Moreover, the 2-cycles can be labeled by the couples of labels (b, c), (b, d) and (c, d). We want to compute the weighted generating function of closed polygons, with respect to the weights given by the labels a, b, c, d. Let us set, for each n ≥ 1: Xn = |{2-cycles with labels b, c}|

Yn = |{2-cycles with labels b, d}|

Wn = |{2-cycles with labels c, d}|

Ising Model on Schreier Graphs

295

One can easily check by using self-similar formulae for the generators, that the following equations hold:  n−2  Xn = Wn−1 + 2 Yn = Xn−1   Wn = Yn−1 . In particular, one gets

 n−2  Xn = Xn−3 + 2 Yn = Xn−1   Wn = Xn−2 ,

with initial conditions X1 = 0, X2 = 1 and X3 = 2. One gets the following values:  2n+1 −2   7 Xn =

2n+1 −4 7   2n+1 −1 7

 n 2 −1  if n ≡ 0(3)  7 n if n ≡ 1(3) Yn = 2 7−2   2n −4 if n ≡ 2(3) 7

 2n−1 −4  if n ≡ 0(3)  7 n−1 if n ≡ 1(3) Wn = 2 7 −1   2n−1 −2 if n ≡ 2(3) 7

if n ≡ 0(3) if n ≡ 1(3) , if n ≡ 2(3)

and, consequently, Theorem 4.1. For each n ≥ 1, the weighted generating function of closed polygons in Σn is  2n+1 −2 2n −1 2n−1 −4 cl  if n ≡ 0(3)  Γn (a, b, c, d) = (1 + bc) 7 (1 + bd) 7 (1 + cd) 7   

Γcl n (a, b, c, d) = (1 + bc) Γcl n (a, b, c, d)

= (1 + bc)

2n+1 −4 7 2n+1 −1 7

(1 + bd)

2n −2 7

(1 + bd)

2n −4 7

(1 + cd)

2n−1 −1 7

(1 + cd)

2n−1 −2 7

if n ≡ 1(3)

if n ≡ 2(3) .

Proposition 4.2. Let wn be the number of edges labeled w in a random closed 2 polygon in Σn , where w = a, b, c, d. Denote by µn,w and σn,w the mean and the variance of wn . Then, • for each n ≥ 1, an = 0; • the means and the variances of the random variables bn , cn , dn are given in the following table:

µn,b 2 σn,b µn,c 2 σn,c µn,d 2 σn,d

n ≡ 0(3)

n ≡ 1(3)

n ≡ 2(3)

3 (2n − 1) 14 3 (2n − 1) 28 5·2n−2 −3 7 5·2n−2 −3 14 3·2n−1 −5 14 3·2n−1 −5 28

3 n−1 (2 − 1) 7 3 n−1 (2 − 1) 14 5 n−1 − 1) 14 (2 5 n−1 (2 − 1) 28 3 n−1 − 1) 14 (2 3 n−1 (2 − 1) 28

3·2n −5 14 3·2n −5 28 5·2n−1 −3 14 5·2n−1 −3 28 3 n−2 (2 − 1) 7 3 n−2 (2 − 1) 14

• the random variables bn , cn , dn are asymptotically normal, as n → ∞.

296

D. D’Angeli, A. Donno and T. Nagnibeda

Proof. It is clear that an edge labeled by a never belongs to a closed polygon of Σn , so that an = 0. Let us choose, for instance, n ≡ 0(3). Putting cl Γcl n (b) := Γn (1, b, 1, 1) = 2

2n−1 −4 7

we can obtain the mean µn,b and the variance of the function log(Γcl n (b)). We get

2 σn,b

(1 + b)

3·2n −3 7

,

of bn by studying the derivatives

    
Γcl 3 n n (b)  µn,b = log(Γcl (b)) = (2 − 1) . b=1 = n cl Γn (b)  14 Taking once more derivative, one gets   cl cl 2   

Γcl 3 n (b)Γn (b) − (Γn (b))  cl n log(Γn (b)) = b=1 = 28 (1 − 2 ). 2 (Γcl (b)) n Hence,  

3 n 2 σn,b = log(Γcl + µn,b = (2 − 1). n (b)) 28 b −µ Now let Bn = nσn,bn,b be the normalized random variable, then the moment generating function of Bn is given by E(etBn ) = e−µn,b t/σn,b E(etbn /σn,b ) = e−µn,b t/σn,b

t/σn,b Γcl ) n (e . cl Γn (1)

We get 3

E(etBn ) = e−t( 7 (2

n

−1))

1/2

3

· 2 7 (1−2

n

)

1/2  3·2n −3  28 7 1 + et( 3(2n −1) ) ,

t2

whose limit as n → ∞ is e 2 , showing that the random variable is asymptotically normal. Similar computations can be done for c and d.
4.2. The Schreier graphs of the Basilica group We also compute the weighted generating function of closed polygons for the Basilica group, with respect to the weights given by the labels a and b on the edges of its Schreier graph Σn . We use here the computations from Proposition 2.2. Theorem 4.3. The weighted generating function of closed polygons in the Schreier graph Σn of the Basilica group is n−1 2 −1

Γcl n (a, b)

 

=

n−1

2n−2k−1 2−1   n−2k k 2 1+a 1 + b2 2k

k=1



k=1

n−1 2 2

× 1+a

2  n−1 2  n+1  1 + b2 2 1 + b2 2

for n ≥ 5 odd and n 2 −1

Γcl n (a, b)

=



k=1

for n ≥ 4 even.

−1  2n−2k−1 2 n

1+a

2k

k=1

1 + b2

k

2n−2k 

n

1 + a2 2

 n 2 1 + b2 2

Ising Model on Schreier Graphs

297

Proposition 4.4. The means and the variances of the densities an and bn are given in the following table: n ≥ 5 odd µn,a

2

n−2

n ≥ 4 even 2n−2

2 σn,a

(n + 1)2n−4

(n + 2)2n−4

µn,b

2n−1

2n−1

2 σn,b

(n + 3)2n−3

(n − 2)2n−3

4.3. The Schreier graphs of H (3) Let us denote by Υlr n (a, b, c) the weighted generating function of the subgraphs that belong to the set Ln , defined in Subsection 3.1 (the exponent lr stands for leftright, to say that we are considering subgraphs whose vertices have even degree, except for the left-most and the right-most vertices of Σn which have odd degree.) ru Analogously, we define Υlu n (a, b, c) and Υn (a, b, c), where the exponents lu and ru stand for left-up and right-up, respectively. By using the self-similar expressions for the generators given in Subsection 3.1, we find that these functions satisfy the following system of equations (we omit the arguments a, b, c):  3  cl lu ru Γn+1 = Γcl + abcΥlr  n n Υn Υn      lu ru cl lu 3 Υn+1 = aΥlr n Υn Γn + bc Υn (9) lu lr cl ru 3  Υru  n+1 = cΥn Υn Γn + ab (Υn )     lr ru cl lr 3 Υn+1 = bΥlu n Υn Γn + ac Υn lr lu with the initial conditions Γcl 1 (a, b, c) = 1 + abc, Υ1 (a, b, c) = ac + b, Υ1 (a, b, c) = ru a + bc, Υ1 (a, b, c) = c + ab.

Proposition 4.5. The mean and the variance for wn , with w = a, b, c, are: µn,w =

3n − 1 4

2 σn,w =

3n − 1 . 8

The random variables wn with w = a, b, c are asymptotically normal. Proof. If we put a = b = 1, the system (9) reduces to  cl  3 lu ru Γn+1 = Γcl + cΥlr  n n Υn Υn   Υlu = Υlr Υru Γcl + c Υlu 3 n+1 n n n n lu lr cl ru 3 Υru = cΥ Υ Γ + (Υ )  n+1 n n n   nlr 3  lr lu ru cl Υn+1 = Υn Υn Γn + c Υn with the initial conditions lr lu ru Γcl 1 (1, 1, c) = Υ1 (1, 1, c) = Υ1 (1, 1, c) = Υ1 (1, 1, c) = 1 + c.

(10)

298

D. D’Angeli, A. Donno and T. Nagnibeda One can prove, by induction on n, that the solutions of the system (10) are

lr lu ru Γcl n (1, 1, c) = Υn (1, 1, c) = Υn (1, 1, c) = Υn (1, 1, c) = (1 + c)

3n −1 2

for each n.

log(Γcl n (1, 1, c))

By studying the derivatives of the function with respect to c, one gets: 3n − 1 3n − 1 2 µn,c = σn,c . = 4 8 Symmetry of the labeling of the graph ensures that the same values arise for the random variables an , bn .
4.4. The Sierpi´ nski graphs The Sierpi´ nski graphs Ωn being not regular, they cannot be realized as Schreier graphs of any group. There exist however a number of natural, geometric labelings of edges of Ωn by letters a, b, c (see [4]). Let us first consider the labeling that is obtained by considering the labeled Schreier graph Σn of the Hanoi Towers group and then by contracting the edges connecting copies of Σn−1 in Σn ; we call this the “Schreier” labeling of Ωn . Remark 4.6. The “Schreier” labeling on Ωn can be constructed recursively, as follows. Start with the graph Ω1 in the picture below; then, for each n ≥ 2, the graph Ωn is defined as the union of three copies of Ωn−1 . For each one of the out-most (corner) vertices of Ωn , the corresponding copy of Ωn−1 is reflected with respect to the bisector of the corresponding angle. • a

c • Ω1 a

• c

•

b

c a

Ω2 b •

•

b

•

b

•

a

•

c

• a • b Ω3 • c • a •

c

b a c •

• b

a •

•

a

b

a

b •

• a c

c b b

c

c

•

c

•

b a a

•

c b

•

Ising Model on Schreier Graphs

299

lu ru Let Υlr n (a, b, c), Υn (a, b, c) and Υn (a, b, c) be defined as for the Schreier graphs Σn of the Hanoi Towers group in the previous subsection. Then one can easily check that these functions satisfy the following system of equations:

 cl  3 lu ru Γn+1 = Γcl + Υlr  n n Υn Υn    Υlu = Υlr Υru Γcl + Υlu 3 n+1 n n n n ru lu lr cl ru 3  Υ = Υ Υ Γ + (Υ )  n+1 n n n   nlr 3  lr lu ru cl Υn+1 = Υn Υn Γn + Υn lr lu with the initial conditions Γcl 1 (a, b, c) = 1 + abc, Υ1 (a, b, c) = ac + b, Υ1 (a, b, c) = ru a + bc, Υ1 (a, b, c) = c + ab.

Proceeding as in Subsection 4.3, we find: lr lu ru Γcl n (1, 1, c) = Υn (1, 1, c) = Υn (1, 1, c) = Υn (1, 1, c) = 2

3n−1 −1 2

n−1

(1 + c)3

,

which implies the following Proposition 4.7. The mean and the variance for the random variable wn , with w = a, b, c, for Ωn with the “Schreier” labeling are: µn,w =

3n−1 2

2 = σn,w

3n−1 . 4

The random variables an , bn , cn are asymptotically normal. It is interesting to compare these computations with those for a different labeling of Ωn , that we call the “rotation-invariant” labeling of Sierpi´ nski graphs, defined recursively as follows. (Compare the construction to the recursive description of the “Schreier labeling” in Remark 4.6.) Let Ω2 be the weighted graph in the following picture. • a

b

Ω2 • c

b •

a

•

c

a

c •

b

•

Then define, for each n ≥ 3, Ωn as the union of three copies of Ωn−1 , rotated by kπ/3 with k = 0, 1, 2.

300

D. D’Angeli, A. Donno and T. Nagnibeda For n = 3, one gets the following graph. • a b • •

a • •

•

a

a

b

b c

•

•

• a

• a b

c c

b

a

c c

b Ω3

•

c

•

b

b •

c

a

c c a

•

•

b

It turns out that the weighted generating function of closed polygons is easier to compute for Ωn with the “rotation-invariant” labeling, than with the “Schreier” labeling. More precisely, we have the following Theorem 4.8. For each n ≥ 2, the weighted generating function of closed polygons for the graph Ωn with the “rotation-invariant” labeling is Γcl n (a, b, c) = ((a+bc)(b+ac))

7·3n−2 4

n−2

ψ13

(a, b, c)

n 

ψk3

n−k

(a, b, c)·(ψn+1 (a, b, c)−1)

k=2

where ψ1 (a, b, c) =

1+c 1 4

,

ψ2 (a, b, c) =

1 + ab 1

((a + bc)(b + ac)) ((a + bc)(b + ac)) 2 2 2 2 2 2 2 2 2 a b c − a b c + a b + 4abc + c − c + 1 + a2 c + b2 c ψ3 (a, b, c) = (a + bc)(b + ac)

,

and, for each k ≥ 4, 2 ψk (a, b, c) = ψk−1 (a, b, c) − 3ψk−1 (a, b, c) + 4.

Proof. Consider the graph Ωn . For each n ≥ 2, define the sets Pn and Ln as in Subsection 3.1 and let Γcl n (a, b, c) and Υn (a, b, c) be the associated weighted generating functions. By using the symmetry of the labeling, one can check that these functions satisfy the following equations   cl 3 Γcl + Υ3n−1 (a, b, c) n (a, b, c) = Γn−1 (a, b, c) (11) Υn (a, b, c) = Υ3n−1 (a, b, c) + Υ2n−1 (a, b, c)Γcl n−1 (a, b, c) with the initial conditions  2 2 2 2 2 2 2 2 2 Γcl 2 = (1 + c)(1 + ab)(a b c − a b c + a b + 4abc − abc − ab + c − c + 1) Υ2 = (1 + c)(1 + ab)(a + bc)(b + ac).

Ising Model on Schreier Graphs

301

As in the proof of Theorem 3.5, one shows by induction on n that the solutions of the system are  $ n−2 n−k 7·3n−2 4 Γcl ψ13 (a,b,c) nk=2 ψk3 (a,b,c) · (ψn+1 (a,b,c) − 1) n = ((a + bc)(b + ac)) $ n−2 n−k 7·3n−2 n Υn = ((a + bc)(b + ac)) 4 ψ13 (a,b,c) k=2 ψk3 (a,b,c)
Remark 4.9. Although the labels a and b are not symmetric to the label c in the “rotation-invariant” labeling, computations show that the functions Γcl n (a, 1, 1), cl Γcl n (1, b, 1) and Γn (1, 1, c) are the same in this case as in the case of the “Schreier” labeling. It follows that the values of the mean and the variance of the random variables an , bn, cn remain the same as in the “Schreier” labeling, see Proposition 4.7. 4.5. Correspondences via Fisher’s Theorem In [6] M. Fisher proposed a method of computation for the partition function of the Ising model on a (finite) planar lattice Y by relating it to the partition function of the dimers model (with certain weights) on another planar lattice Y ∆ constructed from Y . (The latter partition function can then be found by computing the corresponding Pfaffian given by Kasteleyn’s theorem.) This method uses the expression (1) for the partition function in terms of the generating function of closed polygons in Y . The new lattice Y ∆ is constructed in such a way that Ising polygon configurations on Y are in one-to-one correspondence with dimer configurations on Y ∆ . In order to have equality of generating functions however, the edges of Y ∆ should be weighted in such a way that the edges coming from Y have the same weight tanh(βJi,j ) as in the RHS of (1), and other edges have weight 1. Applying Fisher’s construction to Sierpi´ nski graphs, one concludes easily that ∆ ˜ if Y = Ωn for some n ≥ 1, then Y = Σn+1 , the (n + 1)st Schreier graph of the Hanoi Towers group H (3) with three corner vertices deleted. Note that the corner vertices are the only vertices in Σn with loops attached to them, and so it is anyway natural to forget about them when counting dimer coverings. The construction consists in applying to Y the following substitutions, where edges labeled by e in Y ∆ are in bijection with edges in Y , and should be assigned weight tanh(βJi,j ). Other edges should be assigned weight 1. e • =⇒

e

•

•

• e

•

•

=⇒

•

• e

•

•

e

e

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D. D’Angeli, A. Donno and T. Nagnibeda

The correspondence between closed polygons in Y and dimer coverings of Y ∆ is as follows: if an edge in Y belongs to a closed polygon, then the corresponding e-edge in Y ∆ does not belong to the dimer covering of Y ∆ associated with that closed polygon, and vice versa. The following pictures give an example of a closed polygon in Ω2 and of the ˜ 3: associated dimer covering of Σ • • • • •

• • •

•

•

• •

•

• •

•

•

• •

• •

•

• •

•

•

•

•

• •

If, for a certain n ≥ 1, the Sierpi´ nski graph Ωn is considered with the ˜ n+1 given by Fisher’s construc“Schreier” labeling, then the labeling of the graph Σ tion will be a restriction of the usual Schreier labeling of Σn+1 . More precisely, only the edges that connect copies of Σn−1 but not copies of Σn will be labeled (other edges have weight 1), and the labels are the same as in the standard labeling of Σn+1 as a Schreier graph of the group H (3) . The following picture represents ˜ 3 as Y ∆ with Y = Ω2 with the “Schreier” labeling. Σ 1 • a c• • • 1 1 1 1 • • • • 1 b 1 1 1 • • 1 1 1 1 a• 1 •b b• 1 •c • • • • 1 1 1 1 1 • • 1 a 1 • 1 • 1 •c• Remark 4.10. One can wonder what Fisher’s construction gives for {Σn }n≥1 . It ˜ n+1 , the turns out that if Y = Σn , the nth Schreier graph of H (3) , then Y ∆ = Σ same as for Y = Ωn , the nth Sierpi´ nski graph. Acknowledgment We are grateful to R. Grigorchuk for numerous inspiring discussions and to S. Smirnov for useful remarks on the first version of this paper.

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303

References [1] L. Bartholdi and R.I. Grigorchuk, On the spectrum of Hecke type operators related to some fractal groups, Tr. Mat. Inst. Steklova, 231 (2000), Din. Sist., Avtom. i Beskon. Gruppy, 5–45; translation in Proc. Steklov Inst. Math. 2000, no. 4 (231), 1–41. [2] I. Bondarenko, Groups generated by bounded automata and their Schreier graphs, PhD Thesis Texas A&M, 2007, available at http://txspace.tamu.edu/bitstream/ handle/1969.1/85845/Bondarenko.pdf?sequence=1 [3] R. Burioni, D. Cassi and L. Donetti, Lee-Yang zeros and the Ising model on the Sierpi´ nski Gasket, J. Phys. A: Math. Gen., 32 (1999), 5017–5027. [4] D. D’Angeli, A. Donno and T. Nagnibeda, The dimer model on some families of self-similar graphs, preprint. [5] D. D’Angeli, A. Donno, M. Matter and T. Nagnibeda, Schreier graphs of the Basilica group, Journal of Modern Dynamics, 4 (2010), no. 1, 167–205. [6] M.E. Fisher, On the dimer solution of planar Ising models, J. Math. Phys., 7 (1966), no. 10, 1776–1781. [7] Y. Gefen, A. Aharony, Y. Shapir and B. Mandelbrot, Phase transitions on fractals. II. Sierpi´ nski gaskets, J. Phys. A, 17 (1984), no. 2, 435–444. [8] R.I. Grigorchuk, Solved and unsolved problems around one group, in: “Infinite groups: geometric, combinatorial and dynamical aspects” (L. Bartholdi, T. ˙ Ceccherini-Silberstein, T. Smirnova-Nagnibeda and A. Zuk editors), Progr. Math., 248, Birkh¨ auser, Basel, 2005, 117–218. [9] R.I. Grigorchuk and A.M. Stepin, Gibbs states on countable groups. Teor. Veroyatnost. i Primenen. 29 (1984), no. 2, 351–354. ˇ c, Self-similarity and branching in group theory, in: [10] R.I. Grigorchuk and Z. Suni´ “Groups St. Andrews 2005, I”, London Math. Soc. Lecture Note Ser., 339, Cambridge Univ. Press, Cambridge, 2007, 36–95. ˙ [11] R.I. Grigorchuk and A. Zuk, On a torsion-free weakly branch group defined by a three-state automaton, International J. Algebra Comput., 12 (2002), no. 1, 223–246. [12] M. Gromov, Structures m´etriques pour les vari´et´es riemanniennes, Edited by J. Lafontaine and P. Pansu, Textes Math´ematiques, 1. CEDIC, Paris, 1981. [13] E. Ising, Beitrag zur Theorie des Ferromagnetismus, Zeit. f¨ ur Physik, 31 (1925), 253–258. [14] W. Lenz, Beitr¨ age zum Verst¨ andnis der magnetischen Eigenschaften in festen K¨ orpern, Physikalische Zeitschrift, 21 (1920), 613–615. [15] F. Lund, M. Rasetti and T. Regge, Dimer and Ising models on the Lobachevsky plane, Theor. Math. Phys. 33 (1977), 1000–1015. [16] M. Matter and T. Nagnibeda, Self-similar groups and Abelian sandpile model on random rooted graphs, preprint (2010). [17] V. Nekrashevych, Self-similar Groups, Mathematical Surveys and Monographs, 117. American Mathematical Society, Providence, RI, 2005. [18] L. Rogers and A. Teplyaev, Laplacians on the basilica Julia set, to appear in Commun. Pure Appl. Anal., 9 (2010), no. 1, 211–231.

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[19] C. Series and Ya. Sinai, Ising models on the Lobachevsky plane, Commun. Math. Phys. 128 (1990), 63–76. [20] E. Teufl and S. Wagner, Enumeration of matchings in families of self-similar graphs, Discrete Applied Mathematics, 158 (2010), no. 14, 1524–1535. Daniele D’Angeli Department of Mathematics Technion–Israel Institute of Technology Technion City Haifa 32 000, Israel e-mail: [email protected] Alfredo Donno Dipartimento di Matematica Sapienza Universit` a di Roma Piazzale A. Moro, 2 I-00185 Roma, Italia e-mail: [email protected] Tatiana Nagnibeda Section de Math´ematiques Universit´e de Gen`eve 2–4, Rue du Li`evre, Case Postale 64 CH-1211 Gen`eve 4, Suisse e-mail: [email protected]

Progress in Probability, Vol. 64, 305–324 c 2011 Springer Basel AG


Aspects of Toeplitz Determinants Igor Krasovsky Abstract. We review the asymptotic behavior of a class of Toeplitz (as well as related Hankel and Toeplitz + Hankel) determinants which arise in integrable models and other contexts. We discuss Szeg˝ o, Fisher-Hartwig asymptotics, and how a transition between them is related to the Painlev´e V equation. Certain Toeplitz and Hankel determinants reduce, in certain double-scaling limits, to Fredholm determinants which appear in the theory of group representations, in random matrices, random permutations and partitions. The connection to Toeplitz determinants helps to evaluate the asymptotics of related Fredholm determinants in situations of interest, and we review the corresponding results. Mathematics Subject Classification (2000). Primary 47B35; Secondary 70H06. Keywords. Toeplitz matrices, random matrices.

1. Introduction Let f (z) be a function integrable over the unit circle C with Fourier coefficients  2π 1 fj = f (eiθ )e−ijθ dθ, j = 0, ±1, ±2, . . . 2π 0 Then the n-dimensional Toeplitz determinant of a Toeplitz matrix with symbol f (z) is given by Dn (f ) = det(fj−k )n−1 (1.1) j,k=0 . Substituting here the expressions for the Fourier coefficients, and using formulae for Vandermonde determinants, one obtains another useful representation:  2π  2π  n  1 iθj iθk 2 Dn (f ) = · · · |e − e | f (eiθj )dθj . (1.2) (2π)n n! 0 0 1≤j
j=1

Toeplitz determinants are closely related to the polynomials orthogonal with weight f (z) on the unit circle. Namely, if Dk (f ) = 0, k = k0 , k0 + 1 . . . , for some

306

I. Krasovsky

k0 ≥ 0, then the polynomials φk (z) = χk z k + · · · , φ!k (z) = χk z k + · · · of degree k, k = k0 , k0 + 1, . . . , satisfying  2π  2π 1 1 φk (z)z −j f (z)dθ = χ−1 δ , φ!k (z −1 )z j f (z)dθ = χ−1 jk k k δjk , 2π 0 2π 0 

z = eiθ ,

j = 0, 1, . . . , k, (1.3)

exist and χk = Dk /Dk+1 , where, by convention, D0 ≡ 1. We see from (1.2) that if f (z) is positive on C, we have Dn (f ) > 0 for all n, and therefore in this case we can set k0 = 0. A Toeplitz determinant can be represented as a Fredholm determinant of an integral operator acting on L2 (C) which belongs to the special class of so-called integrable operators [48]. It can also be written in a different way in terms of a Fredholm determinant of an operator now acting on 2 (n, n + 1, . . . ) [62, 27, 30]. (Note that the symbols f (z) considered in [62, 27, 30] are assumed to be sufficiently smooth.) Another useful property of many Dn (f )’s encountered in applications is the existence of simple differential identities relating the determinant to orthogonal polynomials evaluated at a few special points. The precise form of such identities depends on the given f (z). It turns out that the above properties play a key role in making Toeplitz determinants amenable to a detailed asymptotic analysis, in particular, by RiemannHilbert-Problem methods. In this paper, we will review some asymptotic results on Dn (f ) (and related Hankel, Teoplitz+Hankel, and Fredholm determinants) and briefly mention their applications in integrable models, random matrices, random permutations, group representation theory, and also in various conjectures on Riemann’s ζ and Dirichlet’s L-functions. This review is based, to a large extent, on the recent work of the author with T. Claeys, P. Deift, A. Its, and J. Vasilevska. For other aspects of Toeplitz determinants not mentioned here, the reader is referred to [33, 34, 32, 67, 50, 49, 47] for properties of Toeplitz matrices and determinants, to [34, 108, 109, 110, 89, 64, 98, 19, 91, 90, 105, 72, 69, 14] for generalizations to the continuous, higher-dimensional, and the block-Toeplitz cases (with relations to stationary determinantal processes, integrable models, and entanglement entropy), to [88, 1, 20] for connections with multiple orthogonal polynomials. The paper consists of three parts: in Section 2, the simplest asymptotics with f (z) fixed and n → ∞ are considered; in Section 3, the symbol f (z) is allowed to depend on n in ways which describe a transition between different asymptotic regimes arising in Section 2; in Section 4, the symbol f (z) also depends on n, but in such a way that in the limit n → ∞, Toeplitz determinants turn into certain Fredholm determinants which are important for random matrices and random permutations. Following standard practice, we refer to the large n asymptotics of Sections 3 and 4 as double-scaling limits.

Aspects of Toeplitz Determinants

307

2. Asymptotics for a fixed symbol We assume in this section that f (z) does not depend on the size of the determinant n. We are interested in the asymptotics of Dn (f ) as n → ∞. The following result is basic. Theorem 2.1 (Strong Szeg˝ o limit theorem). Let f (z) be non-zero on C, ln f (z) ∈ L1 (C), and suppose that the sum S(f ) =

∞ 

|k||(ln f )k | , 2

k=−∞

1 (ln f )k = 2π



2π

ln f (eiθ )e−ikθ dθ,

(2.1)

as n → ∞.

(2.2)

0

converges. Then ln Dn (t) = n(ln f )0 +

∞ 

k(ln f )k (ln f )−k + o(1),

k=1

The theorem was initially proved by Szeg˝ o [99, 100] (the leading term in 1915, the next in 1952) under stronger conditions on f (z). The conditions were then weakened by many authors. In the present form, the theorem was proved in [68, 65, 80]. See [97] for a detailed account. A strong motivation to study such asymptotics first came in the end of 1940’s after Onsager’s solution of the 2-dimensional Ising model and his observation that a 2-spin correlation function in the model can be written as a Toeplitz determinant Dn (f ), where n denotes the distance between the spins. For temperatures less than critical (T < Tc ), the symbol of this Toeplitz determinant has an analytic logarithm in a neighborhood of the unit circle and, moreover, (ln f )0 = 0. Therefore, Szeg˝ o’s theorem can be applied, and one concludes that Dn (f ) tends to a constant as n → ∞. Thus the correlation does not decay as the distance increases, which indicates the presence of a long-range order, and hence, a magnetization. As T $ Tc , however, 2 singularities of f (z) approach the unit circle at z = 1, and, at T = Tc merge into a single singularity on C; namely, a jump-type singularity at z = 1 (see, e.g., [94]). For f (z) with such a singularity, the sum (2.1) diverges, and Theorem 2.1 can no longer be applied. In fact, it turns out [94] that in this case Dn (f ) decays as n−1/4 and, therefore, there exists no long-range order. For correlation functions arising in other situations, such as, e.g., the so-called emptiness formation probability in the XY spin chain in a magnetic field [40, 58], one obtains Toeplitz determinants with both jump-type and root-type singularities, and in the most general situation one is led to consider symbols of the form: f (z) = eV (z) z

m j=0

βj

m 

−βj

|z − zj |2αj gzj ,βj (z)zj

,

z = eiθ ,

θ ∈ [0, 2π),

j=0

(2.3)

308

I. Krasovsky

for some m ≥ 0, where zj = eiθj ,

j = 0, . . . , m, 0 = θ0 < θ1 < · · · < θm < 2π;  eiπβj 0 ≤ arg z < θj gzj ,βj (z) = , e−iπβj θj ≤ arg z < 2π

(2.4)

/αj > −1/2,

(2.6)

βj ∈ C,

(2.5)

j = 0, . . . , m,

and V (eiθ ) is a sufficiently smooth function on the unit circle (see below) with Fourier coefficients  2π 1 Vk = V (eiθ )e−kiθ dθ. (2.7) 2π 0 The canonical Wiener-Hopf factorization of eV (z) is given by eV (z) = b+ (z)eV0 b− (z),

b+ (z) = e

∞ k=1

Vk z k

,

b− (z) = e

 −1 k=−∞

Vk zk

. (2.8)

The condition (2.6) on αj ensures the integrability of f . Note that the size of the jump at zj is determined by the parameter βj , and the root-type singularity, by αj . We assume that zj , j = 1, . . . , m, are genuine singular points, i.e., either αj = 0 or βj = 0. However, the absence of a singularity at z = 1, i.e., the case α0 = β0 = 0, is allowed. Singularities of type (2.3) are known as Fisher-Hartwig singularities because of the work [59] where the authors summarized a variety of applications of Toeplitz determinants with such symbols and presented a conjecture about the asymptotic form of Dn (f ) in this case. Due to the subsequent efforts of many workers, we have the following description of the asymptotics. Define the seminorm |||β||| = max |/βj − /βk |,

(2.9)

j,k

where the indices j, k = 0 are omitted if z = 1 is not a singular point, i.e., if α0 = β0 = 0. Note that in the case of a single singularity, we always have |||β||| = 0. First, consider the situation when |||β||| is strictly less then 1. Theorem 2.2. Let f (z) be defined in (2.3), |||β||| < 1, /αj > −1/2, αj ± βj = −1, −2, . . . for j, k = 0, 1, . . . , m, and V (z) satisfies the smoothness conditions (2.11), (2.12) below. Then as n → ∞, ? m > ∞   Dn (f ) = exp nV0 + kVk V−k b+ (zj )−αj +βj b− (zj )−αj −βj j=0

k=1

×n

m

2 2 j=0 (αj −βj )



|zj − zk |2(βj βk −αj αk )



0≤j
×

m 

G(1 + αj + βj )G(1 + αj − βj ) (1 + o(1)) , G(1 + 2αj ) j=0

zk zj eiπ

αj βk −αk βj

(2.10)

Aspects of Toeplitz Determinants

309

where G(x) is Barnes’ G-function [10]. The double product over j < k is set to 1 if m = 0. The branches in (2.10) are determined as follows: b± (zj )−αj ±βj = ∞ exp{(−αj ±βj ) k=1 V±k z ±k }, (zk zj−1 e−iπ )αj βk −αk βj = exp{i(θk −θj −π)(αj βk − αk βj )}. Note that since G(−k) = 0, k = 0, 1, . . . , formula (2.10) no longer represents the leading asymptotics if αj + βj or αj − βj is a negative integer for some j. Such degenerate cases can be handled by carrying the analysis to higher order, but we present no further details here. The smoothness condition on V (z) assumed in Theorem 2.2 is that ∞ 

|k|s |Vk | < ∞

(2.11)

k=−∞

holds for some s such that s>

1+

m  j=0

(0αj )2 + (/βj )2



1 − |||β|||

.

(2.12)

Note that the condition |||β||| < 1 is important here. The Barnes’ G-function first appeared in asymptotic Toeplitz theory in the work of Lenard [92]. Theorem 2.2 was proved by Widom [104] in the case when /αj > −1/2, and all βj = 0, and with a stronger condition on V (z). In [11], Basor extended the result to /αj > −1/2, /βj = 0, and in [12], to αj = 0, |/βj | < 1/2. In [31], B¨ottcher and Silbermann established the result in the case that |/αj | < 1/2, |/βj | < 1/2. In [53], Ehrhardt proved the theorem for the full range of parameters, namely /αj > −1/2, |||β||| < 1, and for C ∞ functions V (z). These results were established by operator-theory methods (see [53] for a review of these and other related results including an extension to /α < −1/2, 2α = −1, −2, . . . , when f is replaced by a suitable distribution). In [44], the authors reprove the theorem by Riemann-Hilbert-Problem methods, and relax the smoothness conditions on V (z) to (2.11), (2.12). Consider now the general case of Fisher-Hartwig symbols f (z) with the restriction |||β||| < 1 removed. Note first that f (z) has several representations of type (2.3) with different sets of parameters βj . Namely, if each βj in (2.3) such that either βj = 0 or αj = 0 is replaced by β!j = βj + nj , where nj are integers n subject to the condition j=0 nj = 0, then the resulting function f (z; n0 , . . . , nm ) is related to f (z) in the following way: f (z; n0 , . . . , nm ) =

m 

−nj

zj

f (z),

j=0

i.e., it differs from f (z) only by a constant. Each f (z; n0 , . . . , nm ) so obtained is called a FH-representation of the symbol. Denote by M the (finite) set of FHm representations for which j=0 (/β!j )2 is minimal. There exists a simple procedure (see [43]) to solve this discrete variational problem and to construct M explicitly.

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! ≤ 1, and we One can show that there is always a FH-representation with |||β||| have the following 2 mutually exclusive possibilities: ! < 1 then it turns out that • If there exists a FH-representation such that |||β||| this FH-representation is the single element of M. In particular, if |||β||| < 1, the set M consists of a single element corresponding to all nj = 0, and Theorem 2.3 below reduces to Theorem 2.2. ! = 1 then M consists of • If there exists a FH-representation such that |||β||| several (at least 2) elements. The set M is called non-degenerate if it contains no representations for which ! αj + βj or αj − β!j is a negative integer for some j. The general result is as follows. Theorem 2.3. Let f (z) be given in (2.3), V (z) satisfy the condition (2.11) above for some sufficiently large s (depending only on αj , βj ), and /αj > −1/2, βj ∈ C, j = 0, 1, . . . , m. Let M be non-degenerate. Then, as n → ∞,  n  m   n  Dn (f ) = zj j  R(f (z; n0 , . . . , nm ))(1 + o(1)) , (2.13) j=0

where the sum is over all FH-representations in M. Each R(f (z; n0 , . . . , nm )) stands for the right-hand side of the formula (2.10), without the error term, corresponding to f (z; n0 , . . . , nm ). An explicit lower bound on s (depending on βj , αj ) similar to (2.12) is given in [43]. This theorem was conjectured by Basor and Tracy [18] and proved in [43]. Hankel and Toeplitz+Hankel determinants are also of interest. Let w(x) be an integrable function on a subset J of R. Then the Hankel determinant with symbol w(x) supported on J is given by  n−1 H j+k Dn (w(x)) = det x w(x)dx . (2.14) J

j,k=0

When J is a finite interval – we then set J = [−1, 1] without loss of generality – Hankel determinants are related to Toeplitz determinants by the following formulae [43], involving the orthogonal polynomials (1.3): w(x) =

f (eiθ ) , | sin θ|

[DnH (w(x))]2 =

π 2n 4

(n−1)2

x = cos θ,

x ∈ [−1, 1];

(χ2n + φ2n (0))2 D2n (f (z)). φ2n (1)φ2n (−1)

(2.15)

(2.16)

A particularly interesting class of Toeplitz+Hankel determinants appearing in the theory of classical groups and its applications to random matrices and

Aspects of Toeplitz Determinants

311

statistical mechanics (see, e.g., [7, 61, 81]) is defined as follows for even f (eiθ ) = f (e−iθ ) (for even f the matrices involved are symmetric): det(fj−k ± fj+k+1 )n−1 j,k=0 . (2.17) They are related to Hankel determinants with symbols on [−1, 1] by the expressions det(fj−k + fj+k )n−1 j,k=0 ,

det(fj−k − fj+k+2 )n−1 j,k=0 ,

2

det(fj−k +

fj+k )n−1 j,k=0

=

2n

−2n+2

πn

DnH (f (eiθ(x))/



1 − x2 ),

 2n H Dn (f (eiθ(x)) 1 − x2 ), n π & ' @ 2 2n −n H 1+x iθ(x) , = Dn f (e ) πn 1−x & ' @ 2 2n −n H 1−x iθ(x) = Dn f (e ) . πn 1+x

(2.18)

2

det(fj−k − fj+k+2 )n−1 j,k=0 =

(2.19)

det(fj−k + fj+k+1 )n−1 j,k=0

(2.20)

det(fj−k − fj+k+1 )n−1 j,k=0

(2.21)

Asymptotic formulae for Hankel and Toeplitz+Hankel determinants with Fisher-Hartwig singularities, whose derivation was based on the above theorems for Toeplitz determinants, an asymptotic Riemann-Hilbert-Problem analysis of the polynomials (1.3), and the above relations, are presented in [43]. For other asymptotic results, see [16, 17, 8, 13, 36]. For related asymptotic results on an important class (see Section 4) of Toeplitz determinants when the symbol is supported on an arc of the unit circle, see [107, 87, 85, 46, 36]. Theorem 2.3 and asymptotic formulae for Hankel and Toeplitz+Hankel determinants find applications, e.g., for correlation functions in the XY spin chain in a magnetic field mentioned above, in the theory of the impenetrable Bose gas [61, 96], in random matrix conjectures for average values of Riemann’s ζ-function, and Dirichlet’s L-functions [82, 66, 35]. For a more detailed discussion of the material presented in this section so far, the reader is referred to [43]. A related area of interest is the asymptotic analysis of Hankel determinants whose symbol has Fisher-Hartwig singularities and is supported on the whole real line, or the half-line. In particular, in the Gaussian Unitary Ensemble of random matrix theory, the correlation function of products of powers of the absolute values of the characteristic polynomial is precisely such a Hankel determinant DnH (w): $m 2 namely, the symbol is supported on R and given by w(x) = exp(−x ) j=1 |x − µj |2αj , µj ∈ R, /αj > −1/2. This determinant is also related to the 1-dimensional impenetrable Bose gas and conjectures for mean values of Riemann’s ζ-function on the critical line. For a discussion of the results in this area, see [86, 70, 71]. For analysis of some other Hankel determinants appearing in random matrix models, see [21, 55]. For a recent application of Hankel determinants in the six-vertex model see [74, 22, 23, 24, 25].

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Note that the importance of Fisher-Hartwig singularities appears to stem from the following feature. The asymptotics of the orthogonal polynomials at the location of such a singularity are described by the confluent hypergeometric function [43]. The two independent parameters of such functions are related to the parameters α and β of the singularity (for β = 0 confluent hypergeometric functions reduce to Bessel functions). The location of the singularity corresponds to the single finite branch point of the confluent hypergeometric functions. As hypergeometric functions (which depend on 3 parameters) have 2 finite branch points, we would not be able to confine ourselves to hypergeometric functions if we wanted to consider a singular point which generalizes Fisher-Hartwig in some essential way. Roughly speaking, a Fisher-Hartwig singularity is the most general hypergeometric singular point. Modifications, of course, are possible: e.g., the end points of the interval [−1, 1] can be regarded as modified Fisher-Hartwig singularities for a Hankel determinant with symbol on [−1, 1] as discussed above.

3. Transition asymptotics A natural question to ask is how the transition between various asymptotic regimes of the previous section occurs. Consider once again the 2-spin correlation function for the 2-dimensional Ising model discussed above, which is a Toeplitz determinant. As T $ Tc a transition between the Szeg˝o asymptotics and the Fisher-Hartwig asymptotics takes place. It was first investigated in [112, 93, 101], and the authors found that if T → Tc and n → ∞ in such a way that x ≡ (Tc −T )n is fixed, then the determinant is given in terms of Painlev´e III (reducible to Painlev´e V) functions. This transition corresponds to the emergence of one Fisher-Hartwig singularity with α = 0, β = −1/2 at z0 = 1. The condition that x ≡ (Tc − T )n is fixed was removed in [37] where uniform asymptotics were obtained for any α, /α > −1/2, β ∈ C in terms of Painlev´e V functions. Namely, consider the following symbol ft (z) = (z − et )α+β (z − e−t )α−β z −α+β e−iπ(α+β) eV (z) ,

α ± β = −1, −2, . . . (3.1) where t ≥ 0 is sufficiently small (in the above example of the Ising model, t = const(Tc − T )), V (z) is analytic in a neighborhood of C, and α, β ∈ C with /α > − 12 . The singularities of the symbol are at the points e±t . If t = 0 the symbol possesses a Fisher-Hartwig singularity at z = 0 and Theorem 2.2 applies to Dn (f0 ). If t > 0 then ft (z) is analytic in a neighborhood of C, and Szeg˝ o’s Theorem 2.1 applies. We have [37] Theorem 3.1. Let α, β ∈ C with /α > − 12 and let sδ denote a sector −π/2 + δ < arg x < π/2 − δ, 0 < δ < π/2. Let ft be given by (3.1) and consider the Toeplitz determinants Dn (ft ) defined by (1.1) corresponding to this symbol. There exists a finite set {x1 , . . . , xk } ∈ sδ (with k = k(α, β) and xj = xj (α, β) = 0) such that there holds the following expansion as n → ∞ with the error term uniform for 0 < t < t0 (with t0 sufficiently small) as long as 2nt remains bounded away from

Aspects of Toeplitz Determinants

313

the set {x1 , . . . , xk }: ln Dn (ft ) = nV0 + (α + β)nt +

   ∞  e−tk e−tk k Vk − (α + β) V−k − (α − β) k k k=1

G(1 + α + β)G(1 + α − β) + ln + Ω(2nt) + o(1), G(1 + 2α) where G(z) is Barnes’ G-function, and  2nt σ(x) − α2 + β 2 Ω(2nt) = dx + (α2 − β 2 ) ln 2nt. x 0

(3.2)

(3.3)

The function σ(x) is a particular solution to the Jimbo-Miwa-Okamoto σ-form [76, 77] of the Painlev´e V equation '2 &  2 2  2 dσ dσ d σ dσ x 2 = σ−x +2 + 2α dx dx dx dx  2   dσ dσ dσ −4 +α+β +α−β . (3.4) dx dx dx This solution has the following asymptotics for x > 0:  α2 −β 2 2 2 1+2α  C(α, β)}(1 + O(x)), α − β + 2α {x − x 2 2 1+2α σ(x) = α − β + O(x) + O(x ) + O(x1+2α ln x),      −1+2α −x −1 x e Γ(α−β)Γ(α+β) 1 + O x1 ,

x → 0, 2α ∈ /Z x → 0, 2α ∈ Z x → +∞, (3.5)

with

Γ(1 + α + β)Γ(1 + α − β) Γ(1 − 2α) 1 , (3.6) Γ(1 − α + β)Γ(1 − α − β) Γ(1 + 2α)2 1 + 2α where Γ(z) is Euler’s Γ-function. The path of integration in (3.3) is such as to avoid the set {x1 , . . . , xk } and is contained within the sector sδ . C(α, β) =

Note that (3.4) is the σ-form of the Painlev´e V equation   1 1 1 (u − 1)2 B Cu u(u + 1) 2 uxx = + u x − ux + Au + + +D , (3.7) 2u u − 1 x x2 u x u−1 with the parameters A, B, C, D given by A=

1 (α − β)2 , 2

1 B = − (α + β)2 , 2

C = 1 + 2β,

1 D=− . 2

(3.8)

The points xj refer to possible poles of σ(x). In the case when α is real and β is purely imaginary, one can show [37] that σ(x) is real analytic for x > 0, and the path of integration can therefore be chosen along the real axis. If one takes the limit as t → 0 on the r.h.s. of (3.2), one obtains the correct form of the appropriate Fisher-Hartwig asymptotics as given by Theorem 2.2.

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Since the asymptotics of Dn (f ) are known both at t = 0 and t = t0 , one obtains an amusing identity for the Painlev´e function σ(x): Ω(+∞) = − ln

G(1 + α + β)G(1 + α − β) . G(1 + 2α)

(3.9)

Methods used in [37] to prove Theorem 3.1 can be adapted to describe other transition regimes, e.g., two singularities approaching each other along the unit circle, or emergence of an arc on which f (z) = 0. These situations arise for other correlation functions in integrable models [58] and appear in the application of random matrix theory to the theory of L-functions [35].

4. Asymptotics for Fredholm determinants We now consider another type of double-scaling limit for Toeplitz determinants which yields interesting Fredholm determinants and, after certain analysis, allows us to obtain asymptotics of the latter. Note that this approach combined with Riemann-Hilbert-Problem techniques allows us to obtain the full asymptotics of these Fredholm determinants including the multiplicative constants which resisted other methods: see [84] for a short review of the approach and [85, 41, 42, 9, 46] for details. For a review of other applications of Riemann-Hilbert problems to Toeplitz and Fredholm determinants see [48]. For analysis of some other Fredholm determinants, see [111, 83, 38] and references in the introduction. Let f (eiθ ; n) = 1 on the arc 2s/n ≤ θ ≤ 2π −2s/n, 0 < s < n, and f (z; n) = 0 on the rest of the unit circle. Then the Fourier coefficients are f0 = 1 − 2s/(nπ), fj = − sin(2sj/n) , j = 0. In the limit of growing n and, accordingly, a closing arc, πj (s)

lim Dn (f (z; n)) = det(I − Ksine ),

n→∞

(4.1)

(s)

where Ksine is the trace-class operator on L2 (−s, s) with kernel Ksine (x, y) =

sin(x − y) . π(x − y)

(4.2)

In the Gaussian Unitary Ensemble of random matrix theory (and many other random matrix ensembles [45, 56]), the Fredholm sine-kernel determinant det(I − (s) Ksine ) describes, in the bulk scaling limit, the probability that an interval of length (s) 2s contains no eigenvalues. Of interest are the asymptotics of det(I − Ksine ) when s is large. (s)

Theorem 4.1. Let Ksine be the operator acting on L2 (−s, s), s > 0, with kernel (4.2). Then as s → +∞,  2   1 s  (s) det(I − Ksine ) = csine s− 4 exp − 1 + O(s−1 ) , csine = 21/12 e3ζ (−1) , 2 (4.3) and ζ
(x) is the derivative of Riemann’s zeta function.

Aspects of Toeplitz Determinants

315

(s)

We note that s(d/ds) ln det(I − Ksine ) satisfies [75, 40] a form of the Painlev´e V equation. In particular, this fact enables one to reconstruct the full asymptotic series of the logarithmic derivative in the inverse powers of s from the first few terms provided the existence of such asymptotic expansion is established. However, the multiplicative constant csine is not determined this way. Theorem 4.1 was conjectured by Dyson [51] who used, in particular, (4.1) and an earlier result of Widom on Toeplitz determinants with a symbol which vanishes on a fixed arc of the unit circle [107]. The leading asymptotic term was proved by Widom [106], and the lower-order terms apart from csine , in other words the (s) expansion for the derivative (d/ds) ln det(I − Ksine ), by Deift, Its, and Zhou [40] using Riemann-Hilbert methods. Application of a more detailed Riemann-Hilbert analysis to Toeplitz determinants allowed the authors in [85, 41] to extend the result of Widom [107] to varying arcs, and the relation (4.1) then produced the (s) asymptotics of det(I −Ksine ) including csine , which completed the proof of Theorem 4.1. An alternative proof of the theorem was given independently by Ehrhardt [52] who used (different) methods of operator theory. Recall that f (z; n) was defined above on the arc whose end-points converge to z = 1 as n → ∞. We now modify the definition of f (z; n) by placing a Fisher-Hartwig singularity at z = 1. Namely, consider the symbol F (z; n) = |z − 1|2α z β e−iπβ , z = eiθ , on the arc 2s/n ≤ θ ≤ 2π − 2s/n, 0 < s < n, and F (z; n) = 0 on the rest of the unit circle. We then obtain [46] lim

n→∞ (α,β,s)

where Kch

(α,β,s)

Kch

Dn (F (z; n)) (α,β,s) = det(I − Kch ), Dn (F (z; ∞))

(4.4)

is the trace-class operator on L2 (−s, s) with kernel

(u, v) =

1 Γ(1 + α + β)Γ(1 + α − β) A(u)B(v) − A(v)B(u) , 2πi Γ(1 + 2α)2 u−v

(4.5)

where A(x) = gβ (x)|2x|α e−ix φ(1 + α + β, 1 + 2α, 2ix), 1/2

1/2

 gβ (x) =

B(x) = gβ (x)|2x|α eix φ(1 + α − β, 1 + 2α, −2ix), e−πiβ , x > 0, , eπiβ , x < 0.

α, β ∈ C,

/α > −1/2,

α ± β = −1, −2, . . .

Here φ(a, c, z) is the confluent hypergeometric function (see, e.g., [3]) φ(a, c, z) = 1 + (α,β,s)

∞  a(a + 1) · · · (a + n − 1) z n . c(c + 1) · · · (c + n − 1) n! n=1

(4.6)

The kernel Kch appears in the representation theory of the infinite-dimensional (α,β,s) unitary group [29, 26], and the logarithmic derivative (d/ds) ln det(I − Kch ) is related to a solution of the Painlev´e V equation. If we set α = β = 0, the kernel reduces to the sine-kernel (4.2).

316

I. Krasovsky (α,β,s)

Theorem 4.2. Let Kch (4.5). Then as s → +∞, det(I −

(α,β,s) Kch )

=

be the operator acting on L2 (−s, s), s > 0, with kernel √

πG2 (1/2)G(1 + 2α) 2 G(1 + α + β)G(1 + α − β)  2   s − 14 −α2 +β 2 ×s exp − + 2αs 1 + O(s−1 ) , 2 2α2

(4.7)

where G(x) is Barnes’ G-function. This theorem was proved in [46] using the relation (4.4) and a RiemannHilbert analysis. The theorem √ reduces to Theorem 4.1 if α = β = 0 (recall that 2 ln G(1/2) = (1/12) ln 2 − ln π + 3ζ
(−1)). A particular case of the determinant Dn (F (z; n)) with β = 0 is related, via a Hankel determinant and the formula (2.16), to the following Bessel-kernel (a,s) determinant det(I − KBessel ) on (0, s), where the kernel √ √ √ √ √ √ yJa ( x)Ja
( y) − xJa ( y)Ja
( x) (a,s) , (4.8) KBessel (x, y) = 2(x − y) and Ja (x) is Bessel function. In the Jacobi Unitary Ensemble of random matrix theory (and many other ensembles with a so-called hard edge), the Bessel-kernel (a,s) determinant det(I − KBessel ) describes, in the (left) edge scaling limit, the probability that the interval (0, s) contains no eigenvalues. In other words, it describes the distribution of the extreme (smallest) eigenvalue. (a,s)

Theorem 4.3. Let KBessel be the operator acting on L2 (0, s), s > 0, with kernel (4.8) where /a > −1. Then as s → +∞,  s √  2 (a,s) det(I − KBessel ) = cBessel (a)s−a /4 exp − + a s 1 + O(s−1/2 ) , 4 (4.9) G(1 + a) cBessel (a) = . a/2 (2π) These asymptotics were conjectured by Tracy and Widom [102] and proved in [46]. An alternative proof of the particular case |/a| < 1 is given in [54] by methods of operator theory. Finally, we consider the case of the so-called Airy kernel. Let w(x) = e−4xn be supported on J = [0, 1 + s(2n)−2/3 ] for a fixed s ∈ R, and let DnH (w) be the corresponding Hankel determinant. Then    s (s) H lim Dn 1 + = det I − K (4.10) Airy , n→∞ (2n)2/3 (s)

where KAiry is the trace-class operator on L2 (s, +∞) with kernel Ai (x)Ai
(y) − Ai (y)Ai
(x) . x−y Here Ai (x) is the Airy function (see, e.g., [3]). (s)

KAiry (x, y) =

(4.11)

Aspects of Toeplitz Determinants

317

In the Gaussian Unitary Ensemble of random matrix theory (and many other (s) ensembles with a so-called soft edge), the Airy-kernel determinant det(I − KAiry ) describes, in the (right) edge scaling limit, the probability that the interval (s, +∞) contains no eigenvalues. In other words, it describes the distribution of the extreme (largest) eigenvalue. (s)

Theorem 4.4. Let KAiry be the operator acting on L2 (s, +∞), s ∈ R, with kernel (4.11). Then as s → −∞,  |s|3  (s) −1/8 FT W (s) ≡ det(I − KAiry ) = cAiry |s| exp − 1 + O(|s|−3/2 ) , 12 (4.12) 1/24 ζ  (−1) cAiry = 2 e , The distribution FT W (s) is known as the Tracy-Widom distribution. Tracy and Widom showed that [103]   ∞ ) 2 FT W (s) = exp − (x − s)u (x)dx , (4.13) s

where u(x) is the Hastings-McLeod solution of the Painlev´e II equation u

(x) = xu(x) + 2u3 (x) ,

(4.14)

specified by the following asymptotic condition: u(x) ∼ Ai (x)

as x → +∞.

(4.15)

The asymptotics of the logarithmic derivative (d/ds) ln FT W (s) follow, up to a constant (which is in fact zero), from (4.15) and the known asymptotics of the Hastings-McLeod solution at −∞. The constant cAiry (as well as cBessel above) was conjectured by Tracy and Widom using numerical computations and an analogy with the Dyson formula (4.3). The full proof of Theorem 4.4 was given in [42] using (4.10). (s) The determinant det(I − KAiry ) also describes the distribution of the longest increasing subsequence of random permutations. Namely, let π = i1 i2 · · · iN be a permutation in the group SN of permutations of 1, 2, . . . , N . Then a subsequence ik1 , ik2 , . . . ikr , k1 < k2 < · · · < kr , of π is called an increasing subsequence of length r if ik1 < ik2 < · · · < ikr . Let N (π) denote the length of a longest increasing subsequence of π and let SN have the uniform probability distribution. Then N (π) is a random variable, and √ FT W (s) = lim Prob {π ∈ SN : (lN (π) − 2 N )N −1/6 ≤ s} (4.16) N →∞

This result was obtained by Baik, Deift, and Johansson [6] from the double-scaling √ limit N → ∞, n ≤ N ∼ λ, of the Toeplitz determinant ∆n,λ = Dn (exp{ λ(z + z −1 )}). As shown earlier by Gessel [63], this determinant is precisely the following

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I. Krasovsky

generating function: ∆n,λ =

∞  N =0

un (N )

λN , N !2

(4.17)

un (N ) = #(permutations π in SN with N (π) ≤ n). An alternative proof of Theorem 4.4 based on this determinant was given in [9]. By the Robinson-Schensted-Knuth correspondence (see, e.g., [2]) a permutation π is related to a pair of Young tableaux (of the same shape) of integer plane partitions of N . The number lN (π) is the length of the first row of the related tableaux. We also note that random permutations are related to last passage percolation and random vicious walks [60, 4, 5]. There are many related results and extensions of the above results on random partitions and permutations, which, in particular, involve asymptotic analysis of special Toeplitz, Hankel, and Toeplitz+Hankel determinants. This large and growing research area has many connections to geometry, group representation theory, and integrable models. For details and a selection of results, see [6, 78, 79, 28, 95, 7, 8, 73, 57] and references therein. Acknowledgement I thank Florian Sobieczky for inviting me to the Alp-workshop 2009. I am also grateful to Percy Deift for useful comments.

References [1] M. Adler, P. van Moerbeke, P. Vanhaecke. Moment matrices and multi-component KP, with applications to random matrix theory. Comm. Math. Phys. 286 (2009), no. 1, 1–38. [2] G.E. Andrews. The theory of partitions. Cambridge University Press 1984. [3] Handbook of Mathematical Functions (ed. M. Abramowitz, I. Stegun), NY: Dover, 1965. [4] J. Baik. Random vicious walks and random matrices. Comm. Pure Appl. Math. 53 (2000), 1385–1410. [5] J. Baik. Riemann-Hilbert problems for last passage percolation. Contemp. Math. 326 (2003) 1–21. [6] J. Baik, P. Deift, K. Johansson, On the distribution of the length of the longest increasing subsequence of random permutations. J. Amer. Math. Soc. 12 (1999), 1119–1178. [7] J. Baik and E.M. Rains. Algebraic aspects of increasing subsequences. Duke Math. J. 109 (2001), 1–65. [8] J. Baik and E. Rains. The asymptotics of monotone subsequences of involutions. Duke Math. J. 109 (2001), 205–281.

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