In and Out of Equilibrium 2 (Progress in Probability) (No. 2)

Progress in Probability Volume 60

Series Editors Charles Newman Sidney I. Resnick

In and Out of Equilibrium 2 Vladas Sidoravicius Maria Eulália Vares Editors

Birkhäuser Basel · Boston · Berlin

Editors: Vladas Sidoravicius Instituto de Matemática Pura e Aplicada (IMPA) Estrada Dona Castorina, 110 Jardim Botanico CEP 22460-320 Rio de Janeiro, RJ Brasil e-mail: [email protected]

Maria Eulália Vares Centro Brasileiro de Pesquisas Físicas Rua Dr. Xavier Sigaud, 150 Urca 22290-180 Rio de Janeiro, RJ Brasil e-mail: [email protected]

2000 Mathematics Subject Classification: 03D10, 14Q99, 20K01, 32G15,37A50, 82BXX, 82CXX, 86A10, 90B22, 96A17, 60FXX, 60GXX, 60H99, 60JXX, 60KXX, 62GXX, 62MXX, 62P12

Library of Congress Control Number: 2008927086 Bibliographic information published by Die Deutsche Bibliothek. Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

ISBN 978-3-7643-8785-3 Birkhäuser Verlag AG, Basel · Boston · Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use whatsoever, permission from the copyright owner must be obtained. © 2008 Birkhäuser Verlag AG Basel · Boston · Berlin P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Cover picture: Courtesy Tati Howell Printed on acid-free paper produced from chlorine-free pulp. TCF ∞ Printed in Germany ISBN 978-3-7643-8785-3

e-ISBN 978-3-7643-8786-0

987654321

www.birkhauser.ch

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

ix

M. Abadi Poisson Approximations via Chen-Stein for Non-Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

E.D. Andjel and M. Sued An Inequality for Oriented 2-D Percolation . . . . . . . . . . . . . . . . . . . . . . . . . .

21

V. Beffara Is Critical 2D Percolation Universal? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31

G. Ben Arous, V. Gayrard and A. Kuptsov A New REM Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

59

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky The Biham-Middleton-Levine Traffic Model for a Single Junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

97

J. van den Berg, R. Brouwer and B. V´ Vagvolgyi ¨ Box-Crossings and Continuity Results for Self-Destructive Percolation in the Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 N. Berger and O. Zeitouni A Quenched Invariance Principle for Certain Ballistic Random Walks in i.i.d. Environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 J. Bertoin Homogenenous Multitype Fragmentations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 T.P. Cardoso and P. Guttorp A Hierarchical Bayes Model ffor Combining Precipitation Measurements from Different Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 A. Dembo and A.-S. Sznitman A Lower Bound on the Disconnection Time of a Discrete Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

vi

Contents

P. Dupuis, K. Leder and H. Wang On the Large Deviations Properties of the Weighted-Serve-the-Longest-Queue Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 A. Galves and F. Leonardi Exponential Inequalities for Empirical Unbounded Context Trees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

257

N.L. Garcia and T.G. Kurtz Spatial Point Processes and the Projection Method . . . . . . . . . . . . . . . . . .

271

G.L. Gilardoni An Improvement on Vajda’s Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

299

G.R. Grimmett Space-Time Percolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 O. H¨ a ¨ggstr¨m Computability of Percolation Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 A.E. Holroyd, L. Levine, K. M´sz´ ´ aros, ´ Y. Peres, J. Propp and D.B. Wilson Chip-Firing and Rotor-Routing on Directed Graphs . . . . . . . . . . . . . . . . . . 331 D. Ioffe and S. Shlosman Ising Model Fog Drip: The First Two Droplets . . . . . . . . . . . . . . . . . . . . . . . 365 I. Kaj and M.S. Taqqu Convergence to Fractional Brownian Motion and to the Telecom Process: the Integral Representation Approach . . . . . . . . . . . . . . 383 H. Kesten and V. Sidoravicius Positive Recurrence of a One-dimensional Variant of Diffusion Limited Aggregation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 429 A. Le Ny Gibbsian Description of Mean-Field Models . . . . . . . . . . . . . . . . . . . . . . . . . . 463 V. Limic and P. Tarr`es What is the Difference Between a Square and a Triangle? . . . . . . . . . . . . 481 S.R.C. Lopes Long-Range Dependence in Mean and Volatility: Models, Estimation and Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

497

J. Machta, C.M. Newman and D.L. Stein Percolation in the Sherrington-Kirkpatrick Spin Glass . . . . . . . . . . . . . . . . 527

Contents

vii

J. Quastel and B. Valk´ A Note on the Diffusivity of Finite-Range Asymmetric Exclusion Processes on Z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 R.B. Schinazi On the Role of Spatial Aggregation in the Extinction of a Species . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 551 M. Wschebor Systems of Random Equations. A Review of Some Recent Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

559

Preface This volume celebrates the tenth edition of the Brazilian School of Probability (EBP), held at IMPA, Rio de Janeiro, from July 30 to August 4, 2006, jointly with the 69th Annual Meeting of the Institute of Mathematical Statistics. It was indeed an exceptional occasion for the local community working in this field. The EBP, first envisioned and organized in 1997, has since developed into an annual meeting with two or three advanced mini-courses and a high level conference. This volume grew up from invited or contributed articles by researchers that during the last ten years have been participating in the Brazilian School of Probability. As a consequence, its content partially reflects the topics that have predominated in the activities during the various editions of the School, with a strong appeal that comes from statistical mechanics and areas of concentration that include interacting particle systems, percolation, random media and disordered systems. All articles of this volume were peer-refereed. The scientific committee of the 10th EBP comprised D. Brillinger, C. Newman, Y. Peres, C.-E. Pfister, I.M. Skovgaard and V. Yohai. The organizing committee consisted of F. Cribari, C.C. Dorea, L.R. Fontes, M. Fragoso, A.C. Frery, S.R. Lopes, and ourselves. We are indebted to all of them, for their help in shaping and organizing the school. We are also grateful to the colleagues responsible for the IMS meeting. Having such a large meeting jointly with the school required a significant effort and patience for structuring a reasonable schedule. We acknowledge the support constantly given by IMPA and also by “Instituto do Milenio” (IMAGIMB) to the school, and the financial support given by the sponsors: Capes, CNPq, Faperj, Fapesp, Finep, ICTP, NSF, Prosul and RDSES. We are also grateful to the staff of IMPA, for their constant help before and during the joint event. Rio de Janeiro, February 2008

Vladas Sidoravicius Insitituto de Matem´ ´ atica Pura e Aplicada Maria Eul´ a´lia Vares Centro Brasileiro de Pesquisas F´sicas ´

Progress in Probability, Vol. 60, 1–19 c 2008 Birkhauser  ¨ Verlag Basel/Switzerland

Poisson Approximations via Chen-Stein for Non-Markov Processes Miguel Abadi Abstract. Consider a stationary stochastic process X1 , . . . , Xt . We study the distribution law of the number of occurrences of a string of symbols in this process. We consider three different ways to compute it. 1) When the string is the first block generated by the process, called Nt . 2) When the string is the first block generated by an independent copy of the process, called Mt . 3) When the string a is previously fixed, called Nt (a). We show how this  laws can be approximated by a Poisson law with an explicit error for each approximation. We then derive approximations for the distribution of the first occurrence time of the string, according also to the above three different ways to chose it. The base of the proofs is the Chen-Stein method which is commonly used to prove approximations in total variation distance in the Markov context. We show how to apply it in non-Markovian mixing processes. Mathematics Subject Classification (2000). 60F05, 60G10, 60G55, 37A50. Keywords. Number of occurrences, occurrence time, Poisson approximation, exponential approximation, Chen-Stein method, mixing processes.

1. Introduction A very well-known result in probability theory is the convergence of the binomial distribution with parameters n, p to the Poisson distribution with parameter λ as np → λ. That is, if Xn are independent identically distributed random n variables (r.v.) with P(Xn = 1) = p = 1 − P(Xn = 0) for n ≥ 1 and if Nn = i=1 1{Xn =1} then e−λ λk lim P (N . Nn = k) = np→λ k! In the lasts decades there was a lot of work generalizing this limit in several senses, motivated for applications: MA is partially supported by CNPq, grant 308250/2006-0.

2

M. Abadi A. dependent processes, B. occurrence of fixed observables other than singletons, C. rate of convergence of the above limit.

Our motivation is to present results which extends previous ones in the above directions plus a new one. More specifically, with respect to B, we consider not a fixed observable but a random one, chosen according to the first outcomes of the process (Definition 5) or according to the first outcomes of an independent identically distributed process (Definition 4). More specifically, we prove that the number of occurrences of a string of n symbols, properly re-scaled, can be well approximated by a Poisson law. Actually, we present three results which differ in the way the string is chosen. In the first two ways the string is randomly chosen as follows: i. The string is the first block generated by the process. ii. The string is the first block generated by an independent copy the process. These results are based on the third case. iii. The string a is previously fixed.  Yet, with respect to A, our result holds for two kind of dependent processes. We consider firstly ψ-mixing processes (Definition 1). Secondly, we consider βmixing processes (Definition 2). The result in the second case has a larger error but it is obtained under a much weaker hypothesis. ψ-mixing process includes for instance, irreducible and aperiodic finite state Markov chains (with exponential function ψ.), also Gibbs states with a potential with exponential variations are exponentially ψ-mixing. See Bowen (1975) for definitions and properties. Also, chains of infinite order are exponentially ψmixing (see Fern´´andez et al. (2001) for definition). We refer to Doukhan (1995) and Bradley (2005) for examples and properties of the many definitions of mixing processes, among them ψ-mixing and β-mixing (which are not ψ-mixing) decaying at any rate. With respect to C, we use the Chen-Stein method to get an upper bound of the total variation distance between the true law of counting occurrences and a Poisson law. This method is a powerful tool for proving convergence to Poisson distribution. It was firstly introduced by Stein (1972) for proving convergence of a sum of dependent random variables in a generalization of the Central Limit Theorem. Chen (1975) adapted the ideas of Stein to study the question of convergence in law to a Poisson law for dependent r.v. As far as we know, the Chen-Stein method is the first one who provided error terms for the rate of convergence to the Poisson law. Some advantages of using it include: • it applies to a very general kind of process, • not even stationarity is required, • it is very easy to use,

Poisson via Chen-Stein

3

• it provides sharp error terms, • small modifications of the proof adapt to other cases. The main difficulty in applying the Chen-Stein method is to compute two quantities. They are usually called b2 , which measure short correlations of the observable and b3 which measures large ones. The goal is then, to find conditions (on the process and/or the observable) in which the bounds are small. There are a number of works which apply the Chen-Stein method for Markov process where these quantities can be well controlled. These references can be found in Alvarez (2005). Arratia, Goldstein and Gordon (1989) present a clear introduction to the Chen-Stein method. In a further paper, the same authors (1990) apply this method to a wide range of examples in which b3 is zero, that is the processes have finite memory. In this paper we show how to apply it for non-Markovian processes with infinite memory. We apply the method firstly to ψ-mixing processes. Then, even when we get larger bounds, we also show how to apply it under a much weaker condition, the so-called weak Bernoulli or β-mixing condition. The Chen-Stein method allows to present an important enhance with respect to previous works. It is the velocity of convergence (with respect to the total variation distance). In our results, this error is explicitly expressed in terms of the measure of the observable and the rate of mixing of the process. We obtain as immediate corollaries that the first occurrence time of a string chosen also as in: i. denoted by Rn ; ii. denoted by Wn and iii. denoted by τa (when properly re-scaled) can be well approximated by an exponential law. For τarelated results in different mixing contexts can be found in Galves and Schmitt (1997), Hirata et al. (2001), Abadi (2004). A survey can be found in Abadi and Galves (2001). For Wn and Rn there is only one result we are aware (Wyner (1999)) proved actually for two-folded ψ-mixing processes with a larger error bound. Rn and Wn where also studied in other context by Ornstein and Weiss (2002) and Shields (1996). They proved respectively that 1 1 lim − log Rn = h and lim − log Wn = h , n→∞ n→∞ n n where h is the entropy of the process. Both limits are in almost-sure sense. A significant difference between them is that the first case holds for any ergodic source while the second one holds for weak Bernoulli processes but fails for more general processes (for instance, the so-called very weak Bernoulli processes.) This results are now also consequence of our corollaries by a direct application of the Borel-Cantelli lemma. The paper is organized as follows. In Section 2 we establish our framework. In Section 3 we briefly describe the Chen-Stein method following Arratia, Goldstein and Gordon (1989). In Section 4 we state our results. In Section 5 we give the proofs.

4

M. Abadi

2. The framework Let C be a finite set. Put Ω = C N . For each ω = (ωm )m∈N ∈ Ω and m ∈ N, let Xm : Ω → C be the mth coordinate projection, that is Xm (ω) = ωm . We denote by T : Ω → Ω the one-step-left f shift operator. For a finite sequence x = (x0 , . . . , xn−1 ) ∈ C n we say that the set  {X0 = x0 , . . . , Xn−1 = xn−1 } , is a string. We denote by F the σ-algebra over Ω generated by strings. Moreover we denote by FI the σ-algebra generated by strings with coordinates in I, I ⊆ N. We consider a stationary probability measure P over F . We shall assume that there are no singletons of probability 0. Let j be an integer. Due to stationarity and in order to simplify notation we will write P(x) instead of P(X Xjj+n−1 = x). We will only write P(X Xjj+n−1 = x)    when it is needed to make the text clearer. Definition 1. Consider the sequence ψ = (ψ(l))l≥0 defined by    P(C | B)  − 1 . ψ(l) = sup  P(C) B,C The supremum is taken over the sets B and C, such that B ∈ F{0,.,n} , C ∈ F{i|i≥n+l} and P(B)P(C) > 0, n ∈ N. The process (Xm )m∈N is ψ-mixing if the sequence ψ = (ψ(l))l≥0 is decreasing and converges to zero. A weaker notion of dependence is the following. Definition 2. Consider the sequence β = (β(l))l≥0 defined by  1 β(l) = sup |P(B ∩ C) − P(B)P(C)| . 2 B,C B∈Q,C∈R

The supremum is taken over all the finite partitions Q of F{0,.,n} and R of F{i|i≥n+l} , n ∈ N. The process (Xm )m∈N is β-mixing or weak Bernoulli if the sequence β = (β(l))l≥0 is decreasing and converges to zero. We shall use the following notation. For any two measurables A and B, P (A ; B) = P (A ∩ B). The conditional measure of B given A is denoted as usual P(B | A) = P(B ; A)/P(A). We shall write {Xrs = xsr } = {Xr = xr , . . . , Xs = xs } . we write T −k (x) = {Xkk+n−1 = xn−1 }. For x = xn−1 0 0  As usual, the mean of a random variable X will be denoted by E[X]. Wherever it is not ambiguous we will write C and c for different positive constants even in the same sequence of equalities/inequalities.

Poisson via Chen-Stein

5

3. The Chen-Stein method Definition 3. Let X and Y to r.v. with distribution concentrated in the non-negative integers. We define the total variation distance between the distribution laws of X and Y as the following quantity: ||L(X) − L(Y )|| = sup |P(X ∈ K) − P(Y ∈ K)| . K∈N

Let I be a finite or countable set (of indexes). For each α ∈ I, letVα be a Bernoulli r.v. with 0 < P(V Vα = 1) = pα = 1 − P(V Vα = 0) < 1. Let W = α∈I Vα . Assume that 0 < λ = EW = α∈I pα < ∞. For each α ∈ I take a set Bα ⊆ I with α ∈ Bα . Let Z be a Poisson r.v. with EZ = λ. Define b1 =

 

pα pβ ,

α∈I β∈Bα

b2 =





E [V Vα Vβ ] ,

α∈I α=  β∈Bα

and b3 =



   E E Xα − pα | FBαc  .

α∈I

Denote with . the total variation distance. The following theorem holds.  Theorem 1 (Arratia, Goldstein and Gordon, 1989). Let W = α∈I Vα be the number of occurrences of dependent events. Let Z be a r.v. with Poisson distribution. Suppose that EZ = EW = λ. then  ||L(W ) − L(Z)|| ≤ 2 1 ∧ 1.4λ−1/2 (b1 + b2 + b3 ) . Terms b1 and b2 refer to the size of the “neighborhood” Bα around α (for each α ∈ I) in which the process has strong dependence. The term b3 refers to the dependence of the process outside the neighborhoods. The best bound is obtained by a trade off between the size of the neighborhoods and the loss of memory of the process. Yet, the term b2 also refers to short correlations of the event, that is, inside the neighborhoods. This term can be controlled only if the event does not recur very fast.

4. Results 4.1. Poisson statistics In this section we present the (approximated) Poisson statistics for the counting Mt , Nt and Nt (a) described above. The definitions and results follow. In what follows Z denotes a r.v. with Poisson distribution of mean 1.

6

M. Abadi

Due to the bound of the Chen-Stein method we introduce the following notation used in our theorems.  √ f (t) = t 1 ∧ 1.4t−1/2 = t ∧ 1.4 t . Definition 4. Let (Xm )m∈N and (Z Zm )m∈N be two independent, identically distributed processes. For t > 0 integer define t 

Mt =

1{X i+n−1 =Z n−1 } . i

0

i=1

So, Mt counts the number of occurrences of the random sequence Z0n−1 , in the independent process (Xm )m∈N , up to time t. We have the following theorem. Theorem 2 (First independent random string). Let (Xm )m∈N be a stationary process. Then     Mt/P(Z n−1 ) ) − L(Z) ≤ Cf (t)ε(n) , (1) L(M 0

where • if the process is ψ-mixing



ε(n) := inf

Δ≥n

• if the process is β-mixing



ε(n) := inf

Δ≥n

Δ

inf

 Δ maxn P(a) + ψ(Δ) a∈C  

 max

0≤g≤n−1

,

an−1 ∈C n−g g

{P(an−1 )} + β(g) g

 + β(Δ)

.

Definition 5. Let (Xm )m∈N be a process. For t > 0 integer define Nt =

t 

1{X i+n−1 =X n−1 } . i

0

i=1

So, Nt counts the number of occurrences of the initial random sequence X0n−1 , up to time t. Theorem 3 (First random string). Let be a stationary ψ-mixing or β-mixing process. Then     Nt/P(X n−1 ) ) − L(Z) ≤ Cf (t)ε(n) . L(N 0

ε(n) is defined in Theorem 2, according to the case. The main results of this paper are Theorem 2 and Theorem 3. The basic tool for their proofs is the next theorem which refers to the Poisson statistics of a fixed n-string.

Poisson via Chen-Stein

7

Definition 6. Let a ∈ C n . For t > 0 integer let  t  1{X i+n−1 =a} . Nt (a) = i   i=1 So, Nt (a) counts the number of occurrences of the fixed string a up to time t.   To compute the error for the Poisson statistics of Nt (a) we need the following  definition. Definition 7. Let a ∈ C n . We define the first overlapping position or periodicity of  ) defined as follows: a as the number O(a  

O(a) = min k ∈ {1, . . . , n} | an−1−k . = an−1 0 k  n−1 (We consider a−1 by definition.) 0 = an Remark 1. For future purpose it is useful to notice the following. Suppose n can be written as n = O(a)[n/O(a)] + r, then we have two cases: If O(a) ≤ n/2 then    O(a ( )−1

a0 

2O(a ( )−1

= aO(a (  )  If O(a) > n/2 then 

[n/O(a ( )]O(a ( )−1

= · · · = a([n/O(a ( )]−1)O(a  ( )  

and

r−1 an−1 . [n/O(a ( )]O(a ( ) = a0  

n−O(a ( )−1

= an−1 O(a ( ) .  We now can state the Poisson approximation theorem. a0



Theorem 4 (Fixed string). Consider the stationary process (Xm )m∈N with a ∈ C n .  Then     Nt/P(a L(N ( ) ) − L(Z) ≤ Cf (t)ε(a) ,   where • if the process is ψ-mixing 

, ε(a) := inf ΔP(a) + ψ(Δ) + nP an−1 n−O(a ( ) Δ≥n    • if the process is β-mixing   β(g) + P(an−1 ε(a) := n inf ) . n−O(a ( )+g 0≤g≤n−1 O(a)P(a)     Remark 2. Consider ε(a) in the ψ-mixing case. The factor in brackets consists of  not possible to say which term is smaller. For instance, two terms. A priori it is assume the system loses memory exponentially fast with a large positive constant, and assume O(a) is small. Then the error is basically nP(an−1 n−O(a ( ) ). On the other  hand, assume the system loses memory polynomially fast. Assumealso that O(a) =  n. Then the error is given basically by inf Δ≥n {ΔP(a) + ψ(Δ)}. 

8

M. Abadi

Remark 3. P(an−1 n−O(a ( ) ) is small when O(a) is large. (Recall the meaning of O(a) in   Definition 7.) Colletet al. (1999) proved that for exponentially ψ-mixing processes there exist positive constants C and c such that P(a ∈ C n ; O(a) ≤ n/3) ≤ C exp−cn .   Abadi (2001) extended the above inequality to φ-mixing processes when 1/3 is replaced by some constant s ∈ (0, 1). Abadi and Vaienti (2008) proved the above inequality for ψ-mixing processes for any value of s (with c = c(s).) This shows that Theorem 4 holds for typical (in the sense of O(a)) strings. Taking limit on  that the Poisson limit law the length of the strings along infinite sequences, we get holds almost everywhere. 4.2. Corollaries: occurrence times In this section we derive a number of corollaries about the distribution of the first occurrence of a string. They are direct consequences of the Poisson approximation. Consider the zero-level sets of Mt , Nt and Nt (a). It corresponds to the law of the first visit to the observable considered. Denotethem Wn , Rn and τa respec tively. The zero-level set of the Poisson law is the exponential law. Definition 8. Given n ∈ N, we define the waiting time Wn : Ω → N ∪ {∞} as the following r.v. defined on the product space (Ω, F, P) × (Ω, F, P). Wn = inf{i ≥ 1 : Xii+n−1 = Z0n−1 } . So, Wn is the position of the first time the string z0n−1 of the process (Z Zm )m∈N appears in (Xm )m∈N . We recall that (Xm )m∈N and (Z Zm )m∈N are independent and identically distributed processes. Corollary 1 (Waiting time). Under the conditions of Theorem 2, the following inequality holds     P Wn P(Z n−1 ) > t − e−t  ≤ Cf (t)ε(n) . 0

Definition 9. Given n ∈ N, we define the repetition time Rn : Ω → N ∪ {∞} as Rn = inf{i ≥ 1 : Xii+n−1 = X0n−1 } . So, Rn is the position of the first repetition of the initial string xn−1 . 0 Corollary 2 (Recurrence time). Under the conditions of Theorem 3, the following inequality holds     P Rn P(X n−1 ) > t − e−t  ≤ Cf (t)ε(n) . 0 Definition 10. Given a ∈ C n , we define the hitting time τa : Ω → N ∪ {∞} as   τa = inf{k ≥ 1 : Xii+n−1 = a} .   Corollary 3 (Hitting time). Under the conditions of Theorem 4, the following inequality holds      P τa P(a) > t − e−t  ≤ Cf (t)ε(a) .   

Poisson via Chen-Stein

9

5. Proofs 5.1. Short correlations In order to control error term b2 in the Chen-Stein method, we need to compute certain short correlations which are controlled by Lemma 1 and Lemma 2 below. Before that we make the following remark. Remark 4. In the proofs we will use the following inequalities which are direct consequences of Definition 1 and Definition 2 respectively. Take any A ∈ F{0,...,n} and B ∈ F{i|i≥m} . For a ψ-mixing process P(A; B) ≤ CP(A)P(B) , with C = 1 + ψ([n − m]+ ) (and [b]+ = max{b, 0}.) For a β-mixing process P(A; B) ≤ β([n − m]+ ) + P(A)P(B) . Lemma 1 (Short correlation of a string). Let (Xm )m∈N be a stationary process. Let a ∈ C n . Let Δ be any positive integer. Then the following inequality holds  Δ+n−1    E 1{X n−1 =a} 1{X j+n−1 =a} ≤ Cδ(a) . 0 j    j=1 where • if the process is ψ-mixing

  δ(a) := P(a) ΔP(a) + nP(an−1 ) , n−O(a ( )     • if the process is β-mixing   β(g) n−1 δ(a) := n inf + P(an−O(a . ( )+g )P(a) 0≤g≤O(a ( )−1 O(a)      Remark 5. Both terms that defines δ(a) in the ψ-mixing case control short correlations. The second one dominates the very short returns which are ruled by the overlap properties of a. The first one dominates the short, but not so much returns. That is short returnsbut without the influence of the overlap properties of a. For typical strings, O(a) = n (see Remark 3) and δ(a) reduces to the first term. In the  the very short returns. β-mixing case, thelargest contribution comes from Proof of Lemma 1. The ψ-mixing case. Assume first Δ > n. We decompose the above sum in terms with j = 1, . . . , O(a) − 1; j = O(a), . . . , n − 1; and j =   n, . . . , Δ + n − 1. One has for any j    E 1{X n−1 =a} 1{X j+n−1 =a} = P X0n−1 = a ; Xjj+n−1 = a . 0 j     Thus, by the definition of O(a) we have that  O(a ( )−1     E 1{X n−1 =a} 1{X j+n−1 =a} = 0 . 0 j   j=1

10

M. Abadi

For j = n, . . . , Δ + n − 1, by Remark 4  P X0n−1 = a ; Xjj+n−1 = a ≤ CP(a)2 .    Then Δ+n−1   P X0n−1 = a ; Xjj+n−1 = a ≤ CΔP(a)2 .    j=n

(2)

For j = O(a), . . . , n − 1, and also by Remark 4     n−1 P X0 = a ; Xjj+n−1 = a = P X0n−1 = a ; Xnj+n−1 = an−1 n−j      ≤ CP(a)P an−1 n−j   ≤ CP(a)P an−1 . n−O(a ( )   Thus

  . P X0n−1 = Xjj+n−1 = a ≤ Cn P(a)P an−1 n−O(a ( )    j=O(a ( )  If Δ ≤ n the same bound holds. This ends the proof of this case. n 

(3)

The β-mixing case. Let us fix an index j ∈ I. By definition of O(a) one has     j+n−1 j+i+n−1 Xj =a Xj +i =a =∅,   for all 1 ≤ i ≤ O(a) − 1. So  O(a ( )−1

  

i=0

 Xjj+i+n−1 =a , +i 

(4)

and its complement are a partition of F{j,...,j+O(a ( )−1+n−1} . Thus, we divide the n indexes of j = O(a), . . . , n − 1 + O(a), in blocks of length O(a). Then we re-write   the left-hand sideof (2) as  m=iO(a ( )

m+O(a ( )−1

 

j=m

 P X0n−1 = a ; Xjj+n−1 = a .  

(5)

  We consider only the first O(a) symbols of the first occurrence of a. The above   probability is bounded by   O(a ( )−1 O(a ( )−1 P X0  = a0  ; Xjj+n−1 = a .  i=1,...,[n/O(a ( )]+1

Poisson via Chen-Stein

11

Now we open a gap of length g in the first occurrence of a. The inner sum in (5)  is bounded by m+O(a ( )−1

 

j=m

  O(a ( )−1−g O(a ( )−1−g P X0  = a0  ; Xjj+n−1 = a . 

The second set in the above probability belongs to the partition (4). Therefore, by Remark 4 and stationarity, the above sum is bounded by O(a ( )−1−g

β(g) + P(X0 

O(a ( )−1−g

= a0 

)O(a)P(a) .   The cardinal of the first sum in (5) is [n/O(a)] + 1. Then the left-hand side of (2)  is bounded by   O(a ( )−1−g β(g) 2n + P(a0  )P(a) . O(a)   Just to unify notation we note the following. Since for positive integers n1 ≤ n2 one has by stationarity n2 −n    1 −j 0 P X0n−1 = a; Xjj+n−1 = a = P X−(j+n−1) = a; X−(n−1) =a ,     j=n1 j=−n2 we can repeat the above argument exchanging the roles of the first and second O(a ( )−1−g

occurrences of a. Then we can change P(a0   bounds. This ends the proof of the lemma.

) by P(an−1 n−O(a ( )+g ) in the above  

Lemma 2 (Short correlation of the process). Let (Xm )m∈N be a stationary process. Let Δ be any positive integer. Then the following inequality holds Δ    E 1{X n−1 =X j+n−1 } ≤ Cδ(n) , 0

j

j=1

where • if the process is ψ-mixing

• if the process is β-mixing δ(n) = Δ

δ(n) = Δ maxn P(a) , a∈C  

inf

1≤g≤n−1

{β(g) +

max

an−1 ∈C n−g g

P(an−1 )} . g

Proof of Lemma 2. The ψ-mixing case. Consider the equation X0n−1 = Xjj+n−1 .

(6)

We divide the proof in j = 1, . . . , n/2, j = n/2, . . . , n − 1 and j = n, . . . , Δ. Let us start with j = n, . . . , Δ. We condition on the first n-block. Thus      E 1{X n−1 =X j+n−1 } = E E 1{X n−1 =X j+n−1 } |X0n−1 . 0

j

0

j

12

M. Abadi

The last expression is 

P(X Xjj+n−1 = an−1 |X0n−1 = an−1 )P(an−1 ). 0 0 0

(7)

an−1 ∈C n 0

The leftmost probability is bounded by P(a) using the ψ-mixing property. Thus,  the previous sum is bounded by max {P(a)} . 

an−1 ∈C n 0

Suppose now that j = 1, . . . , n/2. Consider (6) and write n + j = qj + r, with q, j, r integers and 0 ≤ r < j. Then X0n+j−1 consists of repeating the first block of length j up to complete n + j symbols, namely qj−1 X0j−1 = Xj2j−1 = · · · = X(q−1)j

n+j−1 Xn+j−r = X0r−1 .

and

Thus, we condition on the first j-block      E 1{X n−1 =X j+n−1 } = E E 1{X n−1 =X j+n−1 } |X0j−1 . 0

0

j

(8)

j

The last expression is  qj−1 n+j−1 j−1 r−1 P(X Xj2j−1 = · · · = X(q−1)j = aj−1 |X0j−1 = aj−1 0 ; Xn+j−r = a0 0 )P(a0 ). aj−1 ∈C j 0

Again, the leftmost probability is bounded by qj−1 n+j−1 = aj−1 ; Xn+j−r = ar−1 ), CP(X Xj2j−1 = · · · = X(q−1)j 0 0

which is the probability of an n-string. Similarly to the previous case, when j = n/2, . . . , n − 1, equation (6) means that X0n+j−1 has its first, last and central n − 2j-blocks equals and the two remaining j-blocks equals too. Namely n+j−1 X0n−j−1 = Xjn−1 = X2n−j

and

j−1 Xn−j = Xn2n−j−1 .

Thus, we condition on the first j-block and (8) still holds but is this case it is  n+j−1 j−1 j−1 P(X0n−j−1 = Xjn−1 = X2n−j ; Xn−j = Xn2n−j−1 |X0j−1 = aj−1 ). 0 )P(a0 aj−1 ∈C j 0

Now, the leftmost probability is bounded using the ψ-mixing property by qj−1 n+j−1 = aj−1 ; Xn+j−r = ar−1 ), CP(X Xj2j−1 = · · · = X(q−1)j 0 0

which again is the probability of an n-string. This finishes this case. The β-mixing case. We can re-write (7) as  P(X0n−1 = an−1 ; Xjj+n−1 = an−1 ), 0 0 an−1 ∈C n 0

Poisson via Chen-Stein

13

which is bounded opening a gap at position j of length 1 ≤ g ≤ n − 1 by  P(X0n−1 = an−1 ; Xjj+n−1 = an−1 ). g 0 +g an−1 ∈C n 0

We bound the last sum using Remark 4 by  P(X0n−1 = an−1 )P(X Xjj+n−1 = an−1 ) β(g) + g 0 +g an−1 ∈C n 0

≤ β(g) +

max

an−1 ∈C n−g g

P(an−1 ). g

Now consider j = 1, . . . , n − 1. This case is completely similar to the previous one. Equation (8) is equal to  P(X0j−1 = aj−1 ; Xjj+n−1 = an−1 ), 0 0 aj−1 ∈C j 0

which is bounded opening a gap at position j of length g ≤ n − 1 by  P(X0j−1 = aj−1 ; Xjj+n−1 = an−1 ). g 0 +g aj−1 ∈C j 0

We bound the last sum using again Remark 4 by  P(X0j−1 = aj−1 Xjj+n−1 = an−1 ) β(g) + g 0 )P(X +g aj−1 ∈C j 0

≤

β(g) +

max

an−1 ∈C n−g g

This ends the proof of the lemma.

P(an−1 ). g 

5.2. Proofs for Poisson statistics We first proceed with the proof of Theorem 4 since it is the base of the proofs of Theorem 2 and Theorem 3. Proof of Theorem 4. The ψ-mixing case. It is enough to prove upper bounds for Therefore, the quantities b1 , b2 and b3 in the Chen-Stein method.

we define the quantities involved in it. Let the set of indexes I be 1, . . . , t/P(a) . For any j ∈ I  put Vj = 1{X j+n−1 =a} . j  Vj ] = P(a) for every j ∈ I. Since the measure is stationary we have pj = E[V Moreover, fix an integer Δ ≥ n. Take Bj = {j − (Δ + n), . . . , j + Δ + n} ∩ I. Hence t b1 ≤ (2Δ + 2n + 1) P(a)2 ≤ C t Δ P(a) . (9) P(a)   

14

M. Abadi

Further, also by stationarity t/P(a ( )

b2

=

 j=1

≤

  E 1{X i+n−1 =a} 1{X j+n−1 =a} i j   i∈Bj \{j} 

t max P(a) j∈I 

j+Δ+n  i=  j i=j−Δ−n

 P Xii+n−1 = a ; Xjj+n−1 = a  

Δ+n    t 2 P a ; Xii+n−1 = a . (10) P(a) i=1    The quantity above is bounded using Lemma 1 by    . C t ΔP(a) + nP an−1 n−O(a ( )   Finally    t   max E E 1{X j+n−1 =a} − P(a) | FBjc  . b3 ≤ j P(a) j∈I    For the sake of simplicity denote B = Bj . Put Y = {Xi }i∈/ B . The above expectation is       P(Y = y) E 1{X j+n−1 =a} − P(a)|Y = y  j   Y =y      = P(y) (1 − P(a))P(X Xjj+n−1 = a|y) − P(a)P(X Xjj+n−1 = a|y)     y     Xjj+n−1 = a ; y) − P(a)P(y) , = P(X   y

=

We can write {Y = y} = {Y Yl = yl } ∩ {Y Yr = yr } , where Yl = {Xt }t<−|B| and Yr = {Xt }t>|B| (respectively yl and yr ). We get the following triangular inequality     P(a ; y) − P(a)P(y) ≤ P(a ; y) − P(a ; yl )P(yr ) (11)       + P(a ; yl ) − P(a)P(yl ) P(yr )   + P(a) |P(yl )P(yr ) − P(y)| .  The three terms on the right-hand side of the above inequality can be bounded using the ψ-mixing property. For the first one we have   P(a ; y) − P(a ; yl )P(yr ) ≤ ψ(Δ)P(a ; yl )P(yr ) .    In the same way we bound the modulus in the second one   P(a ; yl ) − P(a)P(yl ) ≤ ψ(Δ)P(a)P(yl ) .   

Poisson via Chen-Stein

15

And finally the modulus in the third one is bounded by |P(yl )P(yr ) − P(y)| ≤ ψ(2Δ)P(yl )P(yr ) . The above three inequalities give that the left-hand side of (11) is bounded by ψ(Δ)P(a ; yl )P(yr ) + 2ψ(Δ)P(a)P(yl )P(yr ) .   Thus summing on Y = y   P(a ; y) − P(a)P(y) ≤ 3ψ(Δ)P(a) .    yl yr Summing over all j ∈ I we have b3 ≤ 3 t ψ(Δ) . This ends the proof of this case.

The β-mixing case. Take Bj = j − (n + O(a)), . . . , j + n + O(a) ∩ I. b1 is  ).  bounded identically to the ψ-mixing case by CtnP(a  Due to Lemma 1 we have   β(g) t n inf + P(an−1 )P(a ) . b2 ≤ n−O(a ( )+g P(a) 0≤g≤O(a ( )−1 O(a)      For b3 consider the sum on y of the three terms on the right-hand side of (11). Each one is bounded using the β-mixing condition by β(O(a)) and we get b3 ≤  the proof of the tβ(O(a))/P(a). Thus, we get that b1 ≤ b2 and b3 ≤ b2 . This ends   theorem.  Proof of Theorem 2. The ψ-mixing case. Let K ⊆ N. Conditioning Mt on the initial sequence Z0n−1 of (Z Zm )m∈N and since it has the same distribution as (Xm )m∈N  P(M Mt/P(Z n−1 ) ∈ K) = P(M Mt/P(Z n−1 ) ∈ K |Z Z0n−1 = a)P(X0n−1 = a) . 0 0   a∈C n  Zm )m∈N one has Since (Xm )m∈N is independent of (Z P(M Mt/P(Z n−1 ) ∈ K |Z0n−1 = a ) = P(N Nt/P(a ( ) (a) ∈ K) . 0    By Theorem 1, the left-hand side of (1) is bounded by     Nt/P(a P(N ( ) (a) ∈ K) − P(Z ∈ K) P(a)    a∈C n  (b1 (a) + b2 (a) + b3 (a))P(a) . ≤     a∈C n 

16

M. Abadi

Here we denote with bi (a), i = 1, 2, 3 the bi ’s of Theorem 1 applied to each a ∈ C n .  Using (10) we get that  Δ   t   2 b2 (a)P(a) ≤ P a ; Xii+n−1 = a P(a) P(a) i=1      a∈C n a∈C n    Δ  Δ       P a ; Xii+n−1 = a = 2t P X0n−1 = Xii+n−1 . = 2t   i=1 a∈C n i=1 



And we can bound the last term with Lemma 2. Further, by (9) we have that  a∈C n



b1 (a)P(a) ≤ Δ maxn P(a) . a∈C    

Finally, b3 (a) is independent of a. This ends the proof of this case.     The β-mixing case. a b1 (a)P(a) and a b2 (a)P(a) are bounded identically        to the previous case. To bound a b3 (a)P(a) we use (11) and thus           j+n−1 = a ; yr ) − P(a)P(yl ; yr ) . (12) b3 (a)P(a) ≤ t P(yl ; Xj     a yl a yr   Now consider the general term in the above sum. The triangle inequality gives     j+n−1 = a ; yr ) − P(a)P(yl ; yr ) P(yl ; Xj       ≤ P(yl ; Xjj+n−1 = a ; yr ) − P(yl ; Xjj+n−1 = a)P(yr )       + P(yl ; Xjj+n−1 = a) − P(a)P(yl ) P(yr )   + P(a) |P(yl )P(yr ) − P(yl ; yr )|     We apply the β-mixing property in the sum yl a yr of each of the three terms  first one over the partitions in the right-hand side of the above inequality. The j+n−1 = a} and {Y Yr = yr }. The second one over the partitions {Y Yl = yl } {Y Yl = yl , Xj  and {X Xjj+n−1 = a}. The third one over the partitions {Y Yl = yl } and {Y Yr = yr }.  Thus (12) is bounded by 3tβ(Δ). This ends the proof of the theorem.  Proof of Theorem 3. The ψ-mixing case. We partition {N Nt/P(X n−1 ) ∈ K} in even0

tually short and only long occurrences of X0n−1 . We first need the following nota-

Poisson via Chen-Stein

17

tion. For any n ≤ s ≤ t denote Mst =

t 

1{X i+n−1 =Z n−1 } , i

0

Nst =

i=s t 

t 

1{X i+n−1 =X n−1 } , i

0

i=s

1{X i+n−1 =a} . i  Then we have the following decomposition Nst (a) = 

i=s

t t P(N Nt ∈ K) = P(N NΔ = Nt ∈ K) + P(N NΔ < Nt ∈ K) . t Consider firstly the rightmost probability. Since NΔ < Nt it means that there is some occurrence of a in {1, . . . , Δ − 1}. Thus  Δ  t P(N NΔ < Nt ∈ K) ≤ P(N1Δ ≥ 1) ≤ P(X0n−1 = Xii+n−1 ) . i=1

The last term is bounded by Lemma 2. Thus, we only consider the most-left probability in the above decomposition. t t t P(N NΔ ∈ K) − P(N NΔ = Nt ∈ K) = P(N NΔ ∈ K; N1Δ ≥ 1)

≤ P(N1Δ ≥ 1) . t The last probability is bounded again by Lemma 2. So, consider only P(N NΔ ∈ K). n−1 n−1 t t and Z0 respectively. Now decompose NΔ and MΔ on the initial sequences X0 We have  t t P(N NΔ ∈ K) = P(N NΔ (a) ∈ K ; X0n−1 = a) ,   n a∈C  and   t t t P(M MΔ ∈ K) = P(N NΔ (a) ∈ K ; Z0n−1 = a) = P(N NΔ (a) ∈ K)P(a) .     n n a∈C a∈C   Thus, by the ψ-mixing property   P(N N t ∈ K) − P(M M t ∈ K) ≤ ψ(Δ)P(M M t ∈ K) . (13) Δ

Δ

Δ

Further, the proof of Theorem 2 shows that,      n−1   t/P(X0 )  ≤ ε(n) . P(M M ∈ K) − P (Z Z ∈ K)P(a ) t−ΔP(a ( ) Δ      a∈C n  Here, we denote with Zλ a Poisson r.v. with mean λ. The triangle inequality and the Mean Value Theorem give the inequalities    −(t−ΔP(a ( )) k −t k  (t − ΔP(a )) − e t e          −(t−ΔP(a ( )) −(t−ΔP(a ( )) k  ( )) k k −t k  ≤ e−(t−ΔP(a  (t − ΔP(a)) − e  t  + e  t −e t   −(t−ΔP(a ( )) k−1 ≤ e ΔP(a )(kt + tk ) .  

18

M. Abadi

Summing on k ∈ K and dividing by the k! of the Poisson law we conclude that     Zt−ΔP(a Zt ∈ K) ≤ CΔP(a) . P (Z ( ) ∈ K) − P (Z   Therefore     Zt−ΔP(a Zt ∈ K) P(a) ≤ Δ maxn P(a) . P (Z ( ) ∈ K) − P (Z a∈C    a∈C n   This ends the proof of this case. The β-mixing case. The unique difference between this proof and the previous one is that in this case the left-hand side of (13), by the β-mixing property, is bounded by β(Δ). This ends the proof of the theorem.  5.3. Proofs for corollaries of occurrence times Proof of Corollary 1. It is enough to note that  

Mt/P(Z n−1 ) = 0 = Wn P(Z0n−1 ) > t . 0

This ends the proof of the corollary.



Proof of Corollary 2. It is enough to note that  

Nt/P(X n−1 ) = 0 = Rn P(X0n−1 ) > t . 0

This ends the proof of the corollary.



Proof of Corollary 3. It is enough to note that     P(a ) > t . (a ) = 0 = τ Nt/P(a a ( )     This ends the proof of the corollary.



Acknowledgment The author kindly thanks the referee for a number of suggestions and comments to improve a previous version of the paper. The present work was done with the support of CAPES, institution of the Brazilian government for human resources.

References [1] M. Abadi (2001). Exponential approximation for hitting times in mixing processes. Math. Phys. Elec. J. 7 No. 2. [2] M. Abadi (2004). Sharp error terms and necessary conditions for exponential hitting times in mixing processes. Ann. Probab. 32, 1A, 243–264. [3] M. Abadi and S. Vaienti (2008). Large Deviations for Short Recurrence. Discrete and Continuous Dynamical Systems Series A, 21, No. 3, July 2008. [4] M. Abadi and A. Galves (2001). Inequalities for the occurrence times of rare events in mixing processes. The state of the art. Markov Proc. Relat. Fields. 7 1, 97–112. ´ [5] D. Restrepo Alvarez (2005). Ocorrˆ ˆencia de Eventos Sucessivos em Cadeias de Markov Estacion´ ´ arias. Ph. D. thesis, IME-USP.

Poisson via Chen-Stein

19

[6] D. Arratia, M. Goldstein and Gordon (1989). Two moments suffice for Poisson approximations: the Chen-Stein method Ann. Prob. 17, 9–25. [7] D. Arratia, M. Goldstein and Gordon (1990). Poisson approximation and the ChenStein method. With comments and a rejoinder by the authors. Stat. sci. 5, 403–434. [8] R. Bowen (1975). Equilibrium states and the ergodic theory of Anosov diffeomorphisms. Lecture Notes in Math. 470. Springer-Verlag, New York. [9] R.C. Bradley (2005). Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions Probability Surveys Vol. 2 107–144. [10] L. Chen (1975). Poisson approximation for dependant trials. Ann. Probability 3, 534–545. [11] P. Collet, A. Galves and B. Schmitt (1999). Repetition times for gibbsian sources. Nonlinearity vol. 12 , 1225–1237. [12] P. Doukhan (1995). Mixing. Properties and examples. Lecture Notes in Statistics 85. Springer-Verlag. [13] R. Fern´ ´ andez, P. Ferrari and A. Galves (2001). Coupling, renewal and perfect simulation of chains of infinite order. Can be downloaded from http://www.ime.usp.br/˜ galves/papers/ebp5.ps [14] A. Galves and B. Schmitt (1997). Inequalities for hitting times in mixing dynamical systems. Random Comput. Dyn. 5, 337–348. [15] M. Hirata, B. Saussol and S. Vaienti (1999). Statistics of return times: a general framework and new applications. Comm. Math. Phys. 206, 33–55. [16] D. Ornstein, B. Weiss (2002). Entropy and recurrence rates for stationary random fields. Special issue on Shannon theory: perspective, trends, and applications. IEEE Trans. Inform. Theory 48, No. 6, 1694–1697. [17] P.C. Shields (1996). The ergodic theory of discrete sample paths. AMS, Providence RI. [18] Ch. Stein (1972). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proceedings of the sixth Berkley Symposium on Mathematical Statistics and Probability 2, 583–602. [19] A.J. Wyner (1999). More on recurrence and waiting times. Ann. Appl. Probab. 9, No. 3, 780–796. Miguel Abadi IMECC B.P. 6065 University of Campinas 13083-859 Campinas, SP, Brasil e-mail: [email protected]

Progress in Probability, Vol. 60, 21–30 c 2008 Birkhauser  ¨ Verlag Basel/Switzerland

An Inequality for Oriented 2-D Percolation Enrique D. Andjel and Mariela Sued Abstract. We consider a Bernoulli oriented percolation model on the subgraph of Z2 consisting of the points whose coordinates add up to an even number. In this model, each point (m, n) of this graph is the origin of two oriented bonds whose end points are (m + 1, n + 1) and (m − 1, n + 1). We show that the probability that there is an open path from (0, 0) to (2i, 2n) decreases strictly as i increases from 0 to n. Mathematics Subject Classification (2000). 60K35. Keywords. Oriented percolation, strict inequalities.

1. Introduction Oriented two-dimensional percolation has been extensively studied. Durrett’s 1984 paper [4] reviews the results known at that time and also proves some new results; for more recent work on that subject we refer the reader to [1],[2],[3],[8] and [9].The main goal of this paper is to prove a very intuitive inequality which is missing in the literature. We then derive a similar inequality for the contact process. For background on the contact process we refer the reader to [6] and [7]. In the case of the contact process the inequality is stated in Gray’s paper ([5]); but the proof given there depends on Theorem 1 of that paper whose proof has not been published. Some of the ideas of [5], including the notion of forbidden zones, are also present in this paper but we believe that there are some differences with Gray’s approach. To state our results, let Λ = {(x, y) : x, y ∈ Z, y ≥ 0, x + y ∈ 2Z} and for n ∈ N let Λn = {(x, y) ∈ Λ : y ≤ n} Draw oriented bonds from each point (m, n) in Λ to (m + 1, n + 1) and to (m − 1, n + 1). In this paper we suppose that bonds are open independently of each other and that each bond is open with probability p ∈ (0, 1).

22

E.D. Andjel and M. Sued

A path in Λ is a sequence (x0 , y0 ), . . . , (xn , yn ) of points in Λ such that for all 0 ≤ i < n, |xi+1 − xi | = 1 and yi+1 − yi = 1. We say that a path is open if all the bonds joining successive points of that path are open. Given C ⊂ 2Z × {2i} we say that B ⊂ Λ can be generated from C if for all (a, b) ∈ B there is a path (x0 , y0 ), . . . , (xn , yn ) = (a, b) such that (x0 , y0 ) ∈ C and (xi , yi ) ∈ B for all 0 ≤ i ≤ n. In other words, each point in B can be attained from a point in C by means of an oriented path entirely contained in B. We say that a point (a, b) is to the left (right) of a path γ = (x0 , y0 ), . . . , (xn , yn ) if a < xi (a > xi ) whenever b = yi . Note that with this definition, any point (a, b) with b < y0 or b > yn is to the left and to the right of γ. We say that a subset B of Λ (or a path γ2 ) is to the left (right) of a path γ1 if all the points of B (or of γ2 ) are to the left (right) of γ1 . Suppose two paths γ1 = (x0 , y0 ), . . . , (xn , yn ) and γ2 = (u0 , v0 ), . . . , (um , vm ) (or a path γ1 and a set B) have a nonempty intersection. Then, we will call the last intersection of these paths (or the last intersection of γ1 and B) the point in the intersection with the biggest second coordinate. In the case of two paths as above, if their last intersection is the point (xi , yi ) = (uj , vj ) then, we define a new path: γ1 γ2 = (x0 , y0 ), . . . , (xi , yi ), (uj+1 , vj+1 ), . . . , (um , vm ). In our proofs it is important to consider open paths that do not use bonds in a forbidden zone of our graph. With this in mind we introduce the following notation: For any pair of arbitrary subsets B and D of Λ and any point (m, n) ∈ Λ, Λ\B

{D −→ (m, n)} will denote the event that there exists an open path from some point in D to (m, n) whose bonds do not have a point in B. When B = ∅ we will simply write {D −→ (m, n)} and if D reduces to a point (a, b) we will write {(a, b) −→ (m, n)} In Section 2 we prove: Theorem 1.1. Let A ⊂ −2N. Then for any C ⊂ 2N × {0}, any n ∈ N and any set B which can be generated from C, we have Λ\B

Λ\B

P ((A ∪ {0}) × {0} −→ (−1, 2n + 1)) ≥ P ((A ∪ {0}) × {0} −→ (1, 2n + 1)). Moreover the inequality is strict if A ∩ {−2n − 2, −2n, . . . , −2} =  ∅ and B = ∅. In Section 3 we derive from this theorem: Corollary 1.1. a) For all n > 0 and all i ∈ {0, 1, . . . , n} we have P ((0, 0) −→ (2i, 2n)) > P ((0, 0) −→ (2i + 2, 2n)). b) For all n > 0 and all i ∈ {1, . . . , n + 1} we have P ((0, 0) −→ (2i − 1, 2n + 1)) > P ((0, 0) −→ (2i + 1, 2n + 1)). Corollary 1.2. Let ξt be the nearest neighbor contact process on Z with infection parameter λ > 0. Then, for all t > 0 and i ≥ 0 P (ξt0 (i) = 1) > P (ξt0 (i + 1) = 1) .

An Inequality for Oriented 2-D Percolation

23

To prove these corollaries we apply Theorem 1.1 in the special case B = C = ∅. It is nevertheless necessary to include these sets to carry out the inductive step in the proof of the theorem.

2. Proof of Theorem 1.1 Given B ⊆ Λ, we define −B = {(a, b) : (−a, b) ∈ B}. Consider now the following events: D1 = {(0, 0)

Λ\(B∪−B)

−→

(−1, 2n + 1)}c ∩ {(0, 0)

D2 = {(0, 0)

Λ\(B∪−B)

−→

(−1, 2n + 1)} ∩ {(0, 0)

D3 = {(0, 0)

Λ\(B∪−B)

−→

(−1, 2n + 1)}c ∩ {(0, 0)

D4 = {(0, 0)

Λ\(B∪−B)

(−1, 2n + 1)} ∩ {(0, 0)

−→

Λ\(B∪−B)

−→

Λ\(B∪−B)

−→

(1, 2n + 1)}c ,

Λ\(B∪−B)

−→

Λ\(B∪−B)

−→

(1, 2n + 1)}c ,

(1, 2n + 1)},

(1, 2n + 1)},

Λ\B

F1 = {(A ∪ {0}) × {0} −→ (1, 2n + 1)} and Λ\B

F−1 = {(A ∪ {0}) × {0} −→ (−1, 2n + 1)}. Observe that if B is as in Theorem 1.1, then any path originated in (0, 0) and contained in Λ \ (B ∪ −B) is necessarily to the left of B and to the right of −B. Observe also that any point in {(x, y) : x ≥ 0} which is to the left of B must be to the right of −B. These properties will be used frequently in the sequel. We now prove: Proposition 2.1. If A and B are as in Theorem 1.1, then a) D2 ∩ F1 = ∅ and b) P (D1 ∩ F−1 ) ≥ P (D1 ∩ F1 ). Proof of Proposition 2.1. To show part a) note that if D2 occurs there is an open path γ1 from (0, 0) to (−1, 2n + 1) contained in Λ \ (B ∪ −B). If F1 occurs, there is an open path γ2 from (A ∪ {0}) × {0} to (1, 2n + 1) contained in Λ \ B. But, if we have these two paths they must intersect and γ1 γ2 is an open path from (0, 0) to (1, 2n + 1). The portion of that path above the last intersection of γ1 and γ2 is to right of γ1 , hence to the right of −B. Therefore the whole path is contained in Λ \ (B ∪ −B) which contradicts the definition of D2 . Hence D2 ∩ F1 = ∅. To prove part b) we introduce several random objects: let k = sup{r : 0 ≤ r ≤ n and (0, 0) where, by convention, (0, 0)

Λ\(B∪−B)

−→

Λ\(B∪−B)

−→

(0, 2r)}

(0, 0), let Λ\B

I = {m : k < m ≤ n and (A ∪ {0}) × {0} −→ (0, 2m)}

24

E.D. Andjel and M. Sued

and let m0 = inf I where, by convention m0 = ∞ if I = ∅. Let

∅ if m0 = ∞ H= Λ\B {x < 0 : (A ∪ {0}) × {0} −→ (x, 2m0 )} if m0 < ∞, and let Λ\(B∪−B)

K = {(x, y) : (0, 0) −→ (x, y) and y ≤ 2n + 1}. The next figure shows each of the objects we have defined. In this example A = {−2, −4, −6, −8, −10} and C = {(4, 0)}. Moreover only the open bonds are drawn.

B points

−B points

k

m

0

H points

K points

The H points in the drawing are really the points whose first coordinate is in that set and whose second coordinate is 2m0 . Moreover, (0, 2k) also belongs to K although it is not marked as a K point. We now prove: Lemma 2.1. On F1 ∩ D1 the following statements hold. a) m0 < ∞. b) Any open path γ in Λ2n+1 \ B starting from (A ∪ {0}) × {0} and containing a point in the set {(x, y) : x > 0, y = 2m0 } does not contain the point (1, 2n+1) nor any point in the region {(x, y) : x ≤ 0, y ≥ 2m0 }. Moreover all the points in γ with second coordinate y ≥ 2m0 belong to K.

An Inequality for Oriented 2-D Percolation

25

c) B ∩ {(x, y) : x ≤ 0, y = 2m0 } = ∅. d) K ∩ {(x, y) : x ≤ 0, y = 2m0 } = ∅. e) Any open path γ ∈ Λ2n+1 \ B starting from the set {(x, y) : x ≤ 0, y = 2m0 } and containing a point in K does not contain the point (1, 2n + 1) nor the point (−1, 2n + 1). Remark. Statements b)–e) also hold on {m0 < ∞} ∩ D1 . By part a) {m0 < ∞} ∩ D1 ⊃ F1 ∩ D1 . But, the reader can check that the proofs of parts b)–e) are also valid in this larger set. To follow the proof of this lemma we recommend the reader to draw the paths that intervene in it. Proof of Lemma 2.1. We start proving part a). Suppose F1 occurs and let γ1 be an open path from (A ∪ {0}) × {0} to (1, 2n + 1) contained in Λ \ B. Let (0, 2) be the last intersection of this path with the set {(x, y) : x = 0}. It now suffices to show that on D1 we have  > k . To do so we argue by contradiction: suppose  ≤ k and let γ2 be an open path from (0, 0) to (0, 2k) contained in Λ \ (B ∪ −B). Then γ1 and γ2 must intersect. The portion of γ2 γ1 between the last intersection of γ1 and γ2 and the set {(x, y) : y = 2k} is to the right of the corresponding portion of γ2 , hence to the right of −B and therefore contained in Λ \ (B ∪ −B). Its end point must be (2a, 2k) for some a ≥ 0. Since (0, 2) is the last intersection of γ1 and the set {(x, y) : x = 0}, and γ1 reaches the point (1, 2n + 1), the portion of γ1 strictly above the set {(x, y) : y = 2k} must consist of points whose first coordinate is positive. Hence, this portion is also contained in Λ \ (B ∪ −B) and γ2 γ1 contradicts the definition of D1 . To prove part b), let γ1 be an open path in Λ2n+1 \ B starting from (A ∪ {0}) × {0} and containing the point (2a, 2m0 ) for some a > 0 and let γ2 be an open path from (0, 0) to (0, 2k) contained in Λ \ (B ∪ −B). It now follows from the definition of m0 that these paths must intersect. The portion of γ1 between the last intersection of these paths and the set {(x, y) : y = 2k} is to the right of γ2 , therefore it is contained in Λ\(B ∪−B). Recalling the definition of m0 and the fact that a > 0, we see that the portion of γ1 strictly above the set {(x, y) : y = 2k} and below the point (2a, 2m0 ) must remain in the region {(x, y) : x > 0}. Since it is to left of B, it must be contained in Λ \ (B ∪ −B). We now show that the portion of γ1 above the point (2a, 2m0 ) must remain in the region with strictly positive first coordinate. We argue by contradiction: let (0, 2c) be the first point in γ1 above (2a, 2m0 ) with first coordinate equal to 0. Then the portion of γ1 between (2a, 2m0 ) and (0, 2c) is in Λ\(B ∪ −B) and the open path γ2 γ1 contradicts the definition of k. Since γ1 is in Λ \ B, this implies that the portion of γ1 above the point (2a, 2m0 ) must be in Λ\(B ∪ −B). Hence, the following two assertions hold: 1) The portion of γ2 γ1 above the point (2a, 2m0 ) must remain in the region {(x, y); x > 0}. 2) The whole path γ2 γ1 us contained in Λ\(B ∪ −B).

26

E.D. Andjel and M. Sued

All the statements of b) now follow easily from these two assertions and the definition of D1 . Part c) follows from the definition of m0 and the fact that B is to the right of any path starting from (A ∪ {0}) × {0} and contained in Λ \ B. To prove part d) we argue by contradiction. Consider an open path γ1 in Λ \ B from (A ∪ {0}) × {0} to (0, 2m0 ) and suppose there is an open path γ2 in Λ \ (B ∪ −B) from (0, 0) to (x, 2m0 ) for some x ≤ 0. Then γ2 γ1 is an open path in Λ \ (B ∪ −B) from (0, 0) to (0, 2m0 ) which contradicts the definitions of m0 and k. To prove part e), first note the following: part d) of this lemma and the definition of k imply that K cannot contain points in the region {(x, y) : x ≤ 0, 2m0 ≤ y ≤ 2n+1}. Therefore, if γ contains a point (x, y) in K, its first coordinate must be strictly positive and we can argue as in part b) to conclude that the portion of γ above this point is in Λ \ (B ∪ −B). This implies that this part of γ is in K. Hence, by definition of D1 , it cannot contain the point (1, 2n + 1) nor the point (−1, 2n + 1).  We now return to the proof of part b) of the Proposition 2.1. We will proceed by induction on n. The result is obvious for n = 0. To prove the inductive step, let L(i, R, A ) = {m0 = i} ∩ {K = R} ∩ {H = A }. It follows from part a) of Lemma 2.1 that: F1 ∩ D1 ⊂ ∪1≤i≤n ∪R,A L(i, R, A ) ⊂ D1 where A takes all the possible values of H and R takes all the possible values of K not containing the points (1, 2n + 1) and (−1, 2n + 1). Now, using the convention P (A|∅) = 0, write: P (F F1 ∩ D1 )   P (F F1 |D1 ∩ L(i, R, A ))P (D1 ∩ L(i, R, A )) = 1≤i≤n R,A

=

 

1≤i≤n

P (F F1 |L(i, R, A ))P (L(i, R, A ))

(∗)

R,A

where A and R take the same values as above. From part d) of Lemma 2.1 we see that for each i the sum on R can be restricted to R’s such that R ∩ {(x, 2i) : x ≤ 0} = ∅,

(∗∗)

and from part c) of Lemma 2.1 we see that the sum on i can be restricted to values such that B ∩ {(x, 2i) : x ≤ 0} = ∅. Hence we may assume that all the terms in the right-hand side of (∗) are such that Λ ∩ T−2i (B ∪ R) can be generated from some subset of 2N × {0}, where T−2i (x, y) = (x, y − 2i). It follows from the first statement of part b) of Lemma 2.1 that

Λ\B F1 ∩ L(i, R, A ) = (A ∪ {0}) × {2i} −→ (1, 2n + 1) ∩ L(i, R, A ).

An Inequality for Oriented 2-D Percolation

27

But, since we are summing on R’s not containing the points (1, 2n + 1) and (−1, 2n + 1) we can deduce from part e) of Lemma 2.1 that we also have:

Λ\(B∪R) F1 ∩ L(i, R, A ) = (A ∪ {0}) × {2i} −→ (1, 2n + 1) ∩ L(i, R, A ). Hence P (F F1 |L(i, R, A )) = P



Λ\(B∪R) (A ∪ {0}) × {2i} −→ (1, 2n + 1) |L(i, R, A ) .

The event L(i, R, A ) is independent of the state of the bonds whose initial points (x, y) are not in R and are such that y ≥ 2i. Therefore, 

Λ\(B∪R) P (A ∪ {0}) × {2i} −→ (1, 2n + 1) |L(i, R, A ) = P ((A ∪ {0}) × {2i}

Λ\(B∪R)

−→

(1, 2n + 1)),

which by translation invariance is equal to P ((A ∪ {0}) × {0}

Λ\T T−2i (B∪R)

−→

(1, 2(n − i) + 1)).

Since Λ ∩ T−2i (B ∪ R) can be generated from some subset of 2N × {0}, we can apply the inductive hypothesis to conclude that this last probability is bounded above by: P ((A ∪ {0}) × {0}

Λ\T T−2i (B∪R)

−→

(−1, 2(n − i) + 1)).

Hence, from (∗) we get: P (F F1 ∩ D1 )   Λ\T T−2i (B∪R) P ((A ∪ {0}) × {0} −→ (−1, 2(n − i) + 1))P (L(i, R, A )) ≤ 1≤i≤n R,A

=

 

1≤i≤n

P ((A ∪ {0}) × {2i}

Λ\(B∪R)

−→

(−1, 2n + 1))P (L(i, R, A )).

R,A

Thanks to the remark following Lemma 2.1, we can argue as above to show that the last sum is equal to:   P (F F−1 ∩ L(i, R, A )) = P (F F−1 ∩ D1 , m0 < ∞) 1≤i≤n R,A

≤ P (F F−1 ∩ D1 ).



Proof of Theorem 1.1. Since the sets Di (i = 1, . . . , 4) form a partition of our probability space and F1 contains both D3 and D4 we have: P (F F1 ) = P (D4 ) + P (D1 ∩ F1 ) + P (D2 ∩ F1 ) + P (D3 ). Similarly we get: P (F F−1 ) = P (D4 ) + P (D1 ∩ F−1 ) + P (D3 ∩ F−1 ) + P (D2 ).

28

E.D. Andjel and M. Sued

Since by symmetry P (D2 ) = P (D3 ), the first statement of the theorem follows from parts a) and b) of Proposition 2.1. For the second statement just note that under its additional hypothesis we have that P (D3 ∩ F−1 ) > 0. 

3. Proofs of Corollaries 1.1 and 1.2 Proof of Corollary 1.1. To prove part a) let m0 = inf{m : (0, 0) −→ (2i + 1, 2m + 1)} 

and let H=

∅ if m0 ≥ n {x < 2i + 1 : (0, 0) −→ (x, 2m0 + 1)} if m0 < n,

Since P ((0, 0) −→ (2i + 2, 2n))|m0 ≥ n) = 0, and P (m0 < i) = 0 we have P ((0, 0) −→ (2i + 2, 2n)) =

 n−1  A

=

  n−1 A

P ((0, 0) −→ (2i + 2, 2n)|H = A, m0 = j)P (H = A, m0 = j)

j=i

P ((A ∪ {2i + 1}) × {2j + 1} −→ (2i + 2, 2n))P (H = A, m0 = j),

j=i

where A ranges over all possible values of H. Using Theorem 1.1 with B = C = ∅ and translation invariance we get P ((0, 0) −→ (2i + 2, 2n)) ≤

 n−1  A

P ((A ∪ {2i + 1}) × {2j + 1} −→ (2i, 2n))P (H = A, m0 = j)

j=i

= P ((0, 0) −→ (2i, 2n), m0 < n). Part a) now follows from P ((0, 0) −→ (2i, 2n), m0 > n) > 0 . The proof of part b) is similar and we omit it.



Proof of Corollary 1.2. Corollary 1.2 can be deduced from our results for oriented percolation by means of a discrete time approximation of the contact process. Since the ideas are standard we will only sketch this proof. First modify our percolation model as follows: From each point (2m, 2n) add an oriented  bond from that point to λ (2m, 2n+2). Then, let N be sufficiently large to satisfy: N < 1. Now suppose that probability 1 − 1/N and each of the diagonal each of the new bonds is open with 

λ bonds is open with probability N . All bonds being open independently of each other. Then, observe that in this percolation model we still have the following property: If a path γ1 begins to the right (left) and ends to the left (right) of

An Inequality for Oriented 2-D Percolation

29

another path γ2 then, these two paths must intersect. Thanks to this, Theorem 1.1 and its proof remain valid in this new context. Then, define N ξt (x) as 1 if there is an open path in our percolation model from (0, 0) to (2x, 2[N t]) and as 0 if there is no such path ([N t] being the integer part of N t). Letting N go to infinity N ξt converges to ξt0 , the contact process on Z starting from {0} and we can transfer our results on oriented percolation to the contact process. To show that strict inequality is preserved at the limit, let I = {(0, 0) → (2i + 1, 2k + 1) for some k < [N t]}, and note that it follows from Theorem 1.1 that P ((0, 0) → (2i, 2[N t])) − P ((0, 0) → (2i + 2, 2[N t])) ≥ P (I c ∩ {(0, 0) → (2i, 2[N t]}). Since the right-hand side is easily seen to be bounded away from 0 as N goes to infinity, the result follows. 

References ´ s, A. Stacey (1994) Improved upper bounds for the [1] P. Balister, B. Bolloba critical probability of oriented percolation in two dimensions. Random Structure Algorithms. 5, 573–589. [2] V. Belitsky, P.A. Ferrari, N. Konno, T.M. Liggett (1997) A strong correlation inequality for contact processes and oriented percolation. Stochastic Process. Appli. 67, 213–225. [3] J. van den Berg, O. Haggstrom, J. Kahn (2006) Some conditional correlation inequalities for percolation and related processes. Random Structure algorithms. 4, 417–435. [4] R. Durrett (1984) Oriented percolation in two dimensions. Ann. Probab. 12, 999– 1040. [5] L.F. Gray. (1991) Is the contact process dead? Lectures in Appli. Math. 27, 19–29. [6] T.M. Liggett (1989) Interacting particle systems. Springer-Verlag, New York. [7] T.M. Liggett (1999) Stochastic interacting systems: contact, voter and exclusion processes. Springer-Verlag, Berlin. [8] T. Kuczek (1989) The central limit theorem for the right edge of supercritical oriented percolation. Ann. Probab. 17, 1322–1332. [9] M.V. Menshikov, S.Yu. Popov, V.V. Sisko (2002) On the connection between oriented percolation and contact process. J. Theoret. Probab. 15, 207–221.

30

E.D. Andjel and M. Sued

Enrique D. Andjel LATP-CMI Universit´ ´e de Provence 39 Rue Joliot-Curie F-13453 Marseille cedex 13 France e-mail: [email protected] Mariela Sued Instituto de C´ ´ alculo UBA-Conicet Argentina Intendente Guiraldes 2670 1428 Buenos Aires Argentina e-mail: [email protected]

Progress in Probability, Vol. 60, 31–58 c 2008 Birkhauser  ¨ Verlag Basel/Switzerland

Is Critical 2D Percolation Universal? Vincent Beffara Abstract. The aim of these notes is to explore possible ways of extending Smirnov’s proof of Cardy’s formula for critical site-percolation on the triangular lattice to other cases (such as bond-percolation on the square lattice); the main question we address is that of the choice of the lattice embedding into the plane which gives rise to conformal invariance in the scaling limit. Even though we were not able to produce a complete proof, we believe that the ideas presented here go in the right direction. Mathematics Subject Classification (2000). 82B43, 32G15; 82B20, 82B27. Keywords. Percolation, Conformal invariance, Complex structure.

Introduction It is a strongly supported conjectured that many discrete models of random media, such as e.g. percolation and the Ising model, when taken in dimension 2 at their critical point, exhibit conformal invariance in the scaling limit. Indeed, the universality principle implies that the asymptotic behavior of a critical system after rescaling should not depend on the specific details of the underlying lattice, and in particular it should be invariant under rotations (at least under suitable symmetry conditions on the underlying lattice). Since by construction a scaling limit is also invariant under rescaling, it is natural to expect conformal invariance, as the local behavior of a conformal map is the composition of a rotation and a rescaling. On the other hand, conformally invariant continuous models have been thoroughly studied by physicists, using tools such as conformal field theories. In 2000, Oded Schramm ([14]) introduced a one-parameter family of continuous bidimensional random processes which he called SLE processes, as the only possible scaling limits in this situation, under the assumption of conformal invariance; connections between SLE and CFT are now quite well understood (see, e.g., [7, 1]). However, actual convergence of discrete models to SLE in the scaling limit is known for only a few models. The case on which we focus in this paper is that

32

V. Beffara

of percolation. The topic of conformal invariance for percolation has a long history – see [13] and references therein for f an in-depth discussion of it. In the case of site-percolation on the triangular lattice, it is a celebrated result of Smirnov ([16]) that indeed the limit exists and is conformally invariant. While the proof is quite simple and extremely elegant (see Section 3 below and references therein), it is very specific to that particular lattice, to the point of being almost magical; it is a very natural question to ask how it can be generalized to other cases, and in particular to bond-percolation on the square lattice. Universality and conformal invariance have indeed been tested numerically for percolation in various geometries (see, e.g., [12]), and a partial version of conformal invariance (assuming the existence of the limit) is known in the case of Voronoi percolation (see [4]). In fact, it seems that the question of convergence itself has hardly been addressed by physicists, at least in the CFT community – a continuous, conformally invariant object is usually the starting point of their work rather than its outcome. Techniques such as the renormalization group do give reason to expect the existence of a scaling limit and of critical exponents, but they seem to not give much insight into the emergence of rotational invariance. This is not surprising in itself, for a very trivial reason: Take any discrete model for which you know that there is a conformally invariant scaling limit, say a simple random walk on Z2 , and deform the underlying lattice, in a linear way, so as to change the aspect ratio of its faces. Then the scaling limit still exists (it is the image of the previous one by the same transformation); but obviously it is not rotationally invariant. Since all the rescaling techniques apply exactly the same way before and after deformation, they cannot be sufficient to derive rotational invariance. A trace of this appears in the most general statement of the universality hypothesis (see, e.g., [13, Section 2.4]): To paraphrase it, given any two periodic planar graphs, the scaling limits of critical percolation on them are conjugated by some linear map g. The main question we address in these notes is the following: Given a discrete model on a doubly periodic planar graph, how to embed this graph into the plane so as to make the scaling limit isotropic? If the graph has additional symmetry (as for instance in the case of the square or triangular lattices), the embedding has to preserve this symmetry; so a restating of the same question in the terms of the universality hypothesis would be, absent any additional symmetry for one of the two graphs involved, can one determine the map g? The most surprising thing (to me at least) about the question, besides the fact that it appears to actually be orthogonal to the interests of physicists in that domain, is that its answer turns out to depend on the model considered. In other words, there is no absolute notion of a “conformal embedding” of a general graph. In the case of the simple random walk, the answer is quite easy to obtain, though it does not seem to have appeared in the literature in the form we present it here; in the case of percolation, I could find no reference whatsoever, the closest being

Is Critical 2D Percolation Universal?

33

the discussion and numerical study of striated models in [13] where, instead of looking at a different graph, the parameter p in the model is chosen to depend on the site in Z2 in a periodic fashion – which admittedly is a very related question. The paper is roughly divided into two parts. In the first one, comprised of the firs two sections, we introduce some notation and the general framework of the approach, and we treat the case of the simple random walk. This is enough to prove that the correct embedding is not the same for it as for percolation; we then argue that circle packings might give a way to answer the question in the latter case. In the second part, which is of a more speculative nature, we investigate Smirnov’s proof in some detail, and rephrase it in such a way that its general strategy can be applied to general triangulations. We then describe the two main steps of a strategy that could lead to its generalization, though we were able to perform none of the two.

1. Notation and setup 1.1. The graph We first define the class of triangulations of the plane we are interested in. Let T be a 3-regular finite graph of genus 1 (i.e., a graph that is embeddable in the torus T2 := R2 /Z2 but not in the plane, and having only vertices of degree 3). For ease of notation, we assume that T is equipped with a fixed embedding in T2 , which we also denote by T . The dual T ∗ of T (which we also assume to be embedded in the torus once and for all) is then a triangulation of T2 . Let Tˆ (resp. Tˆ ∗ ) be the universal cover of T (resp. T ∗ ): Then Tˆ and Tˆ ∗ are mutually dual, infinite, locally finite planar graphs, on which Z2 acts by translation. We are interested in natural ways of embedding Tˆ into the complex plane C. Let Ti (the meaning of the notation will become clear in a minute) be the embedding obtained by pulling T back using the canonical projection from R2 to T2 – we will call Ti the square embedding of T . For every α ∈ C \ R, let ϕα : C → C be the R-linear map defined by ϕα (x + iy) = x + αy (i.e., it sends 1 to itself and i to α) and let Tα be the image of Ti by ϕα . For lack of a better term, we will call Tα the embedding of modulus α of T into the complex plane. Notice that the notation Tα depends on the a priori choice of the embedding of T in the flat torus; but, up to rotation and scaling, the set of proper embeddings of Tˆ obtained starting from two different embeddings of T is the same, so no generality is lost (as far as our purpose in these notes is concerned). One very useful restriction on embeddings is the following: Definition 1. We say that an embedding Tα of Tˆ in the complex plane is balanced if each of its vertices is the barycenter (with equal weights) of its neighbors; or, equivalently, if the simple random walk on it is a martingale.

34

V. Beffara

Proposition 2. Let T be a 3-regular graph of genus 1: Then, for every α ∈ H, there is a balanced embedding of Tˆ in the complex plane with modulus α. Moreover, this embedding is unique up to translations of the plane. Proof. We only give a sketch of the proof, because expanding it to a full proof is both straightforward and tedious. The main remark is that any periodic embedding which minimizes the sum S2 , over a period, of the squared lengths of its edges is balanced: Indeed, the gradient, with respect to the position of a given vertex, of S2 is exactly the difference between this point and the barycenter of its neighbors. (This would be true in any Euclidean space.) It is easy to use a compactness argument to prove the existence of such a minimizer. To prove uniqueness up to translation is a little trickier, but since it is not necessary for the rest of this paper, we allow ourselves to give an even sketchier argument. First, one can get rid of translations by assuming that a fixed vertex of Tˆ is put at the origin by the embedding; the set of all possible embeddings of modulus α is then parameterized by 2(|V (T )| − 1) real-valued parameters, which are the coordinates of the locations of the other vertices in one period of Tˆ . In terms of these variables, S2 is polynomial of degree 2. It is bounded below by the squared length of the longest edge in the embedding, which itself is bounded below, up to a constant depending only on the combinatorics of the graph, by the square of the largest of the 2(|V (T )| − 1) parameters; so it goes to infinity uniformly at infinity. This implies that its Hessian (which is constant) is positive definite, so S2 is strictly convex as a function of those variables. This immediately implies the uniqueness of the minimizer.  An essential point is that, even though our proof uses Euclidean geometry, the fact that the embedding is balanced is a linear condition. In particular, if the embedding Ti is balanced, then so are all the other Tα . The corresponding a priori embedding of T itself into the flat torus T2 (which is also unique up to translations) will be freely referred to as the balanced embedding of T into the torus. 1.2. The probabilistic model We will be interested in critical site-percolation on the triangulation Tα∗ ; more specifically, the question we are interested is the following. Let Ω be a simply connected, smooth domain in the complex plane, and let A, B, C and D be four points on its boundary, in that order. For every δ > 0, let Ωδ be the largest connected component (in terms of graph connectivity) of the intersection of Ω with δT Tα , and Ω∗δ be its dual graph. Ωδ should be seen as a discretization of Ω at scale δ. Let Aδ , Bδ , Cδ and Dδ be the vertices of Ωδ that are closest to A, B, C and D respectively. The model we are most interested in is critical site-percolation on Ω∗δ ; however, most of the following considerations remain valid for other lattice models. Let Cδ (Ω, A, B, C, D) be the event that there is an open crossing in Ω∗δ , between the intervals Aδ Bδ and Cδ Dδ of its boundary. Under some symmetry conditions on T , Russo-Seymour-Welsh theory ensures that at criticality, the probability of

Is Critical 2D Percolation Universal?

35

Cδ (Ω, A, B, C, D) is bounded away from both 0 and 1 as δ goes to 0. Its limit was conjectured by Cardy (see [6]) using non-rigorous arguments from conformal field theory; actual convergence was proved, in the case of the triangular lattice (embedded in such a way that its faces are equilateral triangles), by Smirnov (see [15, 3]). We defer the statement of the convergence to a later time. The following definition has become standard: Definition 3. Assume that, for every choice of (Ω, A, B, C, D), the probability of the event Cδ (Ω, A, B, C, D) has a limit fα (Ω, A, B, C, D) as δ → 0 – we will refer to this by saying that the model has a scaling limit. We say that the model is conformally invariant in the scaling limit if, for every conformal map Φ from Ω to Φ(Ω), one has fα (Ω, A, B, C, D) = fα (Φ(Ω), Φ(A), Φ(B), Φ(C), Φ(D)). This is equivalent to saying that fα (Ω, A, B, C, D) only depends on the modulus of the conformal rectangle (Ω, A, B, C, D). (Notice that the extension of Φ to the boundary of Ω, which is necessary for the above definition to make sense, is ensured as soon as Ω is assumed to be regular enough.)

2. Periodic embeddings 2.1. Uniqueness of the modulus Given T , it is natural to ask whether it is possible to choose a value for α which provides conformal invariance in the scaling limit. There are two possible strategies: Either give an explicit value for which “a miracle occurs” (in physical terms, for which the model is integrable – this is what Smirnov did in the case of the triangular lattice), or obtain its existence in a non-constructive way – which is what we are trying to do here. A reassuring fact is that, whenever such an α exists, it is essentially unique: Proposition 4. For every graph T , there are either zero or two values of α such that critical site-percolation on Tα∗ is conformally invariant in the scaling limit. In the latter case, the two values are complex conjugates of each other. Proof. The key remark is the following: Let β be a non-real complex number. Since the event Cδ is defined using purely combinatorial features, one can push the whole picture forward through ϕβ without changing its probability. Let α = ϕβ (α): ϕβ then transforms Ω into ϕβ (Ω) and the lattice Tα into Tα . So, assuming convergence on both sides, one always has fα (Ω, A, B, C, D) = fα (ϕβ (Ω), ϕβ (A), ϕβ (B), ϕβ (C), ϕβ (D)). In the case β = −i, ϕβ is simply the map z → z¯. In that case, the modulus of ¯ C, ¯ B, ¯ A) ¯ is the same as that of (Ω, A, B, C, D), the conformal rectangle (ϕ−i (Ω), D, and clearly the event Cδ is invariant when the order of the corners is reversed. So,

36

V. Beffara

conformal invariance for Tα and the previous remark implies that fα¯ (Ω, A, B, C, D) still only depends on the modulus of the conformal rectangle – in other words, if critical percolation Tα is conformally invariant in the scaling limit, that is also the case on Tα¯ . Now assume conformal invariance in the scaling limit for two choices of the modulus in the upper-half plane; these moduli can always be written as α and α = ϕβ (α) for an appropriate choice of β ∈ H \ {i}. Still using the above remark, all that is needed to arrive to a contradiction is to show that fα does actually depend on the modulus of the rectangle (i.e., that it is not constant), and that there exist two conformal rectangles with the same modulus and whose images by ϕβ have different moduli. For the former point, it is enough to prove that for every choice of ρ, ρ > 0, the probability of crossing the rectangle [0, ρ] × [0, 1] horizontally is strictly larger than that of crossing [0, ρ + ρ ] × [0, 1]. This is obvious by Russo-Seymour-Welsh: The event that there is a vertical dual crossing in δT Tα∗ ∩ [ρ + δ, ρ + ρ ] × [0, 1] is independent of Cδ ([0, ρ] × [0, 1], ρ, ρ + i, i, 0) and its probability is bounded below, uniformly in δ < ρ /10, by some positive ε depending only on ρ and ρ . Hence, still assuming that the limits all exist as δ → 0, fα ([0, ρ + ρ ] × [0, 1], ρ + ρ , ρ + ρ + i, i, 0)  (1 − ε)ffα ([0, ρ] × [0, 1], ρ, ρ + i, i, 0). For the latter point, assume that ϕβ preserves the equality of moduli of conformal rectangles. Let Q = [0, 1]2 be the unit square. By symmetry, the conformal rectangles (Q, 0, 1, 1 + i, i) and (Q, 1, 1 + i, i, 0) have the same modulus; on the other hand ϕβ (Q) is a parallelogram, and by our hypothesis on ϕβ it has the same modulus in both directions. This easily implies that it is in fact a rhombus. If now Q is the square with vertices 1/2, 1 + i/2, 1/2 + i, i/2, ϕβ (Q ) is both a rhombus (by the same argument) and a rectangle (because its vertices are the midpoints of the edges of ϕβ (Q) which is a rhombus). Hence ϕβ (Q ) is a square, and so is ϕβ (Q), and in particular β = ϕβ (i) = i, which is in contradiction with our hypothesis.  When such a pair of moduli exists, we will denote by αperc the one with T positive imaginary part. The same reasoning can be done for various models, and in each case where the scaling limit exists and is non-trivial, there will be a pair of moduli making it conformally invariant; we will distinguish them from each other by using the name of the model as a superscript (so that for instance αRW makes T the simple random walk conformally invariant in the scaling limit – cf. below). When an argument does not depend on the specific model (as is the case in the next subsection), we will use the generic notation αT as a placeholder. 2.2. Obtaining αT by symmetry arguments It should be noted that, because the value of αT (when it exists) is uniquely defined by the combinatorics of T , there are cases where additional symmetry specifies its value uniquely. Indeed, assume Ψ is a graph isomorphism of Tˆ which is neither a translation nor a central symmetry; for every α, it induces a topological

Is Critical 2D Percolation Universal?

37

isomorphism of Tα . Assume without loss of generality that the origin of the plane is chosen to be one of the vertices of Tα ; let z0 = Ψ(0), z1 = Ψ(1) and zα = Ψ(α) (notice that both 1 and α are also vertices of Tα ). Assume α = αperc . Because Ψ is an isomorphism, it preserves site-percolation; T so, in particular, critical site-percolation on Ψ(T Tα ) is conformally invariant in the scaling limit. By Proposition 4, this implies that zα − z0 Ψ(α) − Ψ(0) ∈ {α, α}. ¯ (2.1) = z1 − z0 Ψ(1) − Ψ(0) This condition is then enough to obtain the value of αT . There are two natural examples of that (illustrated in Figures 1 and 2), which we now describe.

Figure 1. The graphs Th (left) and Ts (right), embedded into T2 in a balanced way with a vertex at the origin; empty circles and dotted lines represent the dual graphs. Both are represented using their square embedding, so the triangles in Th are not equilateral.

Figure 2. The same graphs as in Figure 1, with the origin on a vertex of the dual. • Let Th be one period of the honeycomb lattice, embedded into T2 in such a way that every vertex is the barycenter of its neighbors (we will call such an embedding balanced ); since we take T2 to be a square, the coordinates of the vertices of Th are (0, 0), (1/3, 0), (1/2, 1/2) and (5/6, 1/2). There is an

38

V. Beffara isomorphism Ψh of order 3 of Tˆh , corresponding to rotation around (0, 0); on Tα , it sends 0 to z0 = 0, 1 to z1 = (3α − 1)/2 and α to zα = −(1 + α)/2. Since Ψh preserves orientation, Equation (2.1) leads to √ zα − z0 3 1+α =α = =⇒ α = ±i , = z1 − z0 1 − 3α 3 √ in other words αTh = i 3/3. Not surprisingly, this corresponds to embedding the faces of Tˆh∗ as equilateral triangles, and those of Tˆh as regular hexagons. • Let Ts be chosen in such a way that Tˆs∗ has the topology of the centered square lattice; if again the embedding is balanced, the coordinates of the vertices of Ts are (0, 0), (1/2, 0), (1/4, 1/4) and (1/4, 3/4). There is an isomorphism Ψs of order 4 of Tˆs , corresponding to a rotation rotation around the vertex (1/4, 0) of Ts∗ . In that case (−3/4 − α/4) − (1/4 − α/4) −1 zα − z0 = = =α z1 − z0 (1/4 + 3α/4) − (1/4 − α/4) α

= =⇒

α = ±i,

so αTs = i. Again not surprisingly, this corresponds to the usual embedding of the square lattice using – well, squares. Of course, identifying αT in those cases is a long way from a proof of conformal invariance; but it would seem that understanding, in the general case, what αperc T is would be a significant progress in our understanding of the process. 2.3. Embedding using random walks As an aside, in this subsection and the next we describe two natural ways of embedding a doubly periodic graph into the complex plane, which both have something to do with conformal invariance. Let T be a finite 3-regular graph of genus 1, embedded in T2 in a balanced way, and let (Xn )n0 be a simple random walk on it. For simplicity, assume that (Xn ) is irreducible as a Markov chain. (Both 3-regularity and irreducibility are completely unnecessary as far as the results presented here are concerned, and the same reasoning would work in the general case, but notation would be a little tedious.) Since T is finite, (Xn ) converges in distribution to the unique invariant measure, which, because T is 3-regular, is the uniform measure on V (T ); moreover the convergence is exponentially fast. Now pick α ∈ H, and lift (Xn ) to a simple random walk (Z Zn ) on Tα . By the balance condition on the embedding, it is easy to check that (Z Zn ) is a martingale; exponential decay of correlations between its increments is enough to obtain a central limit theorem (cf. for instance [9] and references therein). To write the covariance matrix in a convenient form, we need some notation. For each (oriented) edge e of T , choose z1 (e) and z2 (e) in Tα in such a way that they are neighbors and the edge (z1 (e), z2 (e)) is a pre-image of e by the natural projection from Tα to T ; let eα := z2 (e) − z1 (e) – obviously it does not depend on the choice of z1 (e)

Is Critical 2D Percolation Universal?

39

and z2 (e). Define Σxx α (T ) :=

 1 (eα )2 , |E(T )| e∈E(T )

 1 Σyy (eα )2 , α (T ) := |E(T )| e∈E(T )

 1 Σxy (eα )(eα ). α (T ) := |E(T )| e∈E(T )

It is not difficult to compute the covariance matrix of the scaling limit of the walk: Proposition 5. As n goes to infinity, n−1/2 Zn converges in distribution to a Gaussian variable with covariance matrix   xx Σα (T ) Σxy α (T ) . Σα (T ) := yy Σxy α (T ) Σα (T ) Proof. The walk is centered by definition; the existence of a Gaussian limit is a direct consequence of the exponential decay of step correlations. All that remains to be done is to compute the covariance matrix. We focus on the first matrix entry, the others being similar. We have ⎡  ⎤ 2 n−1 n−1   1 1  2 −1/2 ⎣ ⎦ Var(n Z Zn ) = E (Zk+1 − Zk ) E ((Zk+1 − Zk )) = n n k=0

k=0

(the other terms disappear by the martingale property). We know that Zk+1 − Zk converges in distribution, because the walk on T converges in distribution; its limit is the distribution of eα where e is an edge of T chosen uniformly. By Ces``aro’s Lemma, the expression above then converges to Σxx α ; the computation of the other entries in Σα (T ) it exactly similar.  Even though the previous definition of conformal invariance in the scaling limit does not apply directly in this case, its natural counterpart is to ask for the scaling limit of the walk to be rotationally invariant (i.e., to be standard two-dimensional Brownian motion); this is equivalent to saying that the covariance matrix Σα (T ) is scalar, and since its entries are real, yet another equivalent formulation is  yy xy [Σxx ⇐⇒ (eα )2 = 0. α (T ) − Σα (T )] + i [Σα (T )] = 0 e∈E(T )

The last equation is a second-degree equation in α with real-valued coefficients. If α ∈ R, all the terms are non-negative and at least one is positive, so the equation has no solution in R; letting α go to +∞ along the real line leads to |E(T )| positive terms, at least one of which is of order α2 , so the coefficient in α2 in the equation is not zero. Hence the equation has exactly two solutions which are complex conjugate of each other – the situation is very similar to the

40

V. Beffara

one in Proposition 4. For further reference, we let αRW be the one with positive T imaginary part. One advantage of this choice (besides the fact that it exists for every doubly periodic graph) is that the value of αRW is very easy to compute. T Remark 6. In the more general case of a doubly periodic graph but without the assumptions of 3-regularity and irreducibility (but still assuming that the embed 2 ding is balanced), the condition eα is still necessary and sufficient for the walk to be isotropic in the scaling limit – and the proof is essentially the same, so we do not delve into more detail. Remark 7. Of course, in the cases where T has some additional symmetry, αRW T is the same as that obtained in the previous subsection using symmetry . . . Remark 8. One can also look at a simple random walk on the dual graph Tα∗ , and ask for which values of α this dual walk is isotropic in the scaling limit. As it turns out, the modulus one obtains this way is the same as on the initial graph, in other words αRW = αRW T T∗ . This is a very weak version of universality, and unfortunately there doesn’t seem to be a purely discrete proof of it – say, using a coupling of the two walks. There is another natural way to obtain the same condition. We are planning on studying convergence of discrete objects to conformally invariant limits, so it is a good idea to look for discrete-harmonic functions on Tα (with respect to the natural Laplacian, which is the same as the generator of the simple random walk on Tα ). The condition of balanced embedding is exactly equivalent to saying that the identity map is harmonic on Tα ; it is a linear condition, so it does not depend on the value of α. The main difficulty when looking at discrete holomorphic maps is that the product of two such maps is not holomorphic in general. But we are interested in scaling limits, so maybe imposing that such a product is in fact “almost discrete holomorphic” (in the sense that it satisfies the Cauchy-Riemann equations up to an error term which vanishes in the scaling limit) would be sufficient. Whether the previous paragraph makes sense or not – let us investigate whether the map ζ : z → z 2 is discrete-harmonic. For every z ∈ Tα , we can write 1  2 1  1  2 Δζ(z) = (z − z 2 ) = (z + eα )2 − z 2 = eα 3  3 3 z ∼z e∈Ez (T ) e∈Ez (T )  (the term in zeα vanishes because the embedding is balanced). So, if ζ is discrete-harmonic, summing the above relation over z ∈ T gives the very same  2 condition eα = 0 as before; in other words, αRW is the embedding for which T z → z 2 is discrete-harmonic on average. As a last remark, let us investigate how strong the condition of exact harmonicity of ζ is; so assume that α is chosen in such a way that Δζ is identically

Is Critical 2D Percolation Universal?

41

0. Let e be any oriented edge of T ; let e := τ.e and e := τ 2 .e be the two other edges sharing the same source as e. The balance condition on the embedding plus harmonicity of ζ imply the following system:  eα + e α + e α = 0 (2.2) 2 eα + (e α )2 + (e α )2 = 0. Up to rotation and scaling, one can always assume that eα = 1, so the system reduces to e α + e α = −1 and (e α )2 + (e α )2 = −1. Squaring the first of these two relations and subtracting the second, one obtains e α e α = 1, so e α and e α are the two solutions of the equation X2 + X + 1 = 0 which implies that {e α , e α } = {e±2πi/3 }. To sum it up: Proposition 9. The only 3-regular graph on which the map ζ : z → z 2 is discreteharmonic is the honeycomb lattice, embedded in such a way that its faces are regular hexagons. So, imposing ζ to be harmonic not only determines the embedding, it also restricts T to essentially one graph; but in terms of scaling limits, the condition that ζ is harmonic on the average makes as much sense as the exact condition. 2.4. Embedding using circle packings There is another way to specify essentially unique embeddings of triangulations, which is very strongly related to conformal geometry, using the theory of circle packings. It is a fascinating subject in itself and a detailed treatment would be outside of the purpose of these notes, so the interested reader is advised to consult the book of Stephenson [18] and the references therein for the proofs of the claims in this subsection and much more. We first give a version of a theorem of K¨ o¨be, Andreev and Thurston, specialized to our case. It is a statement about triangulations, which is why we actually apply it to T ∗ instead of directly to T . Notice that we do not assume T to be already embedded into the torus T2 . Theorem 10 (Discrete uniformization theorem [18, p. 51]). Let T ∗ be a finite triangulation of the torus, and let Tˆ ∗ be its universal cover. There exists a locally finite family (Cv )v∈V (Tˆ∗ ) of disks of positive radii and disjoint interiors, satisfying the following compatibility condition: Cv and Cv are tangent if, and only if, v and v are neighbors in Tˆ ∗ . Such a family is called a circle packing associated to the graph Tˆ ∗ . It is essentially unique, in the following sense: If (Cv ) is another circle packing associated to Tˆ∗ , then there is a map ϕ : C → C, either of the form z → az + b or of the form z → a¯ z + b, such that for every v ∈ V (Tˆ ∗ ), Cv = ϕ(Cv ). Remark 11. The “existence” part of the above theorem remains true in a much broader class of graphs; essentially all that is necessary is bounded degree and

42

V. Beffara

recurrence of the simple random walk on it. (One can see that a packing exists by completing the graph into a triangulation.) The “uniqueness” part however fails in general, as is made clear as soon as one tries to construct a circle packing associated to the square lattice . . . A consequence of the uniqueness part of the theorem is the following: Let θ : Tˆ ∗ → Tˆ ∗ be a translation along one of the periods of Tˆ ∗ , and let Cv := Cθ(v) ; according to the theorem, let ϕ be such that Cv = ϕ(Cv ) for all v. Up to composition of ϕ by itself, one can always assume that it is of the form ϕ(z) = az + b. By the assumption of local finiteness of the circle packing, one has |a| = 1; besides, the orbits of θ are unbounded, so those of ϕ are too, and in particular it does not have a fixed point, which implies that a = 1 and b = 0. In other words, ϕ is a translation, i.e., the circle packing associated to Tˆ ∗ is itself doubly periodic. As soon as one is given a circle packing associated to a planar graph, it comes with a natural embedding: Every vertex v ∈ V (Tˆ ∗ ) will be represented by the center of Cv , and if v is a neighbor of v, the edge (v, v ) will be embedded as a segment – which is the union of a radius of Cv and a radius of Cv , because those two disks are tangent. One can then specify an embedding of Tˆ by putting each of its vertex at the center of the disk inscribed in the corresponding triangular face of (the embedding of) Tˆ ∗ ; the collection of all those inscribed disks is in fact a circle packing associated with the graph Tˆ (see Figure 3).

Figure 3. The circle packings associated to the graph Ts (solid lines) and its dual (dotted lines). Of more interest to us is the fact that the embedding is itself doubly periodic, by the previous remarks. Up to rotation, and scaling and maybe complex conjugation, one can assume that the period corresponding to the translation by

Is Critical 2D Percolation Universal?

43

1 (resp. i) in T2 is equal to 1 (resp. α ∈ H). Once again, the value of the modulus α is uniquely determined; for further reference, we will denote it by αCP T . ˆ Yet again, as soon as one additional symmetry is present in T , the value of αCP is the same as that obtained using the symmetry; this is again a direct T consequence of the essential uniqueness of the circle packing. 2.5. “Exotic” embeddings Looking closely at Smirnov’s proof, one notices that essentially the only place where the specifics of the graph are used is in the proof of “integrability” or exact cancellation; we will come back to this in the next section, let us just mention that the key ingredient in the phenomenon can be seen to be the fact that ψ(e) (as will be introduced below) is identically 0. This is equivalent to saying that all the triangles of the triangular lattice are equilateral. A way to try and generalize the proof is to demand that all the faces of Tα∗ be equilateral triangles. Of course this cannot be done by embedding it in the plane, even locally – the total angle around a vertex would be equal to 2π only if the degree of the vertex is 6. But one can build a 2-dimensional manifold MT with conic singularities by gluing together equilateral triangles according to the combinatorics of Tˆ ∗ ; since the average degree of a vertex of T ∗ is equal to 6, the average curvature of the manifold (defined e.g. as the limit of the normalized total curvature in large discs) is 0. The manifold MT is not flat in general (the only case where it is being the triangular lattice), but it is homeomorphic to the complex plane, and one can hope to see it as a perturbation of it on which some of the standard tools of complex analysis could have counterparts – the optimal being to be able to perform Smirnov’s proof within it. This is no easy task, and is probably not doable anyway. To relate MT to the topic of this section, one can try to define a module out of it. A good candidate for that is the following: Assume that MT can be realized as a sub-manifold of R3 (or in Rd for d > 2 large enough), in such a way that the (combinatorial) translations on Tˆ act by global translations of the ambient space, thus forming a periodic sub-manifold. Then there is a copy of Z2 acting on it, and the affine plane containing a given point of MT and spanned by the directions of the two generators of that group is at finite Hausdorff distance from it; in other words, this realization of MT looks like a bounded perturbation of a Euclidean plane. One can then look at the orthogonal projections of the vertices of Tˆ ∗ (seen as points of MT ) onto that plane; this creates a doubly periodic, locally finite family of points of the Euclidean plane. It is not always possible to form an embedding of Tˆ ∗ in the plane from it (with disjoint edges); but it does define a value of α as above. Unfortunately, there are cases when this value of α is not well defined, in the sense that it depends on the choice of MT ; this happens if the (infinite) polyhedron associated to Tˆ ∗ , with equilateral faces, is flexible. The simplest example of this phenomenon is to take T to be two periods of the honeycomb lattice in each direction.

44

V. Beffara

2.6. Comparing different methods of embedding We now have at least two (forgetting about the last one) ways of giving a conformal structure to a torus equipped with a triangulation – which is but another way of referring to the choice of α. Assuming that critical percolation does have a scaling limit, it leads to a third choice αPerc of it. T It would be a natural intuition that all these moduli are the same, and correspond to a notion of conformal embedding of a triangulation (or a 3-regular graph) in the plane; and they all have a claim to that name. But this is not true in general: We detail the construction of a counterexample. Start with the graph Ts and its dual Ts∗ ; and refine one of the “vertical” triangular faces of Ts∗ by adding a vertex in the interior of it, connected to its three vertices. In terms of the primal graph, this correspond to replacing one of its vertices by a triangle – see Figure 4. Let Ts be the graph obtained that way; we will refer to such a splitting as a refinement, and to the added vertex as a new vertex.

Figure 4. Square (but not balanced) embeddings of Ts (solid) and its dual (dotted); the origin is taken as a point of Ts on the left, and as a point of the dual on the right, corresponding to the ones chosen for Figures 1 and 2. In terms of circle packings, this changes essentially nothing; the new vertex of (T Ts )∗ can be realized as a new disc without modifying the rest of the configuration (cf. Figure 5). In terms of random walks, however, adding edges will modify the covariance matrix in the central limit theorem. The computation can be done easily, as explained above, and one gets the following values: # 6 CP RW αCP = i = αRW αTs = i Ts = i = αTs ; T . 7 $ In this particular case, the value of αRW (T T  )∗ is also i 6/7. s

RW

CP

So, α and α are different in general. Is αPerc (provided it exists) one of them? An easy fact to notice is the following: Let T ∗ be a triangulation of the torus and let (T )∗ be obtained from it by splitting a triangle into 3 as in the construction of Ts . Then, consider two realizations of site-percolation at pc = 1/2 on both universal covers, coupled in such a way that the common vertices are in the

Is Critical 2D Percolation Universal?

45

Figure 5. Splitting a face of a triangulation into 3 triangles, and the corresponding modification of its circle packing (added features represented by dashed lines). same state for both models. In other words, start with a realization of percolation on Tˆ ∗ and without changing site states, refine a periodic family of triangles of it into 3, choosing the state of each new vertex independently of the others and of the configuration on Tˆ ∗ . If there is a chain of open vertices in Tˆ ∗ , this chain is also a chain of open vertices in the refined graph – because all the edges are preserved in the refinement. Conversely, starting from a chain of open vertices in the refinement and removing each occurrence of a new vertex on it, one obtains a chain of open vertices in Tˆ ∗ ; the reason for that being that the triangle is a complete graph. Another way of stating the same fact is to say that opening (resp. closing) one of the new vertices cannot join two previously disjoint open clusters (resp. split a cluster into two disjoint components); they cannot be pivotal for a crossing event. Hence, the probability that a large conformal rectangle is crossed is the same in both cases (at least if the choice of the discrete approximation of its boundary is the same for both graphs, which in particular implies that it contains no new vertex), and so is fα for every choice of α (still assuming that it exists, of course). If one is conformally invariant in the scaling limit, the other also has to be. In short, αPerc = αPerc T T . Looking at circle packings instead of percolation, we get the same identity (as was mentioned in the particular case of Ts ), with a very similar proof: Adding a vertex does not change anything to the rest of the picture, and we readily obtain CP αCP T = αT  .

This leads us to the following hope, which we state as a conjecture even though it is much closer to being wishful thinking: Conjecture 12. Let T ∗ be a triangulation of the torus. Then, the critical parameter for site-percolation on its universal cover Tˆ ∗ is equal to 1/2, and for every α ∈ H,

46

V. Beffara

critical site-percolation on Tˆα∗ has a scaling limit. The value of the modulus α for which the model is conformally invariant in the scaling limit is that obtained from the circle packing associated to Tˆ ∗ : αPerc = αCP T T .

3. Critical percolation on the triangular lattice For reference, and as a way of introducing our general strategy, we give in this section a very shortened version of Smirnov’s proof of the existence and conformal invariance of the scaling limit for critical site-percolation on the triangular lattice Th . The interested reader is advised to consult our previous note [3] for an “extended shortening”, or Smirnov’s article [16] for the original proof; see the book of Bollobas ´ and Riordan [5] for a more detailed treatment. Up to cosmetic changes, we follow the notation of [3]. Remark 13. Up to the last paragraph of the section, we are not assuming that the lattice we are working with is the honeycomb lattice; our only assumption is that we have an a priori bound for crossing probabilities of large rectangles which depends on their aspect ratio but not on their size (we “assume Russo-SeymourWelsh conditions”). It is not actually clear how general those are; all the standard proofs require at least some symmetry in the lattice in addition to periodicity, but it is a natural conjecture that periodicity is enough. Here and in the remainder of this paper, τ := e2πi/3 will be the third root of unity with positive imaginary part. Let T be a finite graph of genus 1, Tα an embedding of modulus α of T in the complex plane; let V (T Tα ) (resp. E(T Tα )) be the set of vertices (resp. oriented edges) of Tα . Each vertex z ∈ V (T Tα ) has three neighbours; let Ez (T Tα ) be the set of the three oriented edges in E(T Tα ) having their source at z. That set can be cyclically ordered counterclockwise; if e ∈ Ez (T Tα ) is one of the three edges starting at z, we will denote by τ.e (resp. τ 2 .e) the next (resp. second to next) edge in the ordering. Remark 14. In the particular case of the honeycomb lattice, seeing each edge as a complex number (being the difference between its target and its source), the notation τ.e corresponds to complex multiplication by e2πi/3 – in other words, τ.e = τ e as a product of complex numbers. That is of course not the case in general, but we keep the formal notation for clarity. In what follows, whenever an algebraic expression involves the product of a complex number by an edge of Tα or Tα∗ , as above the edge will be understood as the difference, as a complex number, between its target and its source; we will never use formal linear combinations of edges. The notation τ.e (with a dot) will be reserved for the “topological” rotation within Ez (T Tα ). Let again Ω be a smooth Jordan domain in the complex plane, and let A, B, C and D be three points on its boundary, in that order when following ∂Ω

Is Critical 2D Percolation Universal?

47

counterclockwise. Let Ωδ be the largest connected component of Ω ∩ δT Tα , and let Aδ (resp. Bδ , Cδ , Dδ ) be the point of Ωδ that is closest to A (resp. B, C, D). The main result in Smirnov’s paper ([16]) is the following: Theorem 15 (Smirnov). In the case where Tα is the honeycomb √ lattice, embedded so as to make its faces regular hexagons (i.e., when α = i 3/3), critical sitepercolation has a conformally invariant scaling limit. If Ω is an equilateral triangle with vertices A, B and C, then fα (Ω, A, B, C, D) =

|CD| . |CA|

Knowing this particular family of values of fα is enough, together with conformal invariance, to compute it for any conformal rectangle. The formula obtained for a rectangle is known as Cardy’s formula. To each edge e ∈ E(T Tα ) corresponds its dual oriented edge e∗ ∈ E(T Tα∗ ), ∗ oriented in such a way that the angle (e, e ) is in (0, π). If −e denotes the edge with the same endpoints as e but the reverse orientation, then we have e∗∗ = −e. Define ψ(e) := e∗ + τ (τ.e)∗ + τ 2 (τ 2 .e)∗ (where as above we interpret the edges e∗ , (τ.e)∗ and (τ 2 .e)∗ as complex numbers). It is easy to check that ψ(e) = 0 if, and only if, the face of Tα∗ corresponding to the source of e is an equilateral triangle; so, ψ(e) can be seen as a measure of the local deviation between Tα and the honeycomb lattice. An identity which will be useful later is the following:  ∀z ∈ V (T Tα ), ψ(e) = 0. (3.1) e∈Ez (T Tα )

For every z ∈ Ωδ , let EA,δ (z) be the event that there is a simple path of open vertices of Ω∗δ , joining two points of the boundary of the domain, which separates z and A from B and C; let HA,δ := P [EA,δ (z)]. Define similar events for points B and C by a circular permutation of the letters, and let Sδ (z) := HA,δ (z) + HB,δ (z) + HC,δ (z), Hδ (z) := HA,δ (z) + τ HB,δ (z) + τ 2 HC,δ (z). It is a direct consequence of Russo-Seymour-Welsh estimates that these functions are all H¨ ¨older with some universal positive exponent, with a norm which does not depend on δ, so by Ascoli’s theorem they form a relatively compact family, and as δ → 0 they have subsequential limits which are H¨¨older maps from Ω to C; all that is needed is prove that only one such limit is possible. The key argument is to show that if h (resp. s) is any subsequential limit of (H Hδ ) (resp. (Sδ )) as δ → 0, then h and s are holomorphic; indeed, assume for a moment that they are. Since s is also real-valued, it has to be constant, and its value is 1 by boundary conditions (e.g. at point A). On the other hand, along the boundary arc (Aδ Bδ ) of ∂Ωδ , HC,δ is identically 0, so the image of the arc (AB)

48

V. Beffara

by h is contained in the segment [1, τ ] of C; and similar statements hold mutatis mutandis for the arcs (BC) and (CA). By basic index theory, this implies that h is the unique conformal map sending Ω to the (equilateral) triangle of vertices 1, τ and τ 2 , and that is enough to characterize it and to finish the proof of Theorem 15. So, the crux of the matter, as expected, is to prove that the map h has to be holomorphic. The most convenient way to do that is to use Morera’s theorem, which states that h is indeed holomorphic on Ω if, and only if, its integral along any closed, smooth curve contained in Ω is equal to 0. Let γ be such a curve, and let γδ = (z0 , z1 , . . . , zLδ = z0 ) be a closed chain of vertices of Ωδ which approximates it within Hausdorff distance δ and has O(δ −1 ) points. Because the functions Hδ are uniformly H¨ o¨lder, it follows that % % L δ −1 Hδ (z)dz := Hδ (zk )(zk+1 − zk ) → h(z)dz. γδ

γ

k=0

We want to prove that, for a suitable choice of α, the discrete integral on the left-hand side of that equation vanishes in the scaling limit. If e = (z, z ) is an oriented edge of Ωδ , define PA,δ (e) := P [EA,δ (z )\EA,δ (z)]; define PB,δ and PC,δ similarly. A very clever remark due to Smirnov, which is actually the only place in his proof where specifics of the model (as opposed to the lattice) are used, is that one can use color-swapping arguments to prove that, for every oriented edge, PA,δ (e) = PB,δ (τ.e) = PC,δ (τ 2 .e).

(3.2)

On the other hand, since differences of values of Hδ between points of Ωδ can be computed in terms of these functions P·,δ , the discrete integral above can be rewritten using them: Letting E(γδ ) be the set of edges contained in the domain surrounded by γδ and using (3.2), one gets %  Hδ (z)dz = ψ(e)P PA,δ (e) + o(1). (3.3) γδ

e∈E(γδ )

A similar computation, together with the fact that e∗ +(τ.e)∗ +(τ 2 .e)∗ is identically equal to 0, leads to % Sδ (z)dz = o(1).

(3.4)

γδ

We again refer the reader to [3] for the details of this construction. Notice that it already implies that s is holomorphic, hence constant equal to 1, independently of the value of α; so, whether h is holomorphic or not, it will send ¯ to the triangle of vertices 1, τ and τ 2 anyway. In the case of the triangular lattice Ω embedded in the usual way, ψ(e) is also identically equal to 0, as was mentioned above, so h is itself holomorphic, and the proof is complete. The remainder of these notes is devoted to some ideas about how to extend the general framework of the proof to more general cases; it is not clear how close

Is Critical 2D Percolation Universal?

49

one is to a proof, but it is likely that at least one fundamentally new idea will be required. However, we do believe that the overall strategy which we will now describe is the right angle of attack of the problem. Do not expect to find any formal proof in what follows, though.

4. Other triangulations 4.1. Using local shifts The first natural idea when trying to generalize the construction of Smirnov is to try an apply it to more general periodic triangulations of the plane. Indeed, in all that precedes, up to and including Equation (3.3), nothing is specific to the regular triangular lattice, only Russo-Seymour-Welsh conditions (and their corollary that pc = 1/2) are needed. It is only at the very last step, noticing that ψ was identically equal to 0, that the precise geometry was needed. The key fact that makes hope possible is the following (and it is actually similar to one of the points we made earlier): In the expression of the discrete integral as a sum over interior edges, each term is the product of two contributions: • ψ(e) which depends on the geometry of the embedding, and through that on the value of α; • PA,δ (e) which is only a function of the combinatorics of Ωδ . Even though Ωδ as a graph does depend on the choice of α, one can make the following remark: Applying the transformation ϕβ (for some β ∈ H) to both the domain Ω and the lattice δT Tα does not change Ωα as a graph. In particular it does not change the value of PA,δ (e). One can then see the whole sum as a function β, say SΩ,δ (β). Because ϕβ (z) is a real-affine function of β, so is SΩ,δ ; one can then try to solve the equation SΩ,δ (β) = 0 in β. Using the corresponding ϕβ , one gets a joint choice of a domain, a lattice modulus and mesh, and a curve γ making the discrete contour integral vanish. If the modulus thus obtained actually did not depend on Ω, δ or γ, we would be done – call it αPerc and there is only bookkeeping left to do. However we do not T even know whether it has a limit as δ ↓ 0 . . . An alternative is  as follows. Because the lattice is periodic, it makes sense to first look at the sum ψ(e)P PA,δ (e) over one fundamental domain. If that is small, then over the copy of the fundamental domain immediately to the right of the previous one, the terms ψ(e) are exactly the same, and one is lead to compare PA,δ for two neighboring pre-images of a given edge of T . So, let e be an edge of Ωδ , and let e + δ be its image by a translation of one period to the right. Making the dependency on the shape of the domain explicit in the notation, one can replace the translation of e by a translation of the domain itself and the boundary points in the opposite direction, to obtain Ω−δ Ω PA,δ (e + δ) = PA,δ (e).

(4.1)

50

V. Beffara

To estimate the difference between this term and the corresponding one in Ω, one can consider coupling two realizations of percolation, one on Ωδ and the other in Ωδ − δ, so that they coincide on the intersection between the two. Ω The event corresponding to PA,δ (e) is that there is an open simple path separating the target of e and A from B, and C, and that no open simple path separates the source of e and A from B and C; this is equivalent to the existence of 3 disjoint paths from the 3 vertices of the face at the source of e to the 3 “sides” of the conformal triangle (Ω, A, B, C), two of them being formed of open vertices and the third being formed of closed vertices – cf. Figure 6. For this to happen in Ω but not in Ω− δ, one of these arms needs to go up to ∂Ω but not to ∂(Ω− δ), and the only way for this to be realized is for a path of the opposite color to prevent it; Ω−δ Ω this can be done in finitely many ways, Figure 6 being one of them; PA,δ − PA,δ can then be written as the linear combination of the probabilities of finitely many terms of that form – half of these actually corresponding to the reversed situation, where arms go up to ∂(Ω − δ) but not up to ∂Ω. B

e

C A

Ω−δ Ω Figure 6. A typical case contributing to PA,δ (e) − PA,δ (e). The original domain boundary is represented by a solid line, that of the shifted domain by a dashed-and-dotted line; open (resp. closed) arms from the source of e are represented as solid (resp. dotted) lines, and the additional open path preventing the closed arm from connecting to the boundary of Ωδ − δ is represented as a dashed curve.

In the case corresponding to Figure 6, and all the similar ones, one sees that 3 arms connect the source of e to the boundary of Ω ∩ (Ω − δ), and on at least one point of that boundaries there have to be 3 disjoint arms of diameter of order 1. There are O(δ −1 ) points on the boundary, and the probability that 3 such arms exist from one of them is known – at least in the case of a polygon, which is enough for our purposes – to behave like δ 2 , see, e.g., [17].

Is Critical 2D Percolation Universal?

51

Another possible reason for the non-existence of 3 arms from the source of e to the correct portions of the boundary of Ω − δ (say) is that one of the corresponding arms in Ω actually lands very close to either A, B or C: Preventing it from touching the relevant part of ∂(Ω − δ) requires only one additional arm from a δ-neighborhood of that vertex – i.e., a total of 2 arms of diameter of order 1. The probability for that (see [17] also), still in the case when Ω is a polygon with none of A, B or C as a vertex, behaves like δ. Fortunately, there are only 3 corners on a conformal triangle, so the contribution of these cases is of the same order as previously Putting everything together, one gets an estimate of the form Ω−δ Ω PA,δ (e) = PA,δ (e) [1 + O(δ)] .

(4.2)

Coming back to our current goal, let E be the set of oriented edges in a given period of Ωδ , and let E + δ be its image by the translation of vector δ. Then,   Ω Ω ψ(e)P PA,δ (e) = ψ(e)P PA,δ (e) [1 + O(δ)] e∈E

e∈E+δ

=



Ω ψ(e)P PA,δ (e) + O(δ 2+η )

e∈E

with η > 0; the existence of such an η is ensured by Russo-Seymour-Welsh type arguments again, which ensure that, uniformly in e and δ, for every edge Ω e, PA,δ (e) = O(δ η ). Now, if that is the way the proof starts, what needs to be done is quite clear: • Fix a period E of the graph. Ω • Choose α so that the previous sum, over this period, of ψ(e)P PA,δ (e) is equal to 0. • Use the above estimate to give an upper bound for the same sum on neighboring periods. • Try to somehow propagate the estimate up to the boundary. The last part of the plan is the one that does not work directly, because one needs of the order of δ −1 steps to go from E to ∂Ω, and the previous bound is not small enough to achieve that; one would need a term of the order of O(δ 3+η ). It is however quite possible that a more careful decomposition of the events would lead to additional cancellation, though we were not able to perform it. 4.2. Using incipient infinite clusters Another idea which might have a better chance of working out  is based on the idea of incipient infinite clusters. We are trying to ensure that ψ(e)P PA,δ (e) is equal to o(δ 2 ) over a period for a suitable choice of α; but for it to be exactly equal to 0 depends only on the ratios PA (e)/P PA (e ) within the period considered, and not on their individual values. One can then let δ go to 0, or equivalently let Ω increase to cover the whole space, and look at this ratio.

52

V. Beffara

Proposition 16. There is a map π : E(Tˆ) → (0, +∞) such that the following happens. Let e, e be two edges of Tα , which we identify with Tˆ for easier notation, and let δ = 1. Then, as Ω increases to cover the whole plane, Ω PA, 1 (e) Ω (e ) PA, 1

→

π(e) , π(e )

uniformly in the choices of A, B and C on ∂Ω. The map π is periodic and does not depend on the choice of α. Proof. The argument is very similar to Kesten’s proof of existence of the incipient infinite cluster (see [11]); it is based on Russo-Seymour-Welsh estimates. It will appear in an upcoming paper [2]. Notice that there is no requirement for A, B and C to remain separated from each other; this is similar to the fact that the incipient infinite cluster is also the limit, as n → ∞, of critical percolation conditioned to the event that the origin is connected to the point (n, 0) – which in turn is again a consequence of Russo-Seymour-Welsh theory. The speed of convergence is certainly different with and without such restrictions on the positions of A, B and C, though.  Seeing this proposition, one is tempted to define α by solving the equation  ψ(e)π(e) = 0, (4.3) e∈E

where again the sum is taken over one period of the lattice. Indeed, all that remains in the sum, over the same period of the lattice, of ψ(e)P PA (e) is composed of terms of a smaller order. However, because the limit taken to define π is uniform in the choices of A, B and C, in particular it is invariant by re-labelling of the corners of the conformal triangle; equivalently, taking PB instead of PA leads to the same limit. Combining this remark with Equation (3.2), one gets the following identities: ∀e ∈ E(Tˆ ),

π(e) = π(τ.e) = π(τ 2 .e).

(4.4)

In other words, π(e) only depends on the source of e. For every edge e = (z, z ), let π(z) := π(e): If V is a period of V (Tˆ), one has    ψ(e)π(e) = π(z) ψ((z, z )) = 0 e∈E

z∈V

z  ∼z

by using the remark in Equation (3.1). So, the equation (4.3) is actually always true, and does not help in finding the value of α . . . This is actually good news, because it is the sign of emerging cancellations in the scaling limit, which were not at first apparent; that means that the relevant terms in (3.3) are actually smaller than they look at first sight, which in turn means that making the leading term equal to 0 by the correct choice of α leads to even smaller terms.

Is Critical 2D Percolation Universal?

53

Whether the overall strategy can be made to work actually depends on the speed of convergence in the statement of Proposition 16. In the case of the triangular lattice, one can actually use SLE to give an explicit expansion of the ratio PA (e)/P PA (e ) as Ω increases, at least in some cases; this is the subject of an upcoming paper [2].

5. Other lattices 5.1. Mixed percolation We conclude the speculative part of these notes by some considerations about bond-percolation on the planar square lattice. The combinatorial construction we perform here does apply to more general cases, but the probabilistic arguments which follow do not, so we restrict ourselves to the case of Z2 . The general idea it to map the problem of bond-percolation on Z2 to one of site-percolation on a suitable triangulation of the plane. Then, if the arguments in the previous section can be made to work, one could potentially prove the existence and conformal invariance of a scaling limit of critical percolation on the square lattice. The key remark was already present in the book of Kesten [10]: For any bondpercolation model on a graph, one can construct the so-called covering graph on which it corresponds to site-percolation. More specifically, let G1 be a connected graph with bounded degree; as usual, let E(G1 ) be the set of its edges and V (G1 ) be the set of its vertices. We construct a graph G2 as follows: The set V (G2 ) of its vertices is chosen to be E(G1 ), and we put an edge between two vertices of G2 if, and only if, the corresponding edges of G1 share an endpoint. Notice that even if G1 is assumed to be planar, G2 does not have to be – see Figure 7 for the case of Z2 .

Figure 7. The square lattice Z2 and its covering graph. The graph thus obtained from the square lattice is isomorphic to a copy of the square lattice where every second face, in a checkerboard disposition, is completed into a complete graph with 4 vertices. The next remark is the following: in terms of

54

V. Beffara

site-percolation, a complete graph with 4 vertices behaves the same way as a square with an additional vertex at the center, which is open with probability 1 – with the same meaning as when we looked at refinement of triangles in triangulations, i.e., taking a chain of open vertices in the partially centered square lattice and removing from it the vertices which are face centers leads to a chain of open vertices in the covering graph of Z2 . So, let again Gs be the centered square lattice, as was introduced above, and let q ∈ [0, 1]; split the vertices of Gs into three classes, to defined a nonhomogeneous site-percolation model, as follows. Each vertex is either open or closed, independently of the others, and: • The sites of Z2 are open with probability p = 1/2; we will call them vertices of type I, I or p-sites for short, and denote by V1 the set of such vertices. • The vertices of coordinates (k + 1/2, l + 1/2) with k + l even are open with probability q; we will call them vertices of type II, I or q-sites for short, and denote by V2 the set of such vertices. • The vertices of coordinates (k + 1/2, l + 1/2) with k + l odd are open with probability 1 − q; we will call them vertices of type III, I or (1 − q)-sites for short, and denote by V3 the set of such vertices. We will refer to that model as mixed percolation with parameters p = 1/2 and q, and denote by P1/2,q the associated probability measure. Two cases are of particular interest: • If q = 1/2, the model is exactly critical site-percolation on the centered square lattice Gs . • If q = 0 or q = 1 (the situation is the same in both cases up to a translation), from the previous remarks mixed percolation then corresponds to critical bond-percolation on the square lattice. Besides, all the models obtained for p = 1/2 are critical and satisfy Russo-SeymourWelsh estimates. 5.2. Model interpolation We are now equipped to perform an interpolation between the models at q = 0 and q = 1/2. Let (Ω, A, B, C, D) be a simply connected subset of Z2 equipped with 4 boundary points – say, a rectangle; let U = UΩ,A,B,C,D be the event, under mixed percolation with parameters p = 1/2 and q, that there is a chain of open vertices of Ω joining the boundary arcs (AB) and (CD). To estimate the difference between the probabilities of U for the two models we are most interested in, simply write & 1/2 ∂ P1/2,1/2 [U ] − P1/2,0 [U ] = P1/2,q [U ] dq. (5.1) ∂q 0 If percolation is indeed universal, then one would expect cancellation to occur, hopefully for each value of q; the optimal statement being of the form ∂ lim2 P1/2,q [U ] = 0. sup sup (5.2) Ω↑Z A,B,C,D∈∂Ω q∈(0,1) ∂q

Is Critical 2D Percolation Universal?

55

The main ingredient in the estimation of the derivative in q is, as one might expect, a slight generalization of Russo’s formula; to state it, we need a definition: Definition 17. Consider mixed percolation on Gs , and let E be a cylindrical increasing event for it (i.e., an event which depends on the state of finitely many vertices). Given a realization ω of the model, we say that a vertex v is pivotal for the event E if E is realized for the configuration ω v where v is made open, and not realized for the configuration ωv where v is made closed. We will denote by Piv(E) the (random) set of pivotal vertices for E. Proposition 18. With the above notation, one has ∂ P1/2,q [U ] = E1/2,q [|Piv(U ) ∩ Ω ∩ V2 | − |Piv(U ) ∩ Ω ∩ V3 |] . ∂q Proof. The argument is the same as in the proof of the usual formula (in the case of homogeneous percolation); we refer the reader to the book of Grimmett [8].  As was the case in the previous section, one can relate the event that a given site is pivotal to the presence of disjoint arms in the realization of the model, with appropriate color. More precisely, a q-site (say) v ∈ Ω is pivotal if, and only if, the following happens: v is at the center of a face of Z2 ; two opposite vertices of that face are connected respectively to the boundary arcs (AB) and (CD) by disjoint chains of open vertices; the other two vertices of the face are connected respectively to the boundary arcs (BC) and (AD) by disjoint chains of closed vertices; and none of the chains involved contains the vertex v. To state the previous description more quickly, there is a 4-arm configuration with alternating colors at vertex v, and the endpoints of the arms are appropriately located on ∂Ω – see Figure 8.

B C v

Ω D

A

Figure 8. A four-arm configuration at vertex v making it pivotal for the event U (Ω, A, B, C, D).

56

V. Beffara

The main feature of mixed percolation in the case of the centered square lattice is the following: Starting from a configuration sampled according to P1/2,q and shifting the state of all vertices by one lattice mesh to the right, or equivalently flipping the state of all vertices, or rotating the whole configuration by an angle of π/2 around a site of type I, one gets a configuration sampled according to P1/2,1−q ; on the other hand, rotating the picture by π/2 around a vertex of type II or III leaves the measure invariant. Notice that the existence of 4 arms of alternating colors from a given vertex v is invariant by color-swapping; the configuration in Figure 8 is not though, because the arms obtained after the color change connect the neighbors of v to the wrong parts of the boundary. Nevertheless, one can try to apply the same reasoning as in the previous section, as follows: Let v be the vertex that is one lattice step to the right of v. If v is a vertex of type II, then v is a vertex of type III, and up to boundary terms, one can pair all the q-sites in Ω to corresponding (1 − q)-sites. To estimate the right-hand term in the statement of Proposition 18, let Δ(v) := P [v ∈ Piv(U )] − P [v ∈ Piv(U )]. Our goal will be achieved if one is able to show that Δ(v) = o(|Ω|−1 ); or equivalently, if Ωδ is obtained from a fixed continuous domain by discretization with mesh δ, if one has Δ(v) = o(δ 2 ). In the case of critical site-percolation on the triangular lattice, arguments using SLE processes give an estimate to the probability that a vertex is pivotal, and from universality conjectures it is natural to expect that they extend to the case of mixed percolation on Ts . They involve the 4-arm exponent of percolation, and would read (still in the case of a fixed domain discretized at mesh δ) as P [v ∈ Piv(U )] ≈ δ 5/4 . So, shifting the domain instead of the point as we did in the last section, one would expect an estimate on Δ(v) of the order Δ(v) ≈ δ 9/4 (where the addition of 1 in the exponent corresponds to the presence of a 3-arm configuration at some point on the boundary on either the original domain or its image by the shift). Since 9/4 > 2, that would be enough to conclude. However, this approach does not work directly, because of the previous remark that the shift by one lattice step does change the measure, replacing q by 1 − q. If one is interested in the mere existence of the 4 arms around a vertex, combining the shift with color-flipping is enough to cancel the effect; but the estimate one obtains that way is of the form P [v ∈ Piv(U UΩ,A,B,C,D )] − P [v ∈ Piv(U UΩ,B,C,D,A )] ≈ δP [v ∈ Piv(U UΩ,A,B,C,D )] (5.3)

Is Critical 2D Percolation Universal?

57

(and Russo-Seymour-Welsh estimates are actually enough to obtain a formal proof of this estimate). So, once again, what is missing is a way to estimate how much P [v ∈ Piv(U UΩ,A,B,C,D )] depends on the location of A, B, C and D along ∂Ω; if the dependency is very weak, then the estimate in Equation (5.3) might actually be of the right order of magnitude. Once again, it is likely that the way to proceed is to use a modified version of the incipient infinite cluster conditioned to have 4 arms of alternating colors from the boundary, and that the order of magnitude of Δ(v) will be related to the speed of convergence of conditioned percolation to the incipient clusters; but we were not able to conclude the proof that way. It would seem that this part of the argument is easier to formalize than that of the previous section, though, and hopefully a clever reader of these notes will be able to do just that . . .

References [1] Michel Bauer and Denis Bernard, Conformal field theories of stochastic Loewner evolutions, Comm. Math. Phys. 239 (2003), no. 3, 493–521. [2] Vincent Beffara, Quantitative estimates for the incipient infinite cluster of 2D percolation, in preparation. [3] , Cardy’s formula on the triangular lattice, the easy way, Universality and Renormalization (Ilia Binder and Dirk Kreimer, eds.), Fields Institute Communications, vol. 50, The Fields Institute, 2007, pp. 39–45. [4] Itai Benjamini and Oded Schramm, Conformal invariance of Voronoi percolation, Comm. Math. Phys. 197 (1998), no. 1, 75–107. [5] B´ ´ela Bollob´ as and Oliver Riordan, Percolation, Cambridge University Press, 2006. [6] John Cardy, Critical percolation in finite geometries, J. Phys. A 25 (1992), L201– L206. [7] , Conformal invariance in percolation, self-avoiding walks, and related problems, Ann. Henri Poincar´e 4 (2003), no. suppl. 1, S371–S384. [8] Geoffrey Grimmett, Percolation, second ed., Grundlehren der Mathematischen Wissenschaften, vol. 321, Springer-Verlag, Berlin, 1999. [9] Galin L. Jones, On the Markov chain central limit theorem, Probability Surveys 1 (2004), 299–320. [10] Harry Kesten, Percolation theory for mathematicians, Progress in Probability and Statistics, vol. 2, Birkh¨ ¨ auser, Boston, Mass., 1982. [11] , The incipient infinite cluster in two-dimensional percolation, Probab. Theory Related Fields 73 (1986), no. 3, 369–394. [12] R.P. Langlands, C. Pichet, Ph. Pouliot, and Y. Saint-Aubin, On the universality of crossing probabilities in two-dimensional percolation, J. Statist. Phys. 67 (1992), no. 3-4, 553–574. [13] Robert Langlands, Yves Pouillot, and Yves Saint-Aubin, Conformal invariance in two-dimensional percolation, Bulletin of the A.M.S. 30 (1994), 1–61.

58

V. Beffara

[14] Oded Schramm, Scaling limits of loop-erased random walks and uniform spanning trees, Israel Journal of Mathematics 118 (2000), 221–288. [15] Stanislav Smirnov, Critical percolation in the plane: Conformal invariance, Cardy’s formula, scaling limits, C.R. Acad. Sci. Paris S´´er. I Math. 333 (2001), no. 3, 239–244. [16] , Critical percolation in the plane. I. Conformal invariance and Cardy’s formula. II. Continuum scaling limit, http://www.math.kth.se/ stas/papers/percol.ps, 2001. [17] Stanislav Smirnov and Wendelin Werner, Critical exponents for two-dimensional percolation, Mathematical Research Letters 8 (2001), 729–744. [18] Kenneth Stephenson, Introduction to circle packing, Cambridge University Press, 2005. Vincent Beffara CNRS – UMPA ´ Ecole normale sup´ ´erieure de Lyon 46 all´ ´ee d’Italie F-69364 Lyon cedex 07 France e-mail: [email protected]

Progress in Probability, Vol. 60, 59–96 c 2008 Birkhauser  ¨ Verlag Basel/Switzerland

A New REM Conjecture Gerard ´ Ben Arous, V´´eronique Gayrard and Alexey Kuptsov Abstract. We introduce here a new universality conjecture for levels of random Hamiltonians, in the same spirit as the local REM conjecture made by S. Mertens and H. Bauke. We establish our conjecture for a wide class of Gaussian and non-Gaussian Hamiltonians, which include the p-spin models, the Sherrington-Kirkpatrick model and the number partitioning problem. We prove that our universality result is optimal for the last two models by showing when this universality breaks down. Mathematics Subject Classification (2000). 82B44, 60F99. Keywords. Statistical mechanics, disordered media, spin-glasses.

1. Introduction S. Mertens and H. Bauke recently observed ([Mer00], [BM04], see also [BFM04]) that the statistics of energy levels for very general random Hamiltonians are Poissonian, when observed micro-canonically, i.e., in a small window in the bulk. They are universal and identical to those of the simplest spin-glass model, the Random Energy Model or REM, hence the name of this (numerical) observation: the REM conjecture or more precisely the local REM conjecture. This local REM conjecture was made for a wide class of random Hamiltonians of statistical mechanics of disordered systems, mainly spin-glasses (mean field or not), and for various combinatorial optimization problems (like number partitioning). Recently, two groups of mathematicians have established this conjecture in different contexts: C. Borgs, J. Chayes, C. Nair, S. Mertens, B. Pittel for the number partitioning question (see [BCP01], [BCMN05a], [BCMN05b]), and A. Bovier, I. Kurkova for general spin-glass Hamiltonians (see [BK06a], [BK06b]). We introduce here a new kind of universality for the energy levels of disordered systems. We believe that one should find universal statistics for the energy levels of a wide class of random Hamiltonians if one re-samples the energy levels, i.e., draws a random subset of these energies. Put otherwise, our conjecture is thus that the level statistics should also be universal, i.e., Poissonian, when

60

G. Ben Arous, V. Gayrard and A. Kuptsov

observed on a large random subset of the configuration space rather than in a micro-canonical window. We establish this new universality result (which could be called the re-sampling REM universality or the REM universality by dilution) for general (mean-field) spin-glass models, including the case of number partitioning and for large but sparse enough subsets. This approach has the following interesting property: the range of energies involved is not reduced to a small window as in the local REM conjecture. Thus we can study the extreme value distribution on the random subset, by normalizing the energies properly. Doing so we establish that the Gibbs measure restricted to a sparse enough random subset of configuration space has a universal distribution which is thus the same as for the REM, i.e., a Poisson-Dirichlet measure. To be more specific, we specialize our setting to the case of random Hamiltonians (H HN (σ))σ∈SN defined on the hypercube SN = {−1, 1}N . We want to consider a sparse random subset of the hypercube, say X, and the restriction of the function HN to X. We introduce the random point process  δHN (σ) , (1.1) PN = σ∈X

with a normalization:

HN (σ) − aN (1.2) bN to be chosen. Our conjecture specializes to the following: the asymptotic behavior of the random point process PN is universal, for a large class of Hamiltonians HN , for appropriate sparse random subsets X, and appropriate normalization. We will only study here the simplest possible random subset, i.e., a site percolation cluster X = {σ ∈ SN : Xσ = 1}, where the random variables (Xσ )σ∈SN are i.i.d. and Bernoulli: 2M (1.3) P(Xσ = 1) = 1 − P(Xσ = 0) =: pN = N . 2 Thus the mean size of X is 2M and we will always assume that X is not too small, e.g., that log N = o(2M ). We will sometimes call X a random cloud. In order to understand what the universal behavior should be, let us examine the trivial case where the HN (σ) are i.i.d. centered standard Gaussian random variables, i.e., the case of the Random Energy Model. Then, standard extreme value theory proves that if # $ 1 aN = 2M log 2 + 2 log bN − log 2 and bN = , (1.4) M then PN converges to a Poisson point process with intensity measure √ 1 (1.5) μ(dt) = √ e−t 2 log 2 dt. π (σ) = HN

We will now fix the normalization needed in (1.2) by choosing aN and bN as in (1.4). The basic mechanism of the REM universality we propose is that the influence of correlations between the random variables HN (σ) should be negligible

A New REM Conjecture

61

when the two-point correlation (the covariance) is a decreasing function of the Hamming distance dH (σ, σ ) = #{i ≤ N : σi = σi } (1.6) and when the random cloud is sparse enough. We first establish this universality conjecture for a large class of Gaussian Hamiltonians. This class contains the Sherrington-Kirkpatrick (SK) model as well as the more general p-spin models. It also contains the Gaussian version of the number partitioning problem. Consider a Gaussian Hamiltonian HN on the hypercube SN = {−1, 1}N such that the random variables (H HN (σ))σ∈SN are centered and whose covariance is a smooth decreasing function of the Hamming distance or, equivalently, a smooth N  increasing function ν of the overlap R(σ, σ ) = N1 σi σi : i=1

 2dH (σ, σ ) . (1.7) cov(H HN (σ), HN (σ )) = ν(R(σ, σ )) = ν 1 − N We will always assume that ν(0) = 0 and that ν(1) = 1. The first assumption is crucial, since it means that the correlation of the Hamiltonian vanishes for pairs of points on the hypercube which are at a typical distance. The second assumption simply normalizes the variance of HN (σ) to 1. This type of covariance structure can easily be realized for ν real analytic, of the form:  ν(r) = a2p rp . (1.8) p≥1

Indeed such a covariance structure can be realized by taking mixtures of p-spin models. Let HN,p be the Hamiltonian of the p-spin model given by  1 HN,p (σ) = (p−1)/2 gi1 ,i2 ,...,ip σi1 σi2 . . . σip , (1.9) N 1≤i ,i ,...,i ≤N 1

2

p

where the random variables (gi1 ,i2 ,...,ip )1≤i1 ,i2 ,...,ip ≤N are independent standard Gaussians defined on a common probability space (Ωg , F g , P). Then 1  ap HN,p (σ) (1.10) HN (σ) = √ N p≥1 has the covariance structure given in (1.7)–(1.8) Let us recall that the case where ν(r) = r is the Gaussian version of the number partitioning problem ([BCP01]), the case where ν(r) = r2 is the SK model, and more generally, ν(r) = rp defines the pure p-spin model. Let us normalize HN as above (see (1.4)): HN (σ) − aN , bN and consider the sequence of point processes  δHN (σ) . PN = HN (σ) =

σ∈X

(1.11)

(1.12)

62

G. Ben Arous, V. Gayrard and A. Kuptsov

√ Theorem 1.1 (Universality in the Gaussian case). Assume that M = o( N ) if ν (0) = 0, and that M = o(N ) if ν (0) = 0. Then, P-almost surely, the distribution of the point process PN converges weakly to the distribution of a Poisson point process P on R with intensity given by √ 1 μ(dt) = √ e−t 2 log 2 dt. π

(1.13)

Remark 1.2. The condition log N = o(2M ) is needed in order to get P-almost sure results. We extend this result, in Section 5, to a wide class of non-Gaussian Hamiltonians (introduced in [BCMN05a] for the case of number partitioning). The theorem has the following immediate corollary. Let us fix the realization of the random cloud X. For configurations σ belonging to the cloud we consider the Gibbs’ weights GN,β (σ) of the re-scaled Hamiltonian HN (σ) √



e−βHN (σ) e−β M HN (σ) √ GN,β (σ) =  −βH  ( ) =  . N e e−β M HN ( ) ∈X

(1.14)

∈X

Reordering the Gibbs’ weights GN,β (σ) of the configurations σ ∈ X as a nonincreasing sequence (wα )α≤|X| and defining wα = 0 for α > |X| we get a random element w of the space S of non increasing sequences of non negative real numbers with sum less than one. √ Corollary 1.3 (Convergence to Poisson-Dirichlet). If β > 2 log 2 then P-almost surely under the assumptions of Theorem 1.1 the law of the sequence √ w = (wα )α≥1 converges to the Poisson-Dirichlet distribution with parameter m = 2 βlog 2 on S. The fact that Theorem 1.1 implies Corollary 1.3 is well known, see for instance [Tal03] (pp. 13–19) for a good exposition. It is then a natural question to know if our sparseness assumption is optimal. When the random cloud is denser can this universality survive? We show that our sparseness condition is indeed optimal for the number partitioning problem and for the SK model, and that the universality does break down. Theorem 1.4. [Breakdown of Universality for the number partitioning problem] m ) √ (i) Let ν(r) = r. Suppose that lim sup M(N < ∞. Then P-almost surely, the N distribution of the point process PN √ converges to the distribution of a Poisson point process if and only if M = o( N ).

[Breakdown of Universality for the Sherrington-Kirkpatrick model] l ) 1 < 8 log (ii) Let ν(r) = r2 . Suppose that lim sup M(N N 2 . Then P-almost surely, the distribution of the point process PN converges to the distribution of a Poisson point process if and only if M = o(N ).

A New REM Conjecture

63

We prove this theorem in Section 4 by showing that the second factorial moment of the point process does not converge to the proper value. The case of pure p-spin models, with p ≥ 3, or more generally the case where ν (0) = ν (0) = 0 differs strongly (see Theorem 4.7). The asymptotic behavior of the first three moments is compatible with a Poissonian convergence. Proving or disproving Poissonian convergence (or REM universality) in this case is still open, as it is for the local REM conjecture. The paper is organized as follows. In Section 2 we establish important combinatorial estimates about maximal overlaps of -tuples of points on the random cloud. We give a particular care to the case of pairs ( = 2) and triples ( = 3) which are important for the breakdown of universality results. In Section 3 we establish the universality in the Gaussian case (Theorem 1.1). We then prove, in Section 4, the breakdown of universality given in Theorem 1.4. Finally we extend the former results to a wide non-Gaussian setting in Section 5.

2. Combinatorial estimates In this section we fix an integer  ≥ 1 and study the maximal overlap Rmax (σ 1 , . . . , σ ) = max |R(σ i , σ j )|. 1≤i<j≤

(2.1)

For fixed N ∈ N and R ∈ [0, 1) let us define the following subsets of SN

U (R) = {(σ 1 , . . . , σ ) : Rmax (σ 1 , . . . , σ ) ≤ R}

(2.2)

V (R) = {(σ 1 , . . . , σ ) : Rmax (σ 1 , . . . , σ ) = R}.

(2.3)

and

More generally, let the sequence RN ∈ [0, 1) be given (respectively, the corresponding sequence of Hamming distances dN = N2 (1 − RN )) and introduce the sequence of sets U (RN ) and V (RN ) which we denote for simplicity of notation by UN, and VN, respectively.

we denote by Y X its intersection with X . In the following For a set Y ⊂ SN theorem we study the properties of the sets X UN, = {(σ 1 , . . . , σ ) ∈ X : Rmax (σ 1 , . . . , σ ) ≤ RN }

(2.4)

X VN, = {(σ 1 , . . . , σ ) ∈ X : Rmax (σ 1 , . . . , σ ) = RN }.

(2.5)

and In order to state the main result of this section (Theorem 2.1) we define the function  1−x 1+x 2 log(1 − x) + 2 log(1 + x) if x ∈ [−1, 1], J (x) = (2.6) +∞ otherwise.

64

G. Ben Arous, V. Gayrard and A. Kuptsov

2 Theorem 2.1. Let the sequence RN be such that N RN → ∞. (i) Then P-almost surely X X |U UN, | = E|U UN, |(1 + o(1)).

(2.7)

(ii) If RN = o(1) and M (N ) ≥ log N then there exists α ∈ [0, 1) and C > 0, depending only on , such that P-almost surely X X |V VN, | ≤ C eαN J (RN ) E|V VN, |.

(2.8)

Proof. The proof is based on standard inequalities for i.i.d. random variables that result from exponential Chebychev inequality. We formulate them without proof: let (Xi )1≤i≤n be i.i.d. Bernoulli rv’s with P(Xi = 1) = 1 − P(Xi = 0) = p and let n  Z= Xi . Then, for t > 0, i=1

P(Z − EZ ≥ tEZ) ≤ e−n(p(1+t) log(1+t)+(1−p(1+t)) log P(Z − EZ ≤ −tEZ) ≤ e

1−p(1+t) 1−p

−n(p(1−t) log(1−t)+(1−p(1−t)) log

),

1−p(1−t) 1−p

(2.9) ).

(2.10)

If p = p(n) → 0 and t = t(n) → 0 as n → ∞, the above inequalities imply that, for large enough n,   2 P |Z − EZ| ≥ tEZ ≤ 2e−npt /4 , (2.11) whereas if p(n) → 0, t(n) → ∞, and p(n)t(n) → 0 we get from (2.9) that for large enough n 1

P(Z − EZ ≥ tEZ) ≤ e− 2 np(1+t) log(1+t) .

(2.12)

The proof of Theorem 2.1 relies on the following elementary lemma that again we state without proof. Lemma 2.2. (i) For any sequence RN ∈ [0, 1)

     − 1 N R2N |U UN, | ≥ 2N 1 − 2 e 8 . 2

(2.13)

2 → ∞ and RN = o(1). Then for some C > 0 (ii) Suppose RN satisfies N RN depending only on , C |V VN, | = 2N √ e−N J (RN ) (1 + o(1)). (2.14) N

As an elementary consequence of part (i) of Lemma 2.2 one can prove that:  M(N ) log 2 |R(σ, σ )| ≤ δ , where δ ≡ 4 + logNN . Corollary 2.3. P-a.s. max N N N  σ,σ ∈X

The proof of part (i) of Theorem 2.1 then proceeds as follows. Let us first express the size of the random cloud |X| as a sum of i.i.d. random variables  |X| = 1Xσ =1 . (2.15) σ∈SN

A New REM Conjecture

65

Using (2.11) and the assumption that log N = o(2M ), we see that P-almost surely |X| is given by its expected value, i.e., |X| = 2M (1 + o(1)). Therefore |X | = 2M (1 + o(1)). X X Since UN, ⊂ X and E|U UN, | = p N |U UN, | = 2M (1 + o(1)), then proving part X (i) of Theorem 2.1 is equivalent to proving that the set UN, coincides, up to an M

X error of magnitude o(2 ), with the set X . Let us rewrite UN, as  X X = (U UN, )jj  , (2.16) UN, 1≤j<j  ≤

where we defined

   X (U UN )jj = (σ 1 , . . . , σ ) ∈ X : |R(σ j , σ j )| ≤ RN .

(2.17)

X If we prove that every set (U UN, )jj coincides, up to an error of order o(2M ), with the set X , then the representation (2.16) implies part (i) of the theorem. We therefore concentrate on proving that X |(U UN, )jj | = E|(U UN, )X jj  |(1 + o(1)),

P-a.s.

(2.18)

Without loss of generality we can consider the case of j = 1 and j = 2. By X definition of (U UN, )jj we get  

−2   X |(U UN, )12 | = 1Xσ1 =1 1Xσ2 =1 1Xσ =1  =

σ1 ,σ2 ∈SN : |R(σ1 ,σ2 )|≤RN

 σ1 ∈SN

1Xσ1 =1

σ∈SN



 1Xσ2 =1

σ2 :|R(σ1 ,σ2 )|≤RN



−2

1Xσ =1

. (2.19)

σ∈SN

As we already noted, the sum in the second factor of (2.19) concentrates on its expected value and is equal to 2M( −2) (1+o(1)). Let us thus turn to the first factor in (2.19). X Introduce the set (U UN, )σ1 = {σ 2 ∈ X : |R(σ 1 , σ 2 )| ≤ RN }. Then  X )σ1 | = 1Xσ2 =1 (2.20) |(U UN, σ2 :|R(σ1 ,σ2 )|≤RN

The summands in this sum are i.i.d. and it follows from part (i) of Lemma 2.2 that 2 1 their number is at least 2N (1−2e− 8 N RN ) = 2N (1+o(1)). Applying (2.11) together with the assumption that log N = o(2M ) we obtain from the Borel-Cantelli Lemma that P-a.s., for any σ 1 ∈ SN , X X |(U UN, )σ1 | = (1 + o(1))E|(U UN, )σ1 |.

(2.21)

From (2.19), (2.20) and (2.21) we immediately conclude that X X |(U UN, )12 | = (1 + o(1))E|(U UN, )12 |,

This finishes the proof of part (i) of Theorem 2.1.

P-a.s.

(2.22)

66

G. Ben Arous, V. Gayrard and A. Kuptsov

The proof of part (ii) is quite similar to the proof of part (i). By definition X of VN, we get  X X VN, ⊆ (V VN, )jj , (2.23) 1≤j<j  ≤

where

   X X )jj = (σ 1 , . . . , σ ) ∈ VN, : |R(σ j , σ j )| = RN . (V VN,

(2.24)

We claim that it suffices to prove that P-almost surely X X )jj | ≤ eαN J (RN ) E|(V VN, )jj |. |(V VN,

(2.25)

Indeed, from (2.23) and from the above inequality we obtain that   X X X |V VN, |≤ |(V VN, )jj | ≤ eαN J (RN ) E|(V VN, )jj |. 1≤j<j  ≤

(2.26)

1≤j<j  ≤

Using part (i) of Lemma 2.2 it is easy to establish that for all 1 ≤ j < j ≤    VN, )jj | 1 + o(1) , (2.27) |V VN, | = |(V and therefore X X | = E|(V VN, )jj |(1 + o(1)). E|V VN,

(2.28)

Since (2.26) and (2.28) imply the result we concentrate on the proof of (2.25). Without loss of generality we can take j = 1 and j = 2. Then, by definition X of (V VN, )jj , we get  

−2   X 1Xσ1 =1 1Xσ2 =1 1Xσ =1 |(V VN, )12 | =  =

σ1 ,σ2 ∈SN : |R(σ1 ,σ2 )|=RN



σ∈SN

1Xσ1 =1

σ1 ∈SN





1Xσ2 =1

σ2 :|R(σ1 ,σ2 )|=RN



−2

1Xσ =1

. (2.29)

σ∈SN

As in the proof of part (i) we see that the second part of (2.29) concentrates on its expected value and equals to 2M( −2) (1 + o(1)). We are thus left to treat the first part. Introducing the set X (V VN, )σ1 = {σ 2 ∈ X : |R(σ 1 , σ 2 )| = RN }

it is clear that X )σ1 | = |(V VN,



1Xσ2 =1

(2.30) (2.31)

σ2 :|R(σ1 ,σ2 )|=RN

  There are 2 dNN i.i.d. terms in the above sum. Applying (2.12) with t + 1 = eαN J (RN ) where α ∈ [0, 1) will be chosen later, we obtain 1

X X )σ1 | ≥ eαN J (RN ) E|(V VN, )σ1 |) ≤ e− 2 E|(VVN,)σ1 |(1+t) log(1+t) . P(|(V VN, X

(2.32)

A New REM Conjecture From Stirling’s formula we see that for some C > 0,   C N X E|(V VN, )σ1 | = 2pN = 2M √ e−N J (RN ) (1 + o(1)), dN N

67

(2.33)

and thus the exponent in (2.32) is C 1 X E|(V VN, )σ1 |(1 + t) log(1 + t) = √ 2M e−(1−α)N J (RN ) αN J (RN )(1 + o(1)) 2 2 N αC M log 2−(1−α)N J (RN )− 1 log N 2 e = N J (RN )(1 + o(1)). (2.34) 2 X VN, )σ1 is empty then there is nothing to prove. If for every σ 1 ∈ SN the set (V Otherwise, by Corollary 2.3 we obtain that P-almost surely RN < δN , and since J (x) = 12 x2 (1 + O(x2 )) near the origin we can choose α ∈ [0, 1) in such a way that M log 2 − (1 − α)N J (RN ) − for some γ > 0. Hence



1

1 log N > γ log N 2

(2.35)

e− 2 E|(VVN, )σ1 |(1+t) log(1+t) < ∞, and we obtain from X

N

(2.32) and the Borel-Cantelli Lemma that P-almost surely X X |(V VN, )σ1 | ≤ eαN J (RN ) E|(V VN, )σ1 |.

(2.36)

It is easy to see that (2.36) and (2.29) imply (2.25). This concludes the proof of Theorem 2.1.  X In part (ii) of Theorem 2.1 we studied the properties of the sets VN, ⊂ X for arbitrary  ≥ 2. For  = 2 we can improve on Theorem 2.1.

Theorem 2.4. Suppose that lim sup M N < 1. (i) If for some c1 >

1 2

J (RN ) ≤

M log 2 c1 log N − , N N

(2.37)

then P-almost surely X X |V VN, VN, 2 | = (1 + o(1))E|V 2 |.

(ii) If for positive constants c1 , c2 M log 2 c2 log N M log 2 c1 log N − ≤ J (RN ) ≤ + N N N N then there is a constant c such that P-almost surely X c X VN, |V VN, 2 | ≤ N E|V 2 |.

(iii) If for some c2 >

(2.39)

(2.40)

3 2

M log 2 c2 log 2 + , N N is P-almost surely empty. J (RN ) >

X then the set VN, 2

(2.38)

(2.41)

68

G. Ben Arous, V. Gayrard and A. Kuptsov

X Proof. (i) By definition of VN, 2,  X |V VN, 1Xσ1 =1 2| = σ1 ∈S

 σ2 ∈S

N

1Xσ2 =1 .

(2.42)

1 2 N :|R(σ ,σ )|=RN

X Using (2.30) the inner sum is |(V VN, 2 )σ1 |. Since it is a sum of i.i.d. random variables then for all t = t(N ) = o(1) we get from (2.11) that   X X X X −t2 E|(V VN, 2 )σ1 |/4 . P ||(V VN, VN, VN, (2.43) 2 )σ1 | − E|(V 2 )σ1 || ≥ t E|(V 2 )σ1 | ≤ 2e

Using Stirling’s approximation √ √ 1 1 1 1 2πnn+ 2 e−n+ 12n+1 < n! < 2πnn+ 2 e−n+ 12n we obtain that

 X E|(V VN, 2 )σ1 | = 2pN

N dN

 =O

(2.44)

 eM log 2−N J (RN ) $ . 2 ) N (1 − RN

(2.45)

X c1 −1/2 for Further, from (2.37) and Corollary 2.3 we obtain that E|(V VN, 2 )σ1 | ≥ CN −γ for some small enough γ > 0 in some positive constant C > 0. Choosing t = N (2.43), we conclude from the Borel-Cantelli Lemma that X X |(V VN, VN, 2 )σ1 | = E|(V 2 )σ1 |(1 + o(1)),

P-a.s.

(2.46) M

Using (2.42), (2.46), and the fact that P-almost surely there are 2 (1 + o(1)) configurations in the random cloud X proves (i). (ii) The proof is similar to the proof of part (i). In particular, from the representation (2.42) it is easy to see that it suffices to prove that X c X |(V VN, VN, 2 )σ1 | ≤ N E|(V 2 )σ1 |,

P-a.s.

(2.47)

Choosing 1 + t = N c we get from (2.45) that for some positive constant C 1 X c−c2 −1/2 E|(V VN, log N. (2.48) 2 )σ1 |(1 + t) log(1 + t) ≥ CN 2 Choosing c large enough and applying (2.12) together with the Borel-Cantelli Lemma proves part (ii). (iii) We again use the representation (2.42). Clearly it is enough to prove that X X for all σ1 ∈ X the set (V VN, VN, 2 )σ1 is P-almost surely empty. By (2.45) E|(V 2 )σ1 | ≤ 1 −c2 −1/2 CN for some positive constant C. Thus, choosing 1 + t = E|(VV X ) 1 | → ∞ N,2 σ

we obtain from (2.12)   X   X X −1/2 VN,2 )σ1 | ≥ (1 + t)E|(V VN, . P |(V VN,2 )σ1 | ≥ 1 = P |(V 2 )σ1 | ≤ (t + 1)

(2.49)

By definition of t and by condition (2.41) we get (t + 1)−1/2 ≤ C 1/2 N −c2 /2−1/4 . Applying the Borel-Cantelli Lemma proves (iii).

(2.50) 

A New REM Conjecture

69

In order to estimate the third moment in Theorems 4.1, 4.2, and 4.7 we give a result similar to Theorem 2.4 but for  = 3, i.e., for a given sequence of vectors N N N (R12 , R23 , R31 ) we estimate the cardinal of the set   X 1 2 3 3 1 2 N 2 3 N 3 1 N WN, = (σ , σ , σ ) ∈ X : R(σ , σ ) = R , R(σ , σ ) = R , R(σ , σ ) = R 3 12 23 31 . (2.51) N N N Below we omit the explicit dependence of the sequence (R12 , R23 , R31 ) on N and N N N instead of R12 , R23 and R31 will write R12 , R23 and R31 respectively. In order to formulate the theorem we introduce the following function on R3 : 1+x−y −z 1+x+y +z log(1 + x + y + z) + log(1 + x − y − z) J (2) (x,y,z) = 4 4 1−x+y −z 1−x−y +z + log(1 − x + y − z) + log(1 − x − y + z) (2.52) 4 4 if 1 + x ≥ |y + z| and 1 − x ≥ |y − z| and J (2) (x, y, z) = +∞ otherwise. Theorem 2.5. Suppose lim sup M N < 1. (i) If for some c1 >

1 2

(2)

the sequence R12 satisfies (2.37), and if for some c1 > 1 (2)

J (2) (R12 , R23 , R31 ) ≤

M log 2 c log N + J (R12 ) − 1 , N N

(2.53)

then P-almost surely X X |W WN, WN, 3 | = (1 + o(1))E|W 3 |. (2)

(2.54)

(2)

(ii) If for positive constants c1 , c2 , (2)

M log 2 c log N + J (R12 ) − 1 ≤ J (2) (R12 , R23 , R31 ) N N (2) c log N M log 2 + J (R12 ) + 2 , ≤ N N then there is a constant c such that P-almost surely X c X |W WN, WN, 3 | ≤ N E|W 3 |. (2)

(iii) If for some c2 >

(2.55)

(2.56)

3 2 (2)

M log 2 c log 2 + J (R12 ) + 2 , N N X then P-almost surely the set WN, 3 is empty. J (2) (R12 , R23 , R31 ) >

(2.57)

Proof. From Lemma 2.6 stated below it follows that, for arbitrary configurations σ 1 , σ 2 , σ 3 ∈ SN , the function J (2) (R12 , R23 , R31 ) is well defined. This lemma, whose proof we will omit, is a direct consequence of the fact that the Hamming distance on SN satisfies triangle inequality. Lemma 2.6. For arbitrary configurations σ 1 , σ 2 and σ 3 ∈ SN we have 1 + R12 ≥ |R23 + R31 | and 1 − R12 ≥ |R23 − R31 |.

70

G. Ben Arous, V. Gayrard and A. Kuptsov

The proof of the Theorem 2.5 is similar to that of Theorem 2.4. We begin by X writing the size of the set WN, 3 as    X 1Xσ1 =1 1Xσ2 =1 1Xσ3 =1 (2.58) |W WN, 3| = σ1 ∈SN

σ2 ∈SN : R(σ1 ,σ2 )=R12

σ3 ∈SN : R(σ2 ,σ3 )=R23 R(σ1 ,σ3 )=R31

Let us first estimate the number of terms in the last sum. This means that, given σ 1 , σ 2 ∈ SN with overlap R(σ 1 , σ 2 ) = R12 , we have to calculate the number of configurations σ 3 with R(σ 2 , σ 3 ) = R23 and R(σ 3 , σ 1 ) = R31 . Without loss of generality we can assume that all the spins of σ 1 are equal to 1. Further, let C(σ 1 , σ 2 , σ 3 ) be a 3 × N matrix with rows σ 1 , σ 2 , σ 3 . For a column vector δ ∈ {−1, 1}3 we let nδ be the number of columns of the matrix C that are equal to δ, i.e.,   nδ = {j ≤ N : (σ σj1 , σj2 , σj3 ) = δ}. (2.59) Then the overlaps can be written in terms of nδ , namely ⎧ ⎪ n(1,1,1) + n(1,1,−1) − n(1,−1,1) − n(1,−1,−1) = N R12 , ⎪ ⎨ n(1,1,1) − n(1,1,−1) − n(1,−1,1) + n(1,−1,−1) = N R23 , (2.60) n − n(1,1,−1) + n(1,−1,1) − n(1,−1,−1) = N R31 , ⎪ ⎪ ⎩ (1,1,1) n(1,1,1) + n(1,1,−1) + n(1,−1,1) + n(1,−1,−1) = N. Solving this system of linear equations we find ⎧ = 14 N (1 + R12 + R23 + R31 ), ⎪ ⎪ n(1,1,1) ⎨ n(1,1,−1) = 14 N (1 + R12 − R23 − R31 ), = 14 N (1 − R12 − R23 + R31 ), ⎪ n(1,−1,1) ⎪ ⎩ n(1,−1,−1) = 14 N (1 − R12 + R23 − R31 ).

(2.61)

Specifying the configuration σ 3 is equivalent to specifying a partition of the index set {1, 2, . . . , N } into subsets of seizes n(1,1,1) , n(1,1,−1) , n(1,−1,1) , and n1,−1,−1 . Therefore the number of configurations σ 3 ∈ SN with overlaps R(σ 2 , σ 3 ) = R23 , and R(σ 3 , σ 1 ) = R31 is    n(1,1,1) + n(1,1,−1) n(1,−1,1) + n(1,−1,−1) n(1,1,1) n(1,−1,1)     n(1,1,1) + n(1,1,−1) ! n(1,−1,1) + n(1,−1,−1) ! . (2.62) = n(1,1,1) ! n(1,1,−1) ! n(1,−1,1) ! n(1,−1,−1) ! Applying Stirling’s approximation to (2.62) one obtains that the number of terms in the last summation in (2.58) is of order $ (2) 2 2N eN J (R12 )−N J (R12 ,R23 ,R31 ) 1 − R12 $ , N P (R12 , R23 , R31 ) where P (x, y, z) = (1 + x + y + z)(1 + x − y − z)(1 − x + y − z)(1 − x − y + z). The rest of the proof essentially is a rerun of the proof of Theorem 2.4. We skip the details. 

A New REM Conjecture

71

3. Proof of Theorem 1.1 As in [BK06a] and [BCMN05a], [BCMN05b], the proof of the Poisson convergence is based on the analysis of factorial moments of the point processes PN defined in (1.12). In general, let ξN be a sequence of point processes defined on a common probability space (Ω, F , Q) and let ξ be a Poisson point process with intensity measure μ. Define the th factorial moment E(Z) of the random variable Z to be EZ(Z − 1) . . . (Z −  + 1). The following is a classical lemma that is a direct consequence of Theorem 4.7 in [Kal83]. Lemma 3.1. If for every  ≥ 1 and every Borel set A lim EQ (ξN (A)) = (μ(A)) ,

(3.1)

N →∞

then the distribution of (ξN )N ≥1 converges weakly to the distribution of ξ.  Applying Lemma 3.1 to the sequence of point processes PN = δHN (σ) σ∈X

the following result proves Theorem 1.1. Theorem 3.2. Under the assumptions of Theorem 1.1, for every  ∈ N and every bounded Borel set A lim E(PN (A)) = (μ(A)) ,

N →∞

P-a.s.,

(3.2)

where μ is defined in (1.13). Proof. We start with the computation of the first moment of PN (A) :  E PN (A) = P(H HN (σ) ∈ A).

(3.3)

σ∈X

As we saw in the proof of Theorem 2.1 the size of the random cloud |X| is P-almost surely 2M (1 + o(1)). Since HN (σ), σ ∈ SN , are identically distributed normal random variables with mean −aN /bN and variance 1/bN , the sum in (3.3) can be written as & 2 2 2 e−aN /2 2M √ bN e−x bN /2−aN bN x dx (1 + o(1)), P -a.s. (3.4) 2π A

By the dominated convergence theorem and the definition of aN it follows from (3.4) that the limit of the first moment is μ(A). To calculate factorial moments of higher order we follow [BCMN05a] and rewrite the th factorial moment of PN (A) as   1  E(PN (A)) = P HN (σ ) ∈ A, . . . , HN (σ ) ∈ A (3.5) σ1 ,...,σ

where the sum runs over all ordered sequences of distinct configurations σ 1 , . . . , σ

belonging to X. To analyze it we decompose the set SN into three non-intersecting

72

G. Ben Arous, V. Gayrard and A. Kuptsov

subsets

 

SN = U (RN ) ∪ U (δN )\U U (RN ) ∪ (U U (δN ))c ,

(3.6)

where U is defined in (2.2), δN is defined in Corollary 2.3 and the sequence RN is chosen such that: 2 RN → 0, M ν(RN ) → 0, N RN → ∞. (3.7) √ This is possible since we have assumed that M = o( N ) if ν (0) = 0 and M = o(N ) if ν (0) = 0. We recall here that the function ν is defined in (1.7)–(1.8). Having specified RN , let us analyze the contribution to the sum (3.5) coming from the intersection of X with the sets U (RN ), U (δN )\U U (RN ), and (U U (δN ))c . c

Firstly, by Corollary 2.3 the intersection of the set (U U (δN )) with X is P-a.s. empty and therefore its contribution to the sum (3.5) is zero. Next, let us show that P-a.s.   1  lim P HN (σ ) ∈ A, . . . , HN (σ ) ∈ A = (μ(A)) , (3.8) N →∞

σ1 ,...,σ

where the sum is over all the sequences (σ1 , . . . , σ ) ∈ U (RN ) ∩ X . For every  ∈ N let B(σ 1 , . . . , σ ) denote the covariance matrix of random variables HN (σ 1 ), . . . , HN (σ ). By (1.7) its elements, bij , are given by bij = ν(Rij )

(3.9)

where we wrote R(σ i , σ j ) = Rij . Since RN = o(1) the matrix B(σ 1 , . . . , σ ) is non-degenerate. We therefore get for (σ 1 , . . . , σ ) ∈ U (RN ) ∩ X that   1 b N √ (σ ) ∈ A, . . . , HN (σ ) ∈ A = P HN (2π) /2 det B & & −1 2 −1 2  −1 × . . . e−(x,B x)bN /2−aN bN (x,B 1)−aN (1,B 1)/2 dx, A

(3.10)

A

where B = B(σ 1 , . . . , σ ),  = (x1 , . . . , x )t and 1 = (1, . . . , 1)t . From the definition of aN we further get from (3.10) that −1

   1  1 e( −(1,B 1))(M log 2−1/2 log M) √ P HN (σ ) ∈ A, . . . , HN (σ ) ∈ A = M 2 det B & & −1 2 −1 1 × √ . . . e−(x,B x)bN /2−aN bN (x,B 1) dx. (3.11) ( π) A

A

√ Since the matrix B −1 is positive definite and since aN bN → 2 log 2 we conclude from the dominated convergence theorem that for all bounded Borel sets

A New REM Conjecture A, uniformly in (σ 1 , . . . , σ ) ∈ U (RN ) ∩ X , & & −1 2 −1 1 √ . . . e−(x,B x)bN /2−aN bN (x,B 1) d ( π) A A & & √ 1  → √ . . . e− 2 log 2(x,1) d = (μ(A)) . ( π) A

73

(3.12)

A

( −(1,B −11))(M log 2−1/2 log M)

To evaluate term e in (3.11) we look at ( − (1, B −11)) det B as a multivariate function of (bij )1≤i<j≤ . It is a polynomial of degree  with coefficients depending only on  and without constant term. It implies that | − (1, B −11)| = O(ν(RN )) and therefore, by (3.7), 

e( −(1,B

−1

1))(M log 2−1/2 log M)

= 1 + o(1).

Combining (3.11), (3.12), and (3.13) we may rewrite the sum (3.8) as  1 |U U (RN ) ∩ X |

(μ(A)) (1 + o(1)) = (μ(A)) (1 + o(1)). M M 2 2 1 

(3.13)

(3.14)

σ ,...,σ

Now it follows from Theorem 2.1 (i) and Lemma 2.2 (i) that |U U (RN ) ∩ X | concentrates around its expected value, namely 2M (1 + o(1)), so that (3.14) implies (3.8). Next, let us establish that the contribution from the second set in (3.6) is negligible, i.e., let us prove that   1  lim P HN (σ ) ∈ A, . . . , HN (σ ) ∈ A = 0, (3.15) N →∞

σ1 ,...,σ

  where the sum runs over all the sequences (σ 1 , . . . , σ ) ∈ U (δN )\U U (RN ) ∩ X . To do this we first bound the right-hand side of (3.11). By definition (2.1) and Corollary 2.3, Rmax (σ 1 , . . . , σ ) ≤ δN = o(1). Therefore following the same reasoning as above we obtain from   (3.11) that for some constant C > 0 and for all (σ 1 , . . . , σ ) ∈ U (δN )\U U (RN ) ∩ X  1  eCMν(Rmax ) P HN (σ ) ∈ A, . . . , HN (σ ) ∈ A ≤ (μ(A)) (1 + o(1)). (3.16) 2M For fixed N the overlap takes only a discrete set of values   2k KN = 1 − : k = 0, 1, . . . , N . (3.17) N U (RN ) as a union of sets V (RN,k ), where we denoted We represent the set U (δN )\U ∈ K ∩ (R RN,k = 1 − 2k N N , δN ). Let us fix k and bound the contribution from N the set V (RN,k ) ∩ X , i.e.,   1  P HN (σ ) ∈ A, . . . , HN (σ ) ∈ A . (3.18) (σ1 ,...,σ )∈ V (RN,k )∩X 

74

G. Ben Arous, V. Gayrard and A. Kuptsov

We obtain from Lemma 2.2 (ii) that C |V V (RN,k )| = 2N √ e−N J (RN,k ) (1 + o(1)). (3.19) N In the case M (N ) ≤ log N we can choose the sequence RN in such a way that the set U (δN )\U U (RN ) is empty. Therefore we can assume without loss of generality that M (N ) ≥ log N. Applying part (ii) of Theorem 2.1 we further conclude that C |V V (RN,k ) ∩ X | ≤ 2M √ e−(1−α)N J (RN,k ) , P-a.s. (3.20) N Using (3.16) and the last inequality we bound the sum (3.18) by C √ e−(1−α)N J (RN,k ) eCMν(RN,k ) . (3.21) N 2 ) for RN,k ∈ (RN , δN ). One can easily check that M ν(RN,k ) = o(N RN,k 2 Together with J (x) ≥ x /2 it implies that for some positive constants C1 and C2 we can further bound (3.21) by 2 C1 √ e−C2 N RN,k . (3.22) N As a consequence, we obtain an almost sure bound    1  P HN (σ ) ∈ A, . . . , HN (σ ) ∈ A RN,k ∈ (σ1 ,...,σ )∈  KN ∩(RN ,δN ) V (R  N,k )∩X

C1 ≤√ N



2

e−C2 N RN,k .

(3.23)

RN,k ∈ KN ∩(RN ,δN )

√ Introducing new variables yN,k = N RN,k we rewrite the above sum as 2 2 C1  e−C2 yN,k √ , (3.24) y 2 N,k N √ √ √ where the summation is over the discrete set N KN ∩ ( N RN , N δN ). Since 2 N RN → ∞, then for arbitrary C > 0 and for large N we can further bound this sum by 2 C1  2 e−C2 yN,k √ . (3.25) yN,k ≥C 2 N Interpreting the last sum as a sum of areas of nonintersecting rectangles with one side equal e−C2 yN,k and the other √2N we bound it with the integral & ∞ 2 e−C2 y dy (3.26) C − √2

N

Since the constant C is arbitrary we get that (3.15) is o(1). This finishes the proof of Theorem 3.2 and therefore of Theorem 1.1. 

A New REM Conjecture

75

4. Proof of Theorem 1.4 In order to prove the breakdown of universality in Theorem 1.4 we use a strategy similar to that used in [BCMN05b] to disprove the local REM conjecture for the number partitioning problem and for the Sherrington-Kirkpatrick model when the energy scales are too large. We prove that P-a.s. for every bounded Borel set A 1. the limit of the first factorial moment exists and equals μ(A); 2. the second factorial moment E(PN (A))2 does not converge to (μ(A))2 ; 3. the third moment is bounded. These three facts immediately imply that the sequence of random variables PN (A) does not converge weakly to a Poisson random variable and so the sequence of point processes PN does not converge weakly to a Poisson point process. Part (i) of Theorem 1.4 is thus obviously implied by the following Theorem 4.1 (Breakdown of Universality for the number partitioning problem). Let ν(r) = r. For every bounded Borel set A lim E(PN (A))1 = μ(A),

P-a.s.

(4.1)

) √ Moreover, if lim sup M(N = ε < ∞ then P-a.s. N 2

(i) lim sup E(PN (A))2 = e2ε (ii) lim sup E(PN (A))3 < ∞.

log2 2

(μ(A))2 ,

Similarly, part (ii) of Theorem 1.4 is implied by the following Theorem 4.2 (Breakdown of Universality for the Sherrington-Kirkpatrick model). Let ν(r) = r2 . For every bounded Borel set A lim E(PN (A))1 = μ(A), ) Moreover, if lim sup M(N =ε< N

1 8 log 2

P-a.s.

(4.2)

then P-a.s.

2

(μ(A)) (i) lim sup E(PN (A))2 = √1−4ε ; log 2 (ii) lim sup E(PN (A))3 < ∞. 1 Remark 4.3. Condition ε < 8 log 2 in not optimal and could be improved. The 1 reason for such a choice is that for ε < 8 log 2 the third moment estimate is quite simple.

We will prove in detail Theorem 4.2 but omit the proof of Theorem 4.1 as it is very similar and much simpler. Proof of Theorem 4.2. We successively prove the statement on the first, second, and third moment. 1. First moment estimate. Since all the random variables HN (σ), σ ∈ SN , are identically distributed then E(PN (A))1 = |X|P(HN (σ) ∈ A).

(4.3)

76

G. Ben Arous, V. Gayrard and A. Kuptsov

We saw in the proof of Theorem 2.1 that for M (N ) satisfying log N = o(2M ), |X| = 2M (1 + o(1)) P-a.s. Combined with (3.11), the definition of aN , and the dominated convergence theorem this fact implies that P-a.s. & 2 2 1 1 M √ E(PN (A))1 = 2 (1 + o(1)) M e−x bN /2−aN bN x dx = μ(A)(1 + o(1)). 2 π A

(4.4) Hence (4.2) is proven. ) 2. Second moment estimate. Next assume that lim sup M(N = ε. We now want to N calculate lim sup E(PN (A))2 . For this we rewrite the second factorial moment as    P HN (σ 1 ) ∈ A, HN (σ 2 ) ∈ A , (4.5) E(PN (A))2 = σ1 ,σ2

where the summation is over all pairs of distinct configurations (σ 1 , σ 2 ) ∈ X 2 . 2 We then split the set SN into four non-intersecting subsets and calculate the contributions from these subsets separately (these calculations are similar to those of Theorem 3.2). (1) We begin by calculating the contribution from the set S1X , where

2 : |R(σ 1 , σ 2 )| ≤ τN S1 = (σ 1 , σ 2 ) ∈ SN

(4.6) 2

4 and where the sequence τN is chosen in such a way that N τN → 0 and e−N τN decays faster than any polynomial: this can be achieved by choosing, e.g., τN = 1 . N 1/4 log N First, we get from (3.11) for  = 2

  1 (σ ) ∈ A, HN (σ 2 ) ∈ A = P HN 1 × √ 2 ( π)

& &

e−(x,B

−1

2b12

1 e 1+b12 (M log 2−1/2 log M) $ 2M 2 1 − b212

x)b2N /2−aN bN (   x,B −11)

dx1 dx2 ,

(4.7)

A A 2 HN (σ 1 ), HN (σ 2 )) = R12 . If (σ 1 , σ 2 ) ∈ S1 then b12 = o(1) and where b12 = cov(H 1 we get from (3.12) that uniformly in (σ , σ 2 ) ∈ S1 the second line in (4.7) is just (μ(A))2 (1 + o(1)). Next, we represent the set S1 as a union of sets V2 (RN,k ) with RN,k = 1− 2k N ∈ KN ∩ [0, τN ]. Applying Theorem 2.4 (i) we get that # 2 2M −N J (RN,k ) 2 2 e (1 + o(1)). (4.8) |V V2 (RN,k ) ∩ X | = 2 πN

A New REM Conjecture

77

Therefore, up to a multiplicative term of the form 1 + o(1), the contribution from the set S1X to the sum (4.5) is #  1 2b12 2 2M −N J (RN,k ) 2 1+b12 (M log 2−1/2 log M) 2 e e (μ(A)) 2 22M πN RN,k #  2b12 2 (M log 2−1/2 log M) (μ(A))2 =2 e−N J (RN,k ) e 1+b12 . (4.9) πN RN,k

2 Since b12 (RN,k ) = RN,k > 0 the last sum is monotone in M. As we will see below due to this fact it is sufficient to calculate the upper limit of E(P(A))2 for 1 sequences of the form M (N ) = εN with ε ∈ (0, 8 log 2 ). Thus let M = εN, ε ∈ 1 (0, 8 log 2 ). We obtain

2b12 2 4 (M log 2 − 1/2 log M ) = 2εN RN,k log 2 + O(N RN,k ). 1 + b12

(4.10)

The choice of τN guarantees that for RN,k ≤ τN the last term in the right-hand side of (4.10) is of order o(1). Moreover, for x < 1 we observe that J (x) = 12 x2 +O(x4 ). Thus 2 N RN,k N J (RN,k ) = + o(1). (4.11) 2 Using (4.10) and (4.11) the sum (4.9) becomes #  2 2 1 (μ(A))2 2 e− 2 N RN,k (1−4ε log 2) (1 + o(1)), (4.12) πN RN,k

where the √ summation is over RN,k ∈ KN ∩ [0, τN ]. Introducing new variables yN,k = N RN,k we further rewrite (4.12) as # 2 2  2 − yN √ e 2 (1−4ε log 2) (1 + o(1)). (4.13) πy N N,k 1 It is not difficult to see that for ε < 4 log 2 the sum in (4.13) converges to the integral # &∞ 2 1 π − y2 (1−4ε log 2) . (4.14) e dy = √ 1 − 4ε log 2 2 0

2

(μ(A)) Therefore the contribution from the set S1X is √1−4ε (1 + o(1)). log 2 We can extend this result to the case when M (N ) = εN + o(N ) with ε < 1 8 log 2 . Indeed, assume that εn ↑ ε as n → ∞. Then using the above calculation for M = εn N together with the monotonicity argument we have that for all n ≥ 1

(μ(A))2 . E(PN (A))2 ≥ √ 1 − 4εn log 2

(4.15)

78

G. Ben Arous, V. Gayrard and A. Kuptsov

Taking the limit in n we obtain a lower bound. With exactly the same argument we prove the corresponding upper bound. (2) Next, we estimate the contribution from the set S2X , where

2 S2 = (σ 1 , σ 2 ) ∈ SN : |R(σ 1 , σ 2 )| > τN and R(σ 1 , σ 2 ) satisfies (2.37) . (4.16) Since the set A is bounded an elementary computation yields that for a constant C = C(A), uniformly in (σ 1 , σ 2 ) ∈ S2 , & & −1 2 −1 e−(x,B x) bN /2−aN bN (x,B 1) dx1 dx2 ≤ C. (4.17) A A 1 For fixed N we let RN to be the largest value of the overlap satisfying condition (2.37). Then representing S2 as a union of sets V2 (RN,k ), where RN,k = 1 τN , RN ], and using Theorem 2.4 (i), we conclude that the con1 − 2k N ∈ KN ∩ [τ tribution from the set S2X is, up to a multiplicative term of the form 1 + o(1), bounded by +  2b 2 C −N J (RN,k )+ 1+b12 M log 2 12 $ 2 e . (4.18) 2 πN (1 − RN,k ) 1 − b212 RN,k

This quantity is monotone in M. As a consequence, to show that it is negligible in the limit N → ∞ it suffices to show this fact under the assumption M (N ) = εN. Thus, letting M = εN and using that J (x) ≥ 12 x2 , we can bound the exponent in (4.18) by   2b12 1 −N J (RN,k ) − ε log 2 ≤ − N x2 (1 − 4ε log 2). (4.19) 1 + b12 2 Since 1 − b212 ≥ 1/N and since the number of terms in (4.18) is at most N, we can further bound it, for some positive constant C > 0, by # 2 − 1 N τN2 (1−4ε log 2) CN 2 e 2 . (4.20) π Since ε <

1 8 log 2

the contribution from the set S2X is negligible by definition of τN .

(3) We now analyze the contribution from the set S3X , where

2 S3 = (σ 1 , σ 2 ) ∈ SN : R(σ 1 , σ 2 ) satisfies (2.39) .

(4.21)

2 be the largest overlap value satisfying condition (2.39). To bound the Let RN contribution from the set S3 we represent it as a union of sets V2 (RN,k ), where 1 2 RN,k = 1 − 2k N runs over the set KN ∩ [RN , RN ]. Proceeding as in (2) and using Theorem 2.4 (ii), we bound it by #  2b12 2 C Nc $ 2 e 1+b12 M log 2−N J (RN,k ) . (4.22) 2 πN 1 − b12 RN,k

Again, the sum is monotone in M and thus it is enough to bound it for M = εN.

A New REM Conjecture

79

For RN,k satisfying condition (2.39) we can bound the exponent in (4.22) as follows: 2b12 2b12 εN log 2 − N J (RN,k ) ≤ εN log 2 − N ε log 2 − c1 log N 1 + b12 1 + b12 1 − b12 εN log 2 − c1 log N ≤ −CN ε log 2 − c1 log N, (4.23) =− 1 + b12 where C is some positive constant. Since 1 − b212 ≥ 1/N and since there  are at most N terms in the sum (4.22), we can bound the latter by N c+1−c1 exp −CN ε log 2 . Therefore, P-almost surely the contribution from the set S3X is negligible as N → ∞. (4) To finish the second moment estimate it remains to treat the set

2 S4 = (σ 1 , σ 2 ) ∈ SN : R(σ 1 , σ 2 ) satisfies (2.41) . (4.24) But by Theorem 2.4 (iii), the set S4X is P-almost surely empty. This finishes the proof of assertion (i) of Theorem 4.2. 3. Third moment estimate. To analyze the third factorial moment we use that, by formula (3.5) and definition (2.51), it can be written as X    |W WN, |  −1 √ 3 E PN (A) 3 = e(3−(1,B 1))(M log 2−1/2 log M) 3M 2 det B R12 ,R23 ,R13 & & & −1 2 −1 × e−(x,B x)bN /2−aN bN (x,B 1) dx, (4.25) A A A 3 where the summation runs over all triplets of overlaps (R12 , R23 , R31 ) ∈ KN and 1 2 3 1 2 B = B(σ , σ , σ ), the covariance matrix of the vector (H HN (σ ), HN (σ ), HN (σ 3 )), 2 is the matrix with elements bij = Rij , 1 ≤ i, j ≤ 3. To estimate this sum we rely on three auxiliary lemmas whose proofs we skip since they are simple.

Lemma 4.4. If ε < lim sup

max

N →∞ σ1 ,σ2 ,σ3 ∈X

1 8 log 2

then & & & −1 2 −1 e−(x,B x)bN /2−aN bN (x,B 1) dx < ∞,

P-a.s.

(4.26)

A A A

Lemma 4.5. P-almost surely for all configurations σ 1 , σ 2 , σ 3 ∈ X 2 2 2 3 − (1, B −11) ≤ 2(R12 + R23 + R31 ). Lemma 4.6. For all σ 1 , σ 2 , σ 3 ∈ SN

(4.27)

1 2 2 2 R12 + R23 . (4.28) + R31 4 We are now ready to estimate sum (4.25). In the same spirit as for the second moment calculation, we will split (4.25) into four parts and show that every each of them is bounded. Moreover, by monotonicity argument similar to that used in calculation of the second moment we can restrict our attention to the case when M = εN. J (2) (R12 , R23 , R31 ) ≥

80

G. Ben Arous, V. Gayrard and A. Kuptsov (1) We first calculate the contribution to (4.25) coming from the set



3 S1 = (σ 1 , σ 2 , σ 3 ) ∈ SN (4.29) : max |R12 |, |R23 |, |R31 | ≤ τN ,

where τN = N 1/41log N . Using Theorem 2.5 (i) and Lemma 4.5, we obtain that the contribution from S1 is at most of order  (2) 2 2 2 1 e−N J (R12 ,R23 ,R31 )+2(R12 +R23 +R31 )M log 2 , (4.30) N 3/2 R ,R ,R 12

23

31

where the summation is over R12 , R23 , R31 ∈ KN ∩[−ττN , τN ]. Expanding in Taylor series we obtain that for |R12 |, |R23 |, |R31 | ≤ τN J (2) (R12 , R23 , R31 ) =

 1 2 2 2 4 R + R23 + R31 + O(ττN ), 2 12

and thus (4.30) is bounded by  2 2 2 2 2 2 1 1 e− 2 N (R12 +R23 +R31 )+2εN (R12 +R23 +R31 ) log 2 3/2 N R12 ,R23 ,R31 ⎛ ⎞3  2 1 1 = ⎝√ e− 2 N R (1−4ε log 2) ⎠ < ∞. N R ∈K ∩[−τ ,τ ] N

N

N

(4.31)

(4.32)

N

(2) We next calculate the contribution from the set 3 S2 = (σ 1 , σ 2 , σ 3 ) ∈ SN : R12 , R23 , R21 satisfy (2.53)

and max |R12 |, |R23 |, |R31 | > τN .

(4.33)

Without loss of generality we can assume that |R12 | > τN . Then, using Theorem 2.5 (i) and Lemma 4.6, the contribution from this set is at most of order 

(2)

R12 ,R23 ,R31

e−N J (R12 ,R23 ,R31 ) 2(R212 +R223 +R231 )M log 2 e , N 3/2 P (R12 , R23 , R31 )

(4.34)

where P (x, y, z) = (1+x+y +z)(1+x−y −z)(1−x+y −z)(1−x−y +z) and where 3 the sum is over the triplets (R12 , R23 , R31 ) ∈ KN satisfying (2.53) and |R12 | > τN . Then, from Lemma 4.5, we further get that the sum (4.34) is bounded by  |R12 |>τN

2

1

2

2

e− 4 N (R12 +R23 +R31 ) 2(R212 +R223 +R231 )M log 2 e N 3/2 P (R12 , R23 , R31 ) =

 |R12 |>τN

1

2

2

2

e− 4 N (1−8ε log 2)(R12 +R23 +R31 ) N 3/2 P (R12 , R23 , R31 )

(4.35) 1

2

Since the number of terms in (4.35) is at most (N +1)3 , and since e− 4 (1−8ε log 2)N τN decreases faster than any polynomial it follows that sum (4.35) is of order o(1).

A New REM Conjecture (3) We now turn to the contribution from the set

3 S3 = (σ 1 ,σ 2 , σ 3 ) ∈ SN : R12 , R23 , R31 satisfy (2.55) . By Lemma 2.5 (ii) the contribution from this set is at most of order  N c E|W WX |  −1 √ N,3 e(3−(1,B 1))(M log 2−1/2 log M) , 3M det B 2 R12 ,R23 ,R31

81

(4.36)

(4.37)

where the summation is over the triplets (R12 , R23 , R31 ) ∈ KN satisfying (2.55). X Since E|W WN, 3 | is of order (2)

23M e−N J (R12 ,R23 ,R31 ) N 3/2 P (R12 , R23 , R31 ) we obtain, using Lemmas 4.5 and 4.6, that the sum (4.37) is bounded by 2 2 2 1  N c e− 4 N (R12 +R23 +R31 ) 2εN log 2(R212 +R223 +R231 ) e . N 3/2 P (R12 , R23 , R31 ) R ,R ,R 12

23

(4.38)

(4.39)

31

It is easy to show that the triplet (R12 , R23 , R31 ) that satisfy (2.55) must satisfy either |R23 | > τN or |R31 | > τN . Therefore we can further bound the contribution from the set S3 by the sum 2 2 2 1  N c e− 4 N (R12 +R23 +R31 ) 2εN log 2(R212 +R223 +R231 ) e , (4.40) N 3/2 P (R12 , R23 , R31 ) |R23 |>τN or |R31 |>τN which is o(1) by the same argument as in part (2). (4) To finish the estimate of the third factorial moment we have to estimate the contribution to (4.25) coming from the set

3 S4 = (σ 1 , σ 2 , σ 3 ) ∈ SN : R12 , R23 , R21 satisfy (2.57) . (4.41) By Theorem 2.5 (iii) the set S4X is P-a.s. empty and therefore f its contribution is P-a.s. zero. This finishes the proof of assertion (ii) of Theorem 4.2. The proof of Theorem 4.2 is now complete.  For comparison with the cases ν(r) = rp for p = 1 and p = 2, we give here the asymptotic behavior of the first three factorial moments for the case ν(0) = ν (0) = 0, i.e., for instance for the case ν(r) = rp of pure p-spins when p ≥ 3. This behavior is compatible with a Poisson convergence theorem. Theorem 4.7. Assume that ν(0) = ν (0) = 0. For every bounded Borel set A lim E(PN (A))1 = μ(A), M(N ) N

P-a.s.

(4.42)

1 < 8 log 2 2

then P-a.s. Moreover, if lim sup (i) lim E(PN (A))2 = (μ(A)) ; (ii) lim E(PN (A))3 = (μ(A))3 < ∞. We do not include a proof of this last statement, which again follows the same strategy as the proof of Theorem 4.2.

82

G. Ben Arous, V. Gayrard and A. Kuptsov

5. Universality for non-Gaussian Hamiltonians In this section we extend the results of the previous sections to the case of nonGaussian Hamiltonians. We are able to make this extension only for the pure p-spin models, i.e., ν(r) = rp . In this case we recall that the Hamiltonian is defined as  1 1 gi1 ,...,ip σi1 . . . σip . (5.1) HN (σ) = √ HN,p = √ p N N 1≤i1 ,...,ip ≤N Our assumptions on the random variables (gi1 ,...,ip )1≤i1 ,...,ip ≤N in (5.1) are the same as were made in [BCMN05a] and [BCMN05b] for the number partitioning problem. That is, we assume that their distribution function admits a density ρ(x) that satisfies the following conditions: 1. ρ(x) is even; 0 2. x2 ρ(x)dx = 1; 3. for some  > 0 &∞ ρ(x)1+ dx < ∞; (5.2) −∞

4. ρ(x) has a Fourier transform that is analytic in some neighborhood of zero. We write 1 − log ρˆ(z) = (2π)2 z 2 + c4 (2π)4 z 4 + O(|z|6 ). (5.3) 2 1 . Note that the inequality E(X 4 ) ≥ E(X 2 )2 implies that necessarily c4 < 12 Under these assumptions we will show, using the method introduced by C. Borgs, J. Chayes, S. Mertens and C. Nair in [BCMN05b], that Theorems 1.1 and 1.4 still hold. 5.1. Proof of universality In this subsection we fix p ≥ 1 and prove the analog of Theorem 1.1 in the nonGaussian case assuming that the Hamiltonian is given by (5.1), and that the random variables (gi1 ,...,ip )1≤i1 ,...,ip ≤N satisfy conditions (1)–(4) above. √ Theorem 5.1 (Universality in the Non-Gaussian case). Assume M (N ) = o( N) for p = 1 and M = o(N ) for p ≥ 2. Then P-almost surely the sequence of point processes PN converges weakly to a Poisson point process P on R with intensity given by √ 1 μ(dt) = √ e−t 2 log 2 dt. (5.4) π To prove Theorem 5.1 we essentially prove a local limit theorem. More precisely, for any fixed N let us introduce the Gaussian process ZN on SN that has the same mean and covariance matrix as the process HN (σ) defined in (1.2). We will prove in Theorem 5.2 that P-a.s., for all sequences (σ 1 , . . . , σ ) ∈ X , the joint density of the random variables HN (σ 1 ), . . . , HN (σ ) is well approximated by the 1

joint density of ZN (σ ), . . . , ZN (σ ).

A New REM Conjecture

83

√ Theorem 5.2. Assume M (N ) = o( N ) for p = 1 and M = o(N ) for p ≥ 2. Then P-almost surely for every  ≥ 1 and every bounded Borel set A there exists c > 0 such that uniformly in (σ 1 , . . . , σ ) ∈ X P(HN (σ j ) ∈ A, j = 1, . . . , ) = P(ZN (σ j ) ∈ A, j = 1, . . . , )  M 2  p + O(e−cN ). × 1 + O(Rmax (σ 1 , . . . , σ )) + O Np

(5.5)

Applying Theorem 5.2 together with Theorem 1.1 we get from formula (3.5) that E(PN (A)) = (μ(A)) (1 + o(1)) + O(2M e−cN ) → (μ(A)) , p

(5.6)

which, by Lemma 3.1, implies weak convergence of the sequence of point processes PN to a Poisson point process with intensity measure μ, thus implying Theorem 5.1. We therefore focus on the proof of Theorem 5.2. Proof. First, we obtain from the definition of HN (σ) that     (σ) ∈ (x, x + x) = gi1 ,...,ip σi1 . . . σip ∈ HN 1≤i1 ,...,ip ≤N

√  (aN + xbN , aN + (x + x)bN ) N p .

(5.7)

Following [BCMN05b] we get an integral representation of the indicator function 1HN (σ)∈(x,x+x) &∞  √  √ √  = x bN n sinc f xbN n e2πif gi1 ,...,ip σi1 ...σip − 2πif αN n df,

(5.8)

−∞

where, for brevity, we wrote n = N p , αN = aN + bN (x + x/2), sinc(x) = sin(πx) πx , and the sum in the exponent runs over all possible sequences 1 ≤ i1 , . . . , ip ≤ N. Changing the integration variable in (5.8) from f to −f and applying the resulting formula to the product of indicator functions we arrive at the following representation

1

1HN (σj )∈(xj ,xj +xj ) =

j=1

1

√ xj bN n

j=1

&&& ×

∞ 1

−∞ j=1

√  √  −2πiffj  gi1 ...ip σij ...σij + 2πiffj α(j) n N p 1 sinc fj xj bN n e dffj , (5.9)

(j)

where αN = aN + bN (xj + xj /2). Introducing the variables vi1 ,...,ip =

 j=1

fj σij1 . . . σijp ,

(5.10)

84

G. Ben Arous, V. Gayrard and A. Kuptsov

we rewrite the integral in the above formula as &&&

∞

1

e−2πigi1 ...ip vi1 ,...,ip

−∞ 1≤i ,...,i ≤N 1 p

1

 (j) √ √  sinc fj xj n e2πiffj αN n dffj . (5.11)

j=1

To get an integral representation of the joint density  1  (σ ) ∈ (x1 , x1 + dx1 ), . . . , HN (σ ) ∈ (x , x + dx ) P HN

(5.12)

we have to take the expectation of (5.9) and then let xj → 0 for all j = 1, 2, . . . . As was proved in [BCMN05a] (see Lemma 3.4), the exchange of expectation and integration for p = 1 is justified when the rank of the matrix formed by the row vectors σ 1 , . . . , σ , is . To justify the exchange in our case we introduce an  by N p matrix, Cp (σ 1 , . . . , σ ), defined as follows: for any given set of configurations σ 1 , . . . , σ , the jth row is composed of all N p products, σij1 σij2 . . . σijp over all subsets 1 ≤ i1 , . . . , ip ≤ N. By generalizing the arguments from [BCMN05a] the exchange can then be justified provided that the rank of the matrix Cp (σ 1 , . . . , σ ) is . As we will see in Lemma 5.3 below this holds true P-almost surely when M = o(N ). Given a vector δ ∈ {−1, 1} let nδ be the number of times the column vector δ appears in the matrix Cp : 

 σj1 , . . . , σj ) = δ  . (5.13) nδ = nδ (σ 1 , . . . , σ ) =  j ≤ N p : (σ With this notation we have: Lemma 5.3. Suppose M = o(N ). Then there exists a sequence λN = o(1) such that P-almost surely for all collections (σ1 , . . . , σ ) ∈ X  n   max nδ −  ≤ nλN . (5.14) 2 δ∈{−1,1} Proof. We first prove by induction that the following simple fact holds true: if for

a given sequence of configurations (σ 1 , . . . , σ ) ∈ SN the matrix C1 (σ 1 , . . . , σ ) satisfies condition (5.14) then necessarily the matrix Cp (σ 1 , . . . , σ ) satisfies (5.14) for all p ≥ 1. For p = 1 there is nothing to prove. We now assume that the statement is true for the matrix Cp−1 (σ 1 , . . . , σ ) and prove it for Cp (σ 1 , . . . , σ ). Let σμ , 1 ≤ μ ≤ N denote the columns of the matrix C1 (σ 1 , . . . , σ ). For every column vector σμ let us construct a matrix Cpμ−1 = Cpμ−1 (σ 1 , . . . , σ ) with entries (C Cpμ−1 )ij = (σμ )j (C Cp−1 )ij . nμδ

denote the variable nδ for the matrix For future convenience let the inductive assumption it follows that for all 1 ≤ μ ≤ N  N p−1   maxnμδ −  ≤ N p−1 λN . δ 2

(5.15) Cpμ−1 .

From

(5.16)

A New REM Conjecture

85

Now note that the  × N p matrix Cp (σ 1 , . . . , σ ) can be obtained by concatenating N matrices Cpμ−1 (σ 1 , . . . , σ ) each of size  × N p−1 . Therefore for any sequence of configurations (σ 1 , . . . , σ ) with matrix C1 (σ 1 , . . . , σ ) satisfying (5.14) we have    N p   μ N p−1    μN N p−1  + · · · + n − nδ −  ≤ nδ 1 −   ≤ δ 2 2 2 ≤ N p−1 λN + · · · + N p−1 λN = N p λN (5.17) and the induction is complete. To prove Lemma 5.3 it is thus enough to demonstrate that P-almost surely there are no sequences (σ 1 , . . . , σ ) ∈ X such that C1 (σ 1 , . . . , σ ) violates condition (5.14). Let us prove this by induction in . For  = 1 let us introduce the sets    N   TN = σ ∈ S : max nδ −  ≤ N λN . (5.18) δ∈{−1,1} 2 Then by Chernoff bound   N  1 c |T TN | = 2 ≤ 2N +1 e− 2 N ((1+λN ) log(1+λN )−λN ) . N + i 2

(5.19)

i≥N λN

Let us choose λN = o(1) in such a way that M = o(N λ2N ) and log N = o(N λ2N ). Then using (2.12) we obtain that  1σ = 0, P-a.s. (5.20) c σ∈T TN

which proves the statement for  = 1. Now assume that P-a.s. for all sequences (σ 1 , . . . , σ −1 ) ∈ X ( −1) the matrix C1 (σ 1 , . . . , σ −1 ) satisfies condition (5.14). Since there is only a countable number of sequences (σ 1 , . . . , σ −1 ) we fix a sequence (σ 1 , . . . , σ −1 ) ∈ X ( −1) and prove that P-almost surely there are no configurations σ such that C1 (σ 1 , . . . , σ ) violates (5.14). Let δ ∈ {−1, 1} be given. Define δ 1 (δ) ∈ {−1, 1} −1 as (δ 1 )i = (δ)i for 1 ≤ i ≤  − 1 and also define δ 2 (δ) ∈ {−1, 1} as δ 2 = (δ) . Let us also introduce, for given δ 1 ∈ {−1, 1} −1, the set Nδ1 = {j ≤ N : (σ σj1 , . . . , σj −1 ) = δ 1 }.

(5.21)

By the inductive assumption we conclude that for all δ 1 ∈ {−1, 1}  N   Nδ1 | − −1  ≤ N λN . |N 2 From the definition of Nδ it is not hard to see that for all δ ∈ {−1, 1}   nδ = {j ∈ Nδ (δ) : σ = δ 2 (δ)}.

−1

1

j

(5.22)

(5.23)

Using the above relation together with (2.11) and the assumptions on λN we get that P-almost surely  |N Nδ1 (δ) |  N λN  . (5.24)  nδ − ≤ 2 2

86

G. Ben Arous, V. Gayrard and A. Kuptsov

Therefore from (5.22) and (5.24)  Nδ 1 (δ) | N  N λN |N Nδ 1 (δ) |   |N N λN N    −  ≤ + = N λN . (5.25) nδ −  ≤ nδ − + 2 2 2 2 2 2 The induction is now complete and the lemma is proved.  Lemma 5.3 implies that P-almost surely, for all (σ 1 , . . . , σ ) ∈ X , n nmin (σ 1 , . . . , σ ) = min nδ = (1 + O(λN )),  2 δ∈{−1,1}

(5.26)

and hence, for sufficiently large N , the rank of the matrix Cp (σ 1 , . . . , σ ) is . The exchange of integration and expectation is thus justified. Using once again Lemma 5.3, condition (5.2), and the dominated convergence theorem we obtain that the joint density is

 j  1 √ P HN (σ ) ∈ (xj , xj + dxj ) for j = 1, . . . ,  = bN n dxj j=1

&&& ×

1

∞

−∞ 1≤i ,...,i ≤N 1 p

ρˆ(vi1 ,...,ip )

1

(j) √

e2πiffj αN

n

dffj ,

(5.27)

j=1

(j)

(j)

where we redefined αN = aN + bN xj . We remark for future use that αN = O(aN ) for all 1 ≤ j ≤ . It is straightforward at this point to adapt the saddle point analysis used in [BCMN05b] to calculate the integrals of such type. The only difference is that instead of the matrix C1 (σ 1 , . . . , σ ) with rows formed by row vectors σ 1 , . . . , σ , we use the matrix Cp (σ 1 , . . . , σ ). By analogy with Lemma 5.3 from [BCMN05b] we first approximate the integral in (5.27) by an integral over a bounded domain, i.e., for some c1 > 0 depending on μ1 > 0 &&& ∞ 1

1 (j) √ ρˆ(vi1 ,...,ip ) e2πiffj αN n dffj = −∞ i ,...,i 1 p

&&&

μ1

j=1

1

−μ1 i ,...,i 1 p

ρˆ(vi1 ,...,ip )

1

(j) √

e2πiffj αN

n

dffj + O(e−c1 nmin ).

We next rewrite the integral in the right-hand side of (5.28) as & & & μ1

1 1 e2πinf ·α ρˆ(f · δ)nδ dffj , −μ1

(5.28)

j=1

δ∈{−1,1}

(5.29)

j=1

()  α(1) α  where α = √Nn , . . . , √Nn , f = (f1 , . . . , f ), and where α · f = α1 f1 + · · · + α f is the standard scalar product. Using Lemma 5.3 again we can apply Lemma 5.4 from [BCMN05b] to conclude that given μ1 there are constants c1 (μ1 ) > 0 and μ2 > 0 such that the

A New REM Conjecture

87

following equality holds whenever η1 , . . . , η is a sequence of real numbers with  (j) ηj | ≤ μ2 and ηj αN ≥ 0 for all j = 1, . . . ,  j |η &&&

μ1

1

−μ1 δ

ρˆ(f · δ)nδ

1

(j)

e2πinffj αN dffj

(5.30)

j=1

&&&

μ1

=

1

e2πn(if ·α−η·α)

−μ1

ρˆ(f · δ + iη · δ)nδ

δ

1

1

dffj + O(e− 2 c1 nmin ).

j=1

The values of the shifts η1 , . . . , η are determined by the following system:  nδ δ

(j)

α δj F (iδ · η) = 2πi √N , n n

j = 1, . . . , ,

(5.31)

where we wrote F = − log ρˆ. Since max |R(σ, σ )| is P-almost surely of order o(1) when M = o(N ) we  σ,σ ∈X

can apply Lemma 5.5 from [BCMN05b] and obtain that this system has a unique solution   1 −1  η(α) = B α 1 + O ||α||22 . (5.32) 2π Moreover, for sufficiently small μ1 , & & & μ1

1 1 e2πn(if ·α−η·α) ρˆ(f · δ + iη · δ)nδ dffj −μ1

= e−nGn, (α)



δ

1 2πn

 /2

where Gn, (α) =

j=1



 1 + O(n−1/2 ) + O(a2N /n) + O(Rmax ) ,

 nδ δ

n

(5.33)

F (iδ · η(α)) + 2πη(α) · α.

(5.34)

Therefore   j (σ ) ∈ (xj , xj + dxj ) for j = 1, . . . ,  = P HN



bN √ 2π



1

e−nGn, (α)

dxj

j=1

  1 × 1 + O(n−1/2 ) + O(a2N /n) + O(Rmax ) + O(b N n /2 e− 2 c1 nmin ).

(5.35)

Expanding Gn, we get the approximation nGn, (α) =

 a4 n (α, B −1 α) + O N . 2 n

(5.36)

√ By definition√aN = O( M ). Under the assumptions of Theorem 5.2 we √ get that aN = o( 4 N ) for p = 1, that aN = o( N ) for p ≥ 2, and thus that a4N = o(n). It implies that asymptotically the joint density (5.12) is Gaussian.

88

G. Ben Arous, V. Gayrard and A. Kuptsov

More precisely, it follows from the equations (5.35) and (5.36) that P-a.s., for all collections (σ 1 , . . . , σ ) ∈ X ,    j  −1 bN P HN (σ ) ∈ (xj , xj + dxj ) for j = 1, . . . ,  = √ e−n(α,B α)/2 2π

  4 1 1 a + O(b N n /2 e− 2 c1 nmin ). (5.37) × dxj 1 + O(Rmax ) + O N n j=1 1

By Lemma 5.3 the term O(b N n /2 e− 2 c1 nmin ) is of order o(e−cN ) as N → ∞. This finishes the proof of Theorem 5.2.  p

5.2. Breakdown of universality In this subsection we follow the same strategy as we used in the proof of Theorem 1.4 – we fix a bounded set A and study the first three factorial moments of the random variable PN (A). In the case of the number partitioning problem the following √ theorem implies that the Poisson convergence fails as soon as lim sup M/ N > 0. Theorem 5.4 (Number partitioning problem). Fix p = 1 and let the Hamiltonian ) = ε < ∞ then P-a.s. for every bounded Borel be given by (5.1). If lim sup M(N N set A we have 2

2

(i) lim E(PN (A))1 = μ(A)e−4c4 ε log 2 ; 2 2 2 (ii) lim sup E(PN (A))2 = e2ε log 2−32c4 ε log 2 (μ(A))2 ; (iii) lim sup E(PN (A))3 < ∞. Therefore the limit of the ratio of the second factorial moment to the square of the first is 2 2 2 2 2 2 E(PN (A))2 → e2ε log 2−24c4 ε log 2 = e2ε log 2(1−12c4 ) . 2 E(PN (A))1

(5.38)

1 we conclude that the ratio is strictly larger than Taking into account that c4 < 12 one and thus there is no Poisson convergence for ε > 0.

And in the case of the Sherrington-Kirkpatrick model the failure of Poisson convergence follows from Theorem 5.5 (Sherrington-Kirkpatrick model). Fix p = 2 and let the Hamiltonian ) 1 be given by (5.1). If lim sup M(N = ε < 8 log N 2 then P-a.s. for every bounded Borel set A we have 2

(i) lim E(PN (A))1 = μ(A)e−4c4 ε

2

log2 2 2

;

4 ε log 2 e−32c √ (μ(A))2 ; 1−4ε log 2

(ii) lim sup E(PN (A))2 = (iii) lim sup E(PN (A))3 < ∞.

A New REM Conjecture

89

The ratio of the second factorial moment to the square of the first moment is 2

2

E(PN (A))2 e−24c4 ε log 2 . →√ 2 E(PN (A))1 1 − 4ε log 2

(5.39)

For ε > 0 the above ratio is strictly larger than one and thus convergence to a Poisson point process fails. We will give the proof of Theorem 5.5 only since the case p = 1 is based on essentially the same computations. Proof of Theorem 5.5. As in the proof of Theorem 4.2 we successively prove the statement on the first, second, and third moment. To simplify our computations we will assume that M = εN . (Using the monotonicity argument, the case of general sequences M (N ) can be analyzed just as in Theorem 4.2 of Section 4.) 1. First moment estimate. Following the same steps as in Subsection 5.1 we ap proximate the density of HN (σ) by  1  a2 1  + O(e−c1 n ), e−nGn,1 (αN ) √ 1+O √ (5.40) +O N n 2π N where according to the notation introduced above n = N 2 . In the case M = εN we need a more precise approximation of the function Gn, than given by formula (5.36). Expanding the solution of the system (5.31) as 4c4  nδ 1 −1 B α+ (δ, B −1 α)3 B −1 δ + O(||α||5 ) (5.41) η(α) = 2π 2π n δ

and applying (5.41) we obtain from (5.34) and (5.3) that (2π)2  nδ (δ · η)2 Gn, (α) = − 2 n δ  nδ 4 (δ · η)4 + 2π(η · α) + O(||α||6 ) + c4 (2π) n δ  nδ 1 (δ, B −1 α)4 + O(||α||6 ). = (α, B −1 α) + c4 2 n

(5.42)

δ

Using the approximation for Gn,1 given by formula (5.42), we obtain  a6 α4 α2 nGn,1 = N + c4 N + O N2 , 2 n n

(5.43)

where αN = aN + bN x. Since α4N /n = 4c4 ε2 log2 2(1 + o(1)), we see that up to an ε-dependent multiplier the density of HN (σ) is given by the normal density, more precisely, the density of HN (σ) is   1  α2 2 2 2 1 √ e−αN /2 e−4c4 ε log 2 1 + O √ + O(e−c1 n ). (5.44) +O N n 2π N

90

G. Ben Arous, V. Gayrard and A. Kuptsov

Therefore the first factorial moment of PN (A) is 2

μ(A)e−4c4 ε

log2 2

(1 + o(1)) + o(1)

(5.45)

and part (i) is proven. 2. Second moment estimate. When analyzing the second moment   1  E(PN (A))2 = P HN (σ ) ∈ A, HN (σ 2 ) ∈ A ,

(5.46)

(σ1 ,σ2 )∈X 2

it is useful to distinguish between “typical” and “atypical” sets of configurations (σ 1 , . . . , σ ) ∈ X , a notion introduced in [BCMN05a]. 2 → ∞ and consider the  × n Take some sequence θN → 0 such that N θN 1

matrix Cp (σ , . . . , σ ), introduced in Subsection 5.1. Then all but a vanishing

obey the condition fraction of the configurations (σ 1 , . . . , σ ) ∈ SN   n  max  nδ −  ≤ nθN . (5.47) 2 δ∈{−1,1} When M = o(N ), Lemma 5.3 guarantees that for a properly chosen sequences θN , P-almost surely, all the sampled sets (σ 1 , . . . , σ ) ∈ X obey condition (5.47). It is no longer the case when M = εN and thus we have to consider the contribution from the sets violating (5.47). Fix p = 2 and  = 2. Let a sequence θN → 0 be given, and define    I= P HN (σ 1 ) ∈ A, HN (σ 2 ) ∈ A , (5.48) σ1 ,σ2

where the sum runs over all pairs of distinct configurations (σ 1 , σ 2 ) ∈ X 2 satisfying condition (5.47) (the so-called “typical” configurations). Also define    II = P HN (σ 1 ) ∈ A, HN (σ 2 ) ∈ A , (5.49) σ1 ,σ2

where the sum is over all pairs of distinct configurations (σ 1 , σ 2 ) ∈ X 2 violating (5.47) (the “atypical” configurations). For later use we introduce the quantities Ig and IIIg – the analogs of the variables I and II in the case where the random variables (gi1 ,i2 )1≤i1 ,i2 ≤N are i.i.d. standard normals. Lemma 5.6. Let θN = and

√

1 . N log2 N

1 If ε ∈ (0, 4 log 2 ) then P-almost surely II = o(1) 2

I=

2

e−32c4 ε log 2 (μ(A))2 √ (1 + o(1)). 1 − 4ε log 2

(5.50)

Proof. To prove that II is o(1) let us bound the quantity II by IIIg and show that 1 IIIg is almost surely negligible for θN = √N log 2 N . We start with the proof of the second statement, for which we will need the following simple observation.

A New REM Conjecture

91

Consider a sequence λN → 0 such that N λN → ∞ and define the set of configurations σ ∈ {−1, 1}N with almost equal number of spins equal to 1 and to −1 :     TN = σ ∈ SN : #{σi = 1} − #{σi = −1} ≤ N λN . (5.51) It is not hard to prove that configurations σ 1 , σ 2 ∈ TN with overlap |R(σ 1 , σ 2 )| ≤ λN must satisfy (5.47) with θN = λ2N . Therefore the set of pairs (σ 1 , σ 2 ) ∈ X 2 violating condition (5.47) with θN = λ2N is contained in the set   (σ 1 , σ 2 ) ∈ X 2 : |R(σ 1 , σ 2 )| > λN or σ 1 ∈ TNc or σ 2 >∈ TNc . (5.52) 1 Thus to prove that IIIg is o(1) for θN = √N log 2 N it suffices to prove that  P(HN (σ 1 ) ∈ A, HN (σ 2 ) ∈ A) = o(1), (5.53) σ1 ,σ2

where the summation√is over all pairs of distinct configurations contained in the set (5.52) with λN = θN = N 1/41log N . Let us prove (5.53). Since we already proved in Section 4 that the contribution from the set 1 2

(σ , σ ) ∈ X 2 : |R(σ 1 , σ 2 )| > λN (5.54) to the sum (5.46) is negligible, it is enough to consider the sum (5.46) restricted to the set   (σ 1 , σ 2 ) ∈ X 2 : σ 1 or σ 2 ∈ (T (5.55) TNc )X and |R(σ 1 , σ 2 )| < λN .  2 N −N J (λN ) (1 + O(λ2N )) and By Stirling’s formula we obtain that |T TNc | = πN 2 e using this fact one can prove, proceeding as in part (i) of Theorem 2.4, that # 2 M −N J (λN ) c X c X |(T TN ) | = E|(T 2 e TN ) |(1 + o(1)) = (1 + o(1)). (5.56) πN Thus for large enough N the contribution from the set (5.55) to the sum (5.46) is bounded by # 2b12 2 2M −N J (λN ) C 1+b (M log 2−1/2 log M) 2 2 e e 12 , (5.57) 2M πN 2  where the constant C is from (4.17). Using that N J (λN ) = 12 N λ2N + O log14 N we can further bound (5.57) by #

2 − 1 N λ2N (1−4ε log 2) e 2 , (5.58) πN which is o(1) by the choice of λN and ε. Our next step is to bound the sum II by IIIg . For this purpose we need to give an estimate of the joint density of HN (σ 1 ), HN (σ 2 ) that would be valid also 1 2 for pairs (σ , σ ) violating condition (5.47). As we already noted for such (σ 1 , σ 2 ) the results of Subsection 5.1 cannot be applied directly since it is no longer true 2C

92

G. Ben Arous, V. Gayrard and A. Kuptsov

that max |R(σ, σ )| is o(1). Fortunately, we have only 2 × 2 covariance matrix  σ,σ ∈X

B(σ 1 , σ 2 ) and using this fact we can easily adapt the results of Subsection 5.1 to the case where max |R(σ, σ )| is not o(1). We start with formula (5.27) which, in  σ,σ ∈X

the case  = 2, can be rewritten as  j  P HN (σ ) ∈ (xj , xj + dxj ) for j = 1, 2 && ∞  √  1 (1) (2) = b2N n dx1 dx2 ρˆ(vi1 ,i2 )e2πi n f1 αN +f2 αN df1 dff2 . (5.59) −∞ 1≤i ,i ≤N 1 2

We can rewrite the integral in the above expression as && ∞ 1 ρˆ(f · δ)nδ e2πinf ·α df1 dff2 ,

(5.60)

where δ ∈ {−1, 1}2. Since the function ρˆ is even we obtain 1 ρˆ(f · δ)nδ = ρˆ(f1 + f2 )n(1,1) +n(−1,−1) ρˆ(f1 − f2 )n(1,−1) +n(−1,1) .

(5.61)

−∞ δ

δ

One obvious relation between n(1,1) , n(−1,−1) , n(1,−1) and n(−1,1) is n(1,1) + n(−1,−1) + n(1,−1) + n(−1,1) = n. The other one we obtain by noting that n(1,1) +n(−1,−1) − n(−1,1) − n(1,−1) =



2 σi1 σj1 σi2 σj2 = nR12 .

(5.62)

(5.63)

i,j

Therefore



2 n(1,1) + n(−1,−1) = 12 n(1 + R12 ), 1 2 n(1,−1) + n(−1,1) = 2 n(1 − R12 ).

(5.64)

By Theorem 2.4 we conclude that P-a.s. max |R(σ, σ )| < 1 and therefore, for  σ,σ ∈X

some positive constant c, n(1,1) + n(−1,−1) ≥ n(1,−1) + n(−1,1) ≥ cn. The above inequality allows us to approximate (5.60) by & & μ1 1   e2πn(if ·α−η·α) ρˆ(f · δ + iη · δ)nδ df1 dff2 + O e−c1 n . −μ1

(5.65)

(5.66)

δ

Adapting the proof of Lemma 5.5 from [BCMN05b] we get & & μ1 1 e2πn(if ·α−η·α) ρˆ(f · δ + iη · δ)nδ df1 dff2 −μ1

δ

$ = e−nGn,2 (α)

 1  a2 det B(σ 1 , σ 2 )  1+O √ +O N . 2πn n n

(5.67)

A New REM Conjecture Finally, we obtain that  j  P HN (σ ) ∈ (xj , xj + dxj ) for j = 1, 2 = dx1 dx2 bn e−nGn,2 (α) √  1  a2   det B  1+O √ +O N + O b2N ne−c1 n . × 2π n n From (5.42) we see that for some constant C  a6 n n 16c4 α4N N ≥ (α, B −1 α) + C, nGn,2 = (α, B −1 α) + + O 2 )3 n 2 (1 + R12 n2 2 and thus the joint density of HN (σ 1 ), HN (σ 2 ) is bounded by √  a2  1   b2N det B −n(α,B −1 α)/2−C  e +O N + O b2N ne−c1 n . 1+O √ 2π n n

93

(5.68)

(5.69)

(5.70)

This last bound for the joint density clearly implies that the sum II could be bounded by IIIg plus an error resulting from the second term. But the cumulative error coming from the second term is of order O 22M b2N ne−c1 n which is negligible even in the case lim sup M/N > 0. To prove the second statement of the lemma we will approximate the sum I by Ig , which was already calculated in Section 4. We first notice that for sequences (σ 1 , . . . , σ ) satisfying condition (5.47) Rmax (σ 1 , . . . , σ ) = O(θN ). Furthermore, for configurations (σ 1 , . . . , σ ) obeying condition (5.47) it is possible to derive from (5.42) that  a6  a2 1 a4 Gn, (α) = (α, B −1 α) + c4 (1 + 3( − 1)) N + O N3 + O N2 θN . (5.71) 2 n n n For the details of the derivation we refer to Subsection 5.4 of [BCMN05b] and in particular to formula (5.57) in there. Using formula (5.71) with  = 2 and substituting it into (5.35) we obtain that  α2   1 2 2 I = Ig e−32c4 ε log 2 1 + O √ +O N . (5.72) n n 

This finishes the proof of Lemma 5.6.

To conclude the calculation of the second moment we notice that summing I and II we get the second factorial moment 2

2

e−32c4 ε log 2 (μ(A))2 √ (1 + o(1)). E(PN (A))2 = 1 − 4ε log 2

(5.73)

Assertion (i) of Theorem 5.5 is thus proven. 3. Third moment estimate. To deal with the third moment   1  P HN (σ ) ∈ A, HN (σ 2 ) ∈ A, HN (σ 3 ) ∈ A , E(PN (A))3 = (σ1 ,σ2 ,σ3 )∈X 3

(5.74)

94

G. Ben Arous, V. Gayrard and A. Kuptsov

we use the same strategy as we used to calculate the second moment. In particular, we fix  = 3 and split the sum (5.74) in two parts:   1  I= P HN (σ ) ∈ A, HN (σ 2 ) ∈ A, HN (σ 3 ) ∈ A , (5.75) σ1 ,σ2 ,σ3

where the sum runs over all sequences of distinct configurations (σ 1 , σ 2 , σ 3 ) ∈ X 3 satisfying condition (5.47), and   1  II = P HN (σ ) ∈ A, HN (σ 2 ) ∈ A, HN (σ 3 ) ∈ A , (5.76) σ1 ,σ2 ,σ3

where the sum is over all sequences of distinct configurations (σ 1 , σ 2 , σ 3 ) ∈ X 3 violating (5.47). By exactly the same argument as in the calculation of the second moment the contribution from the “typical” collections, I, is bounded. We therefore concentrate on the analysis of the contribution from the “atypical” collections, II. Since we are interested only in the estimate of the third moment from above  it suffices to bound the joint density P HN (σ j ) ∈ (xj , xj + dxj ) for j = 1, 2, 3 . We start with formula (5.27) which, in the case  = 3, can be rewritten as  j  P HN (σ ) ∈ (xj , xj + dxj ) for j = 1, 2, 3 = b3N n dx1 dx2 dx3 &&& ∞ 1  √  (1) (2) (3) 2πi n f1 αN +f2 αN +f3 αN × ρˆ(vi1 ,i2 ,i3 )e df1 dff2 dff3 . −∞ i ,i ,i 1 2 3

(5.77) We can rewrite the integral in the above expression as &&& ∞ 1 ρˆ(f · δ)nδ e2πinf ·α df1 dff2 dff3 ,

(5.78)

−∞ δ

where δ ∈ {−1, 1}3. Since the function ρˆ is even we obtain 1 ρˆ(f · δ)nδ = ρˆ(f1 + f2 + f3 )n1 ρˆ(f1 + f2 − f3 )n2 δ

× ρˆ(f1 − f2 + f3 )n3 ρˆ(f1 − f2 − f3 )n4 , where

⎧ ⎪ n1 ⎪ ⎨ n2 n3 ⎪ ⎪ ⎩ n4

= n(1,1,1) + n(−1,−1,−1) , = n(1,1,−1) + n(−1,−1,1) , = n(1,−1,1) + n(−1,1,−1) , = n(−1,1,1) + n(1,−1,−1) .

By definition of the matrix C2 (σ 1 , σ 2 , σ 3 ) we have ⎧ 2 n1 + n2 − n3 − n4 = nR12 , ⎪ ⎪ ⎨ 2 n1 − n2 − n3 + n4 = nR23 , 2 n − n + n − n = nR , ⎪ 1 2 3 4 31 ⎪ ⎩ n1 + n2 + n3 + n4 = n.

(5.79)

(5.80)

(5.81)

A New REM Conjecture Solving the system

⎧ n1 ⎪ ⎪ ⎨ n2 n3 ⎪ ⎪ ⎩ n4

= = = =

1 4 n(1 1 4 n(1 1 4 n(1 1 4 n(1

95

2 2 2 + R12 + R23 + R31 ), 2 2 2 + R12 − R23 − R31 ), 2 2 2 − R12 − R23 + R31 ), 2 2 2 − R12 + R23 − R31 ).

From Theorem 2.4 we obtain that P-almost surely   lim sup 1max J R(σ 1 , σ 2 ) ≤ ε log 2. 2 N →∞ σ ,σ ∈X

(5.82)

(5.83)

Since the function J is monotone we obtain from (5.83) and from assumption 1 ε < 8 log 2 that P-a.s. |R(σ 1 , σ 2 )| < lim sup 1max 2 N →∞ σ ,σ ∈X

1 . 2

(5.84)

It implies that min{n1 , n2 , n3 , n4 } ≥ cn for some positive constant c. It allows us to approximate the integral in (5.77) by & & & μ1 1   e2πn(if ·α−η·α) ρˆ(f · δ + iη · δ)nδ df1 dff2 dff3 + O e−cn . (5.85) −μ1

δ

Adapting the proof of Lemma 5.5 from [BCMN05b] we get & & & μ1 1 e2πn(if ·α−η·α) ρˆ(f · δ + iη · δ)nδ df1 dff2 dff3 −μ1

δ

$  a2  1 det B(σ 1 , σ 2 )  +O N . 1+O √ = e−nGn,3 (α) 3/2 n n (2πn)

(5.86)

Next, after a little algebra, one can derive from (5.42) that for some constant C n nGn,3 ≥ (α, B −1 α) + C (5.87) 2 (σ 1 ), HN (σ 2 ), HN (σ 3 ) and this estimate is enough to bound the joint density of HN 1 2 3 by the joint density of ZN (σ ), ZN (σ ), ZN (σ ). Thus Theorem 5.5 is proved.  Acknowledgment ˇ We thank J.Cern´ y ´ for a helpful reading of the first versions of the manuscript. ´ V. Gayrard thanks the Chair of Stochastic Modeling of the Ecole Polytechnique F´ ´ed´erale of Lausanne for financial support. A. Kuptsov gratefully acknowledges the support of NSF grant DMS-0102541 “Collaborative Research: Mathematical Studies of Short-Ranged Spin Glasses.”

References [BFM04] H. Bauke, S. Franz, and S. Mertens, Number partitioning random energy model. Journal of Statistical Mechanics: Theory and Experiment, page P04003, 2004 [BM04] H. Bauke and S. Mertens, Universality in the level statistics of disordered systems. Physical Review E 70, 025102(R), 2004

96

G. Ben Arous, V. Gayrard and A. Kuptsov

[BCMN05a] C. Borgs, J.T. Chayes, S. Mertens, and Ch. Nair, Proof of the local REM conjecture for number partitioning I: Constant energy scales. preprint, 2005. http://arxiv.org/abs/cond-mat/0501760, to appear in Random Structures & Algorithms. [BCMN05b] C. Borgs, J.T. Chayes, S. Mertens, and Ch. Nair, Proof of the local REM conjecture for number partitioning II: Growing energy scales. preprint, 2005. http://arxiv.org/abs/cond-mat/0508600 [BCP01] C. Borgs, J.T. Chayes and B. Pittel, Phase transition and finite-size scaling for the integer partitioning problem Random Structures & Algorithms, 19 (2001), no. 3-4, 247–288. [B06] A. Bovier, Statistical mechanics of disordered systems. Cambridge Series in Statistical and Probabilistic Mathematics 18, Cambridge University Press, 2006. [BK06a] A. Bovier and I. Kurkova, Local Energy Statistics in Disordered Systems: A Proof of the Local REM Conjecture. Commun. Math. Phys. 263, 513–533 (2006). [BK06b] A. Bovier and I. Kurkova, Local energy statistics in spin glasses, Journal of Statistical Physics, (Online). [BK04] A. Bovier and I. Kurkova, Poisson convergence in the restricted k-partitioning problem. WIAS preprint 964, to appear in Random Structures & Algorithms, 2006. [BKL02] A. Bovier, I. Kurkova, and M. L¨owe, Fluctuations of the free energy in the REM and the p-spin SK models. Ann. Probab. 30, 605–651 (2002) [Kal83] O. Kallenberg, Random measures. 3rd, rev. and enl. ed. New York : Academic Press, 1983 [Mer00] S. Mertens, Random costs in combinatorial optimization. Phys. Rev. Lett. 84 (2000), no. 6, 1347–1350. [Tal03] M. Talagrand, Spin glasses: a challenge for mathematicians: cavity and mean c 2003 field models. Berlin; New York: Springer,  G´ ´erard Ben Arous Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland and Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012 e-mail: [email protected] Veronique ´ Gayrard Laboratoire d’Analyse, Topologie, Probabilit´es CMI, 39 rue Joliot-Curie, F-13453 Marseille Cedex e-mail: [email protected], [email protected] Alexey Kuptsov Swiss Federal Institute of Technology (EPFL), CH-1015 Lausanne, Switzerland and Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012 e-mail: [email protected]

Progress in Probability, Vol. 60, 97–116 c 2008 Birkhauser  ¨ Verlag Basel/Switzerland

The Biham-Middleton-Levine Traffic Model for a Single Junction Itai Benjamini, Ori Gurel-Gurevich and Roey Izkovsky Abstract. In the Biham-Middleton-Levine traffic model cars are placed in some density p on a two dimensional torus, and move according to a (simple) set of predefined rules. Computer simulations show this system exhibits many interesting phenomena: for low densities the system self organizes such that cars flow freely while for densities higher than some critical density the system gets stuck in an endless traffic jam. However, apart from the simulation results very few properties of the system were proven rigorously to date. We introduce a simplified version of this model in which cars are placed in a single row and column (a junction) and show that similar phenomena of self-organization of the system and phase transition still occur. Mathematics Subject Classification (2000). 60K35. Keywords. Traffic, phase transition, cellular automata.

1. The BML traffic model The Biham-Middleton-Levine (BML) traffic models was first introduced in [3] published 1992. The model involves two types of cars: “red” and “blue”. Initially the cars are placed in random with a given density p on the N × N torus. The system dynamics are as follows: at each turn, first all the red cars try to move simultaneously a single step to the right in the torus. Afterwards all blue cars try to move a single step upwards. A car succeeds in moving as long as the relevant space above/beside it (according to whether it is blue/red) is vacant. The basic properties of this model are described in [3] and some recent more subtle observations due to larger simulations are described in [4]. The main and most interesting property of the system originally observed in simulations is a phase transition: for some critical density pc one observes, that while filling the torus randomly with cars in density p < pc the system self organizes such that after some time all cars flow freely with no car ever running into another car (see Figure 1), by slightly changing the density to some p > pc not only does system

98

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky

Figure 1. Self organization for p = 0.3: the initial configuration in the left organizes after 520 steps to the “diagonal” configuration on the right which flows freely. not reach a free flow, but it will eventually get stuck in a configuration in which no car can ever move (see Figure 2). Very little of the above behaviour is rigorously proven for the BML model. The main rigorous result is that of Angel, Holroyd and Martin [2], showing that for some fixed density p < 1, very close to 1, the probability of the system getting stuck tends to 1 as N → ∞. In [1] one can find a study of the very low density regime (when p = O( N1 )). First, we introduce a slight variant of the original BML model by allowing a car (say, red) to move not only if there is a vacant place right next to it but also if there is a red car next to it that moves. Thus, for sequence of red cars placed in a row with a single vacant place to its right – all cars will move together (as oppose to only the rightmost car in the sequence for the original BML model). Not only does this new variant exhibits the same phenomena of self-organization and phase transition, they even seem to appear more quickly (i.e., it takes less time for the system to reach a stable state). Actually the demonstrated simulations (Figures 1, 2) were performed using the variant model. Note that the results of [2] appear to apply equally well to the variant model. In the following, we will analyze a simplified version of the BML model: BML on a single junction. Meaning, we place red cars in some density p on a single row of the torus, and blue cars are placed in density p on a single column. We will show that for all p the system reaches optimal speed 1 , depending only on p. For p < 0.5 we will show the system reaches speed 1, while for p > 0.5 the speed cannot be 1, but the system will reach the same speed, regardless of the initial configuration. Moreover, at p = 0.5 the system’s behaviour undergoes a phase transition: we will prove that while for p < 0.5 the stable configuration will have linearly many 1 The

speed of the system is the asymptotic average rate in which a car moves – i.e., number of actual moves per turn.

The Biham-Middleton-Levine Traffic Model for a Single Junction

99

Figure 2. Traffic jam for p = 0.35: the initial configuration on the left gets to the stuck configuration on the right after 400 steps, in which no car can ever move. sequences of cars, for p > 0.5 we will have only O(1) different sequences after some time. We will also examine what happens at a small window around p = 0.5. Note that in the variant BML model (and unlike the original BML model) car sequences are never split. Therefore, the simplified version of the variant BML model can be viewed as some kind of 1-dimensional coalescent process. Much of the proofs below rely on the fact that we model BML on a symmetric torus. Indeed, the time-normalization of Section 2.2 would take an entirely different form if the height and width were not equal. We suspect that the model would exhibit similar properties if the height and width had a large common denominator, e.g., if they had a fixed proportion. More importantly, we believe that for height and width with low common denominator (e.g., relatively prime), we would see a clearly different behaviour. As a simple example, note that in the relatively prime case, a speed of precisely 1 cannot be attained, no matter how low is p, in contrast with Corollary 3.14. This dependence on the arithmetic properties of the dimensions is also apparent in [4].

2. The junction model 2.1. Basic model We start with the exact definition of our simplified model. On a cross shape, containing a single horizontal segment and a single vertical segment, both of identical length N , red cars are placed in exactly pN randomly (and uniformly) chosen locations along the row, and blue cars are similarly placed in pN locations along the column. p will be called the density of the configuration. For simplicity and consistency with the general BML model, we refer to the cars placed on the horizontal segment as red and those on the vertical segment as blue. For simplicity

100

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky

we may assume that the junction, i.e., the single location in the intersection of the segments, is left unoccupied. The segments are cyclic – e.g., a red car located at the rightmost position (N − 1) that moves one step to the right re-emerges at the leftmost position (0). At each turn, all the red cars move one step to the right, except that the red car that is just left of the junction will not move if a blue car is in the junction (i.e., blocking it), in which case also all the red cars immediately following it will stay still. Afterwards, the blue cars move similarly, with the red and blue roles switched. As in the original BML, we look at the asymptotic speed of the system, i.e., the (asymptotic) average number of steps a car moves in one turn. It is easily seen that this speed is the same for all blue cars and all red cars, since the number of steps any two cars of the same color have taken cannot differ by more than N . It is somewhat surprising, perhaps, that these two speed must also be the same. For instance, there is no configuration for which a blue car completes 2 cycles for every 1 a red car completes. 2.2. Time-normalized model Though less natural, it will be sometimes useful to consider the equivalent description, in which two rows of cars – red and blue – are placed one beneath the other, with a “special place” – the junction – where at most one car can be present at any time. In every step first the red line shifts one to the right (except cars immediately to the left of the junction, if it contains a blue car) and then the blue line does the same. Furthermore, instead of having the cars move to the right, we can have the junction move to the left, and when a blue car is in the junction, the (possibly empty) sequence of red cars immediately to the left of the junction moves to the left, and vice verse. Figure 3 illustrates the correspondence between these models. From the discussion above we get the following equivalent system, which we will call the time-normalized junction: −1 N −1 1. Fix S = N − 1 ∈ ZN and fix some initial configuration {Ri }N i=0 , {Bi }i=0 ∈ {0, 1}N representing the red and blue cars respectively,  i.e., Ri = 1 iff there’s a red car at the ith place. We require that Ri = Bi = p · N , and at place S (= N − 1) there is at most one car in both rows2. 2. In each turn: • If place S contains a blue car, and place S − 1 contains a red car (if BS = RS−1 = 1), push this car one step to the left. By pushing the car we mean also moving all red cars that immediately followed it one step to the left, i.e., set RS−1 = 0, RS−i = 1 for i = minj≥1 [RS−j = 0]. • If place S does not contain a blue car and place S − 1 contains both a red and a blue car (if BS = 0 and RS−1 = BS−1 = 1), push the blue car at S − 1 one step to the left (set BS−1 = 0 and BS−i = 1 for i = minj≥1 [BS−j = 0]). • set S = S − 1 2 The

last requirement is that the junction itself can contain only one car at the beginning.

The Biham-Middleton-Levine Traffic Model for a Single Junction

101

Figure 3. On the left: a junction configuration and the analogous configuration beneath it. The junction is marked with an asterisk. On the right: same configuration after 3 turns in both views. Note that indeed the dynamics above guarantee that after a turn place (S −1) – the new junction – contains at most one car. Generally, as long as cars flow freely, the time-normalized system configuration does not change (except for the location of the junction). Cars in the time-normalized system configuration actually move only when some cars are waiting for the junction to clear (in the non-time-normalized system).

3. Analysis of an (N, p) junction Our analysis of the junction will argue that for any p, regardless of the initial configuration, the system will reach some optimal speed, depending only on p. First let us state what is this optimal speed: Theorem 3.1. For a junction with density p, the maximal speed for any configura1 tion is min(1, 2p ). Proof. Obviously, the speed cannot exceed 1. Observe the system when it reaches its stable state, and denote the speed by s. Thus, at time t, a car has advanced ts(1 + o(ts)) steps (on average) which means that it has passed the junction ts/N (1 + o(1)) times. As only one car can pass the junction at any time, the total number of cars passing the junction at time t is bounded by t. Therefore, we have 2pN × ts/N ≤ t which implies s ≤ 1/2p.  We will now show that the system necessarily reaches these speeds from any starting configuration.

102

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky

3.1. The case p < 0.5 We begin by proving that for p < 0.5 the junction will eventually reach speed very close to 1. A system in a stable state is said to be free-flowing if no car is ever waiting to enter the junction. Lemma 3.2. A junction is free-flowing iff the time-normalized junction satisfies: (1) For all 0 ≤ i ≤ N − 1 there is only one car in place i in both rows. (2) For all 0 ≤ i ≤ N − 1 if place i contains a blue car, place (i − 1) mod N does not contain a red car. 

Proof. Obviously, this is just a reformulation of free-flowing.

We will now turn to show that for p < 0.5, following the system dynamics of the time-normalized junction will necessarily bring us to a free flowing state, or at worst an “almost” free flowing state, meaning a state for which the system speed will be arbitrarily close to 1 for large enough N . For this let us consider some configuration and look at the set of “violations” to Lemma 3.2, i.e., places that either contain both a blue and a red car, or that contain a blue car and a red car is in the place one to the left. As the following lemma will show, the size of the set of violations is very closely related to the system speed, and posses a very important property: it is non-increasing in time. More formally, for a configuration R, B we define two disjoint sets for the two types of violations: VB = {0 ≤ i ≤ N − 1 : Ri−1 = Bi−1 = 1, Bi = 0}, VR = {0 ≤ i ≤ N − 1 : Ri−1 = Bi = 1}. Also, let V = VB ∪ VR be the set of all violations. It will be sometimes useful to −1 refer to a set of indicators X = {X(i)}N i=0 , where each X(i) is 1 if i ∈ V and 0 N −1 otherwise, thus |V | = i=0 X(i). For a junction with some initial configuration R, B, let Rt , B t be the system configuration at time t, and let Vt = VB t ∪ VRt be the set of violations for this configuration, and X t be the corresponding indicator vector. Similarly, let S t denote the junction’s position at time t. Lemma 3.3. 1. |V Vt+1 | ≤ |V Vt |. 2. For any t, the system speed is at least (1 +

|V Vt | −1 N )

≥1−

|V Vt | N .

Proof. Property (1) follows from the system dynamics. To see this, examine the three possible cases for what happens at time t: 1. If in turn t place S t does not contain a violation then the configurations do not change during the next turn, i.e., Rt+1 = Rt , B t+1 = B t hence clearly |V Vt+1 | = |V Vt |. 2. If S t ∈ VB t , then the configuration B t+1 changes in two places:

The Biham-Middleton-Levine Traffic Model for a Single Junction

103

(a) BSt t −1 is changed from 1 to 0. Thus, place S t is no longer in VB t , i.e., X t (S t ) changes from 1 to 0. (b) BSt t −i is changed from 0 to 1 for i = minj≥1 (BSt t −j = 0). This may affect X t+1 (S t − i), and X t+1 (S t − i + 1). However, for place S t − i + 1, by changing BSt t −i from 0 to 1 no new violation can be created (since t+1 by definition of i, BSt+1 (S t − i + 1) = 1 iff RSt+1 t −i+1 = 1, so X t −i = 1 t+1 regardless of BS t −i ). For other indices X t+1 (i) = X t (i) since R, B do not change, so between times t and t + 1, we have that X t (S t ) changes from 1 to 0, and at worst only one N −1 other place - X t (S t − i + 1) changes from 0 to 1, so |V Vt+1 | = i=0 X t+1 (i) ≤ N −1 t Vt |. i=0 X (i) = |V 3. Similarly, if place S t ∈ VRt then the configuration Rt+1 changes in two places: t t (a) RSt+1 t −1 is changed from 1 to 0. Thus, X (S ) changes from 0 to 1. (b) RS t −i is changed from 0 to 1 for i = minj≥1 (RSt t −j = 0), affecting X t+1 (S t − i), X t+1 (S t − i + 1). However for place S t − i changing RS t −i does not affect whether this place is a violation or not, so at worst X t (S t − i + 1) changed from 0 to 1. By the same argument we get |V Vt+1 | ≤ |V Vt |. Vt |. Therefore, |V Vt+1 | ≤ |V For property (2) we note that in the time-normalized system, following a k −1 ) where k is the specific car in the system, its “current speed” is NN+k = (1 + N number of times the car was pushed to the left during the last N system turns. We note that if a car at place j is pushed to the left at some time t, by some violation at place S t , this violation can reappear only to the left of j, so it can push the car again only after S t passes j. Hence any violation can push a car to the left only once in a car’s cycle (i.e., N moves). Since by (1) at any time from t onwards the number of violations in the system is at most |V Vt |, then each car is pushed left only |V Vt | times in N turns, so its speed from time t onwards is at least (1 + |VVNt | )−1 > 1 − |VVNt | as asserted.  With Lemma 3.3 at hand we are now ready to prove system self organization for p < 0.5. We will show that for p < 0.5, after 2N system turns |V Vt | = O(1), and hence deduce by part (2) of Lemma 3.3 the system reaches speed 1 − O( N1 ) → 1 (as N → ∞). As the junction advances to the left it pushes some car sequences, thus affecting the configuration to its left. The next lemma will show that when p < 0.5, for some T < N , the number of cars affected to the left of the junction is only a constant, independent of N . Lemma 3.4. Consider a junction with density p < 0.5. There exists some constant p C = C(p) = 1−2p , independent of N , for which: From any configuration R, B with junction at place S there exist some 0 < T < N such that after T turns:

104

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky

(1) For i ∈ {S − T, . . . , S}, X T (i) = 0 (i.e., there are no violations there). (2) For i ∈ {S + 1, . . . , N − 1, 0, . . . , S − T − C}, RiT = Ri0 and BiT = Bi0 (R, B are unchanged there). Proof. First let us consider T = 1. For T = 1 to not satisfy the lemma conclusions, there need to be a car sequence (either red or blue) of length exceeding C, which is pushed left by the junction as it moves. Progressing through the process, if, for some T < N − C, the sequence currently pushed by the junction is shorter then C, then this is the T we seek. Therefore, for the conclusions not to hold, the length of the car sequence pushed by the junction must exceed C for all 0 < T < N . If this is the case, then leaving the junction we see alternating red and blue sequences, all of lengths exceeding C and one vacant place after any blue sequence and before any red one. However, if this is the case for all 0 < T ≤ T then the average number of 2C cars per location in {S − T , . . . , S} at time T must be at least 2C+1 (at least 2C cars between vacant places). Therefore, the total number of cars in {S − T , . . . , S} 2p 2C T = 2pT (recall that C = 1−2p ). at time T is more than 2C+1 Since there are only 2pN cars in the system, this cannot hold for all T up to N . Thus, there must be some time for which the conclusions of the lemma are satisfied.  We are now ready to easily prove the main result for this section. Theorem 3.5. A junction of size N with density p < 0.5 reaches speed of 1 − from any initial configuration.

C(p) N

Proof. Let R, B be some initial configuration, with S = N − 1 and let V the corresponding set of violations and X the matching indicators vector. By Lemma 3.4 there exist T0 > 0 for which X T0 (i) = 0 for i ∈ [N − 1 − T0 , N − 1]. Now starting at RT0 , B T0 and S = N − 1 − T0 reusing the lemma there exist T1 > 0 s.t. X T0 +TT1 (i) = 0 for i ∈ [N −1−T T1, N −1−T T0], and also, as long as N −1−T T0 −T T1 > 2p , X T0 +TT1 (i) = X T0 (i) = 0 for i ∈ [N − 1 − T0 , N − 1] as well. C(p) = 1−2p  Proceeding in this manner until T = Ti ≥ N we will get that after T turns, X T (i) = 0 for all but at most C(p) places, hence by Lemma 3.3 the system speed from this time onward is at least 1 − C(p)  N . We remark one cannot prove an actual speed 1 (rather than 1 − o(1)) for the case p < 12 since this is definitely not the case. Figure 4 (a) demonstrates a general construction of a junction with speed 1 − N1 for all N . Finally we remark that a sharper constant C (p) can be provided, that will also meet a provable lower bound for the speed (by constructions similar to the above). This C (p) is exactly half of C(p). We will see proof of this later in this paper as we obtain Theorem 3.5 as a corollary of a different theorem, using a different approach.

The Biham-Middleton-Levine Traffic Model for a Single Junction

105

p Figure 4. (a) A junction configuration with density p = 13 ( 1−2p = 1) 1 not reaching speed 1 (speed is 1 − N ). (b) A similar construction for p = 2) reaching speed of only 1 − N2 . p = 0.4 ( 1−2p

3.2. Number of segments for p < 0.5 The proof in the previous section were combinatorial in nature and showed that for a density p < 0.5 the system can slow-down only by some constant number independent of N regardless of the configuration. As we turn to examine the properties of the stable configuration of course we can not say much if we allow any initial configuration. There are configurations in which the number of different segments in each row will be Θ(N ) while clearly if the cars are arranged in a single red sequence and a single blue sequence in the first place, we will have only one sequence of each color at any time. However, we will show that for a random initial configuration, the system will have linearly many different segments of cars with high probability. Theorem 3.6. A junction of size N with density p < 0.5, started from a random initial configuration, will have Θ(N ) different segments at all times with high probability (w.r.t. N ). Proof. As we have already seen in the proof of 3.4, as the system completes a full round, since every place (but a constant number of places) contains at most a single car there must be (1−2p)N places in which no car is present. Each two such places that are not adjacent must correspond to a segment in the cars configuration. It is evident by the system dynamics, that the number of places for which Ri = Bi = Ri−1 = Bi−1 = 0 is non-increasing. More precisely, only places for which Ri = Bi = Ri−1 = Bi−1 = 0 in the initial configuration can satisfy this in the future. In a random initial configuration with density p, the initial number of these places is expected to be (1 − p)4 N , and by standard CLT, we get that

106

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky

with high probability this number is at most ((1 − p)4 + ε)N . Thus, the number of different segments in the system configuration at any time is at least ((1 − 2p) − (1 − p)4 − ε)N . However for p very close to 12 this bound may be negative, so this does not suffice. To solve this we note that similarly also for any fixed K, the number of consecutive K empty places in a configuration is non-increasing by the system dynamics, and w.h.p. is at most ((1 − p)2K + ε)N for an initial random state. 2K −ε N different segments in the system. But this guarantees at least (1−2p)−(1−p) K−1 Choosing K(p) (independent of N) sufficiently large s.t. (1 − p)2K < (1 − 2p) we get a linear lower bound on the number of segments from a random initial configuration.  3.3. The case p > 0.5 The proofs of speed optimality and segment structure for p > 0.5 will rely mainly of a the combinatorial properties of a stable configuration. A stable configuration for the system is a configuration that re-appears after running the system for some M turns. Since for a fixed N the number of possible configurations of the system is finite, and the state-transitions (traffic rules) are time independent, the system will necessarily reach a stable configuration at some time regardless of the starting configuration. We will use mainly two simple facts that must hold after a system reached a stable configurations: (a) |V Vt | cannot change – i.e., no violation can disappear. This is clear from Lemma 3.3; (b) Two disjoint segments of car cannot merge to one (i.e., one pushed until it meets the other), since clearly the number of segments in the system is also non-increasing in time. These two facts alone already provide plenty of information about the stable configuration for p > 0.5. We begin with the following twin lemmas on the stable state. Lemma 3.7. Let R, B be a stable configuration with junction at S = 0 and B0 = 0. Assume that there is a sequence of exactly sR consecutive red cars at places [N − sR , N − 1] and sB blue cars at places [N − sB , N − 1], sR , sB ≥ 1. Then: 1. Bi = 0 for i ∈ [N − sR − sB − 1, N − sB − 1]. 2. Ri = 1 for i ∈ [N − sR − sB − 1, N − max(sR , sB ) − 2]. 3. Ri = 0 for i ∈ [N − max(sR , sB ) − 1, N − sR − 1]. Proof. Since it is easy to lose the idea in all the notations, a visual sketch of the proof is provided in Figure 5. By the assumptions: – Bi = 1 for i ∈ [N − sB , N − 1], BN −sB −1 = 0. – Ri = 1 for i ∈ [N − sR , N − 1], RN −sR −1 = 0. To get (1), we note that with the assumption B0 = 0, the blue sequence will be pushed to the left in the next sR turns, so by restriction (b), Bi = 0 for

The Biham-Middleton-Levine Traffic Model for a Single Junction

107

Figure 5. Sketch of proof ideas for Lemmas 3.7 (steps 1–3) and 3.8 (step 4). i ∈ [N − sB − sR − 1, N − sB − 1] since otherwise 2 disjoint blue segments will merge while the sequence is pushed. Thus following the system, after sR turns we will get: Bi = 1 for i ∈ [N − sR − sB , N − sR − 1] and Bi = 0 for i ∈ [N − sR , N − 1], and Bi not changed left to N − sR − sB , and R unchanged. Note that originally R, B contained min(sR , sB ) consecutive violations in places [N − min(sR , sB ), 0] which all vanished after sR turns. Possible violations at places [N − max(sR , sB ), N − min(sR , sB )] remained as they were. From here we get that we must have Ri = 1 for min(sR , sB ) places within [N − sR − sB − 1, N − max(sR , sB ) − 1]. Since for place N − max(sR , sB ) − 1 either R or B are empty by the assumption, we must therefore have Ri = 1 for i ∈ [N − sB − sR − 1, N − max(sR , sB ) − 2], giving (2). If we follow the system for sB more steps we note that any red car in [N − max(sR , sB ) − 1, N − sR − 1] will be pushed left until eventually hitting the red car already proven to be present at N − max(sR , sB ) − 2], thus Ri = 0 for i ∈ [N − max(sR , sB ) − 1, N − sR − 1] giving (3).  The following lemma is completely analogous when reversing the roles of R, B, and can be proven the same way. Lemma 3.8. Let R, B be a stable configuration with junction at S = 0 and B0 = 1. Assume that there is a sequence of exactly sR consecutive red cars at places [N − sR , N − 1] and sB blue cars at places [N − sB + 1, 0], sR , sB ≥ 1. Then: 1. Ri = 0 for i ∈ [N − sB − sR − 1, N − sR − 1]. 2. Bi = 1 for i ∈ [N − sB − sR , N − max(sB , sR ) − 1]. 3. Bi = 0 for i ∈ [N − max(sB , sR ), N − sB ].

108

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky

Figure 6. Lemmas 3.7 (A) and 3.8 (B) combined together yield Lemma 3.9. Putting Lemmas 3.7, 3.8 together we get the following characterization for stable configurations: Lemma 3.9. Let R, B be a stable configuration with junction at S = 0 and B0 = 0. Assume that there is a sequence of exactly sR consecutive red cars at places [N −sR , N −1] and sB blue cars at places [N −sB , N −1]. Denote M = max(sB , sR ) Then 1. There are no additional cars are at [N − M, N − 1]. 2. Place i = N − M − 1 is empty – i.e., Ri = Bi = 0. 3. Starting at N − M − 2 there is a sequence of K1 ≥ min(sB , sR ) places for which Ri = 1; Bi = 0; (i ∈ [N − M − K1 − 1, N − M − 2]). 4. Starting N −M −K1 −2 (i.e., right after the red sequence) there is a sequence of K2 ≥ min(sB , sR ) places for which Bi = 1; Ri = 0; (i ∈ [N − M − K1 − K2 − 1, N − M − K1 − 2]). Proof. Figure 6 outlines the proof, without the risk of getting lost in the indices. For the full proof, first get from Lemma 3.7 that: 1. Bi = 0 for i ∈ [N − sR − sB − 1, N − sB − 1]. 2. Ri = 1 for i ∈ [N − sR − sB − 1, N − max(sR , sB ) − 2]. 3. Ri = 0 for i ∈ [N − max(sR , sB ) − 1, N − sR − 1]. From (1), (3) we get in particular, Bi = 0 for i ∈ [N − max(sR , sB ) − 1, N − sB − 1], and Ri = 0 for i ∈ [N − max(sR , sB ) − 1, N − sR − 1], so indeed no additional cars are at [N − max(sB , sR ), N − 1], and place N − max(sR , sB ) − 1 is empty, proving claims 1, 2 in the lemma. From (2), (3) we get Bi = 0 for i ∈ [N −sR −sB −1, N −max(sR , sB )−2] and Ri = 1 for i ∈ [N −sR −sB −1, N −max(sR , sB )−2], thus places [N −sR −sB −1, N − max(sR , sB ) − 2] contain a sequence of length min(sR , sB ) of red cars with no blue

The Biham-Middleton-Levine Traffic Model for a Single Junction

109

cars in parallel to it. This sequence is possibly a part of a larger sequence of length s R ≥ min(sR , sB ), located at [N − max(sR , sB ) − s R − 1, N − max(sR , sB ) − 2]. Now running the system for max(sR , sB ) turns, we will have the junction at place S = N − max(sR , sB ) − 1, BS = 1, followed by sequences of s R reds and s B = min(sB , sR )(≤ s R ) blues. Applying Lemma 3.8 for the system (rotated by max(sR , sB ), i.e., for N = N − max(sR , sB ) − 1): 1. Ri = 0 for i ∈ [N − s B − s R − 1, N − s R − 1] = [N − max(sR , sB ) − min(sR , sB ) − s R − 2, N − max(sR , sB ) − s R − 2] = [N − sR − sB − s R − 2, N − max(sR , sB ) − s R − 2]. 2. Bi = 1 for i ∈ [N − s B − s R , N − max(s B , s R ) − 1] = [N − sR − sB − s R − 1, N − max(sR , sB ) − max(s B , s R ) − 2] = [N − sR − sB − s R − 1, N − max(sR , sB ) − s R − 2]. 3. Bi = 0 for i ∈ [N − max(s B , s R ), N − s B ] = [N − max(sR , sB ) − s R − 1, N − max(sR , sB )− min(sR , sB )− 1] = [N − max(sR , sB )− s R − 1, N − sR − sB − 1]. In particular from (3) we get that no blue cars are in parallel to the entire red segment in [N − max(sR , sB ) − s R − 1, N − max(sR , sB ) − 2]: We were previously assured this is true up to place N − sR − sB − 1, and for places [N − max(sR , sB ) − s R − 1, N − sR − sB − 2] ⊆ [N − max(sR , sB ) − s R − 1, N − max(sR , sB ) − s B − 1] this holds by (3). Furthermore by (2) we get that a sequence of blue cars which begins from place N − max(sR , sB ) − s R − 2 with no red cars in parallel to it by (1). Note that N − max(sR , sB ) − s R − 2 is exactly to the left of N − max(sR , sB ) − s R − 1 where the red sequence ended. Now clearly choosing K1 = s R ; K2 = min(sR , sB ) we get claims 3, 4, 5 in the lemma.  Putting it all together we can now get a very good description of a stable state. Theorem 3.10. Let R, B be a stable configuration with junction at S = 0 and B0 = 0. Assume that there is a sequence of exactly sR consecutive red cars at places [N −sR , N −1] and sB blue cars at places [N −sB , N −1]. Denote M = max(sB , sR ). Then no additional cars are at [N − M, N − 1], and at places [0, N − M − 1] the configurations R, B satisfies: 1. Each place contains at most one type of car, red or blue. 2. Place N − M − 1 is empty. Each empty place, is followed by a sequence of places containing red cars immediately left to it, which is followed by a sequence of places containing blue cars immediately left to it. 3. Any sequence of red or blue cars is of length at least min(sR , sB ). Proof. This is merely applying Lemma 3.9 repeatedly. Applying Lemma 3.9 we know that there exist K1 , K2 ≥ min(sR , sB ) such that: Place N − M − 1 is empty, followed by K1 consecutive places with only red cars and K2 consecutive places with only blue cars left to it, thus the assertion holds for the segment [T, N −M −1] for T = N − M − 1 − K1 .

110

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky

Figure 7. Repetitively applying Lemma 3.9, we unveil a longer segment in the configuration, [T, N − 1], for which properties of Theorem 3.10 hold.

Figure 8. A typical stable configuration. Now we completely know R, B in [N − M − 1 − K1 , N − M − 1], and this is enough to advance the system for K1 + M turns. The sB blue segment is pushed left sR places, further pushing the K1 red sequence min(sR , sB ) places to the left such that its last min(sR , sB ) cars now overlap with the K2 blue sequence. So after K1 + M turns the system evolves to a state where S = N − K1 − M , BS = 0, and left to S there are K2 > K2 consecutive blue cars and exactly min(sR , sB ) consecutive red cars. Noting that this time M = max(K2 , min(sR , sB )) = K2 , once again we can deduce from Lemma 3.9 that: there are no additional cars in [N − M − K1 − K2 , N − M − K1 − 1] (thus we are assured that the entire blue segment of length K2 does not have red cars parallel to it). Place N − M − K1 − K2 − 1 is empty, followed by some K3 consecutive places with only red cars and K4 consecutive places with only blue cars left to it, for K3 , K4 ≥ min(K1 , K2 ) = min(sR , sB ) thus the assertion holds for the segment [T , N − M − 1] for T = N − M − K1 − K2 − K3 − 1 < T . Repeatedly applying Lemma 3.9 as long as T > 0, we repeatedly get that the assertion holds for some [T, N − M − 1] for T strictly decreasing, so the assertion holds in [0, N − M − 1].  We have worked hard for Theorem 3.10, but it will soon turn to be worthwhile. Let us obtain some useful corollaries.

The Biham-Middleton-Levine Traffic Model for a Single Junction

111

Corollary 3.11. Let R, B be a stable configuration with junction at S = 0 and B0 = 0. Assume that there is a sequence of exactly sR consecutive red cars at places [N −sR , N −1] and sB blue cars at places [N −sB , N −1]. Denote m = min(sB , sR ) Then: 1. 2. 3. 4. 5.

The number of blue segments in the system equals the number of red segments. −1 System speed is at least (1 + m . N) Total number of cars in the system is at most N + m. 2m Total number of cars in the system is at least 2m+1 N + m. Total number of segments in the system is at most N m + 1.

Proof. (1) Follows by the structure described in 3.10, since there is exactly one red and one blue sequence in [N − M, N − 1] (as always M = max(sR , sB )) and an equal number of reds and blues in [0, N − M − 1] since any red sequence is immediately proceeded by a blue sequence. For (2) we note that by 3.10 the configuration R, B contains exactly m violations: the m overlapping places of the segments sR , sB in [N − M, N − 1] are the only violations in the configuration since any places in [0, N − M − 1] contains a single car, and no red car can be immediately to the left of a blue car. By 3.3 the −1 system speed is hence at least (1 + m . N) By 3.10 any place in [0, N − m − 1] contains at most one car, or it is empty, and places [N − m, N − 1] contain both a red car and a blue car. Thus the total number of cars in the system is at most N − m + 2m = N + m giving (3). On the other hand, Since any sequence of a red car or a blue car is of length at least m, an empty place in [0, N − m] can occur only once in 2m + 1 places, and other places contain one car. Thus the number of cars is lower bounded by 2m 2m 2m2 +2m 2m 2m+1 (N − m) + 2m = 2m+1 N + 2m+1 ≥ 2m+1 N + m giving (4). The last property follows the fact that any sequence of cars is of length at least m, and by (3) total number of cars is at most N +m, thus number of different +m N =m + 1.  sequences is at most Nm Theorem 3.12. All cars in the system have the same asymptotic speed. Proof. As we have seen, when it reaches a stable state, the system consists of alternating red and blue sequences of cars. Obviously, the order of the sequences cannot change. Therefore, the difference between the number of steps two different cars have taken cannot be more then the length of the longest sequence, which is less then N . Thus, the asymptotic (w.r.t. t) speed is the same for all cars.  With these corollaries we can now completely characterize the stable state of a junction with p > 0.5, just by adding the final simple observation, that since the number of cars in the model is greater than N , there are violations at all times, including after reaching a stable state. Now let us look at some time when the junction reaches a violation when the system is in stable state. At this point the conditions of Theorem 3.10 are satisfied, thus:

112

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky

1 Theorem 3.13. A junction of size N and density p > 0.5 reaches speed of 2p −O( N1 ) 1 (i.e., arbitrarily close to the optimal speed of 2p , for large enough N ), and contains at most a bounded number (depending only on p) of car sequences.

Proof. We look at the system after it reached a stable state. Since 2pN > N at some time after that conditions of Theorem 3.10 are satisfied for some sR , sB ≥ 1. Let m = min(sR , sB ) at this time. Using claims (3), (4) in Corollary 3.11 we get: 2m N + m ≤ 2pN ≤ N + m. 2m + 1 From here we get N 2m )N = (2p − 1)N + (∗) (2p − 1)N ≤ m ≤ (2p − 2m + 1 2m + 1 and reusing m ≥ (2p − 1)N on the left-hand size we get: 1 . (2p − 1)N ≤ m ≤ (2p − 1)N + 4p − 1 1 a constant independent of N (C = O(1)). So m = (2p − 1)N + K, For C = 4p−1 for K ≤ C. Now by claim (2) in 3.11, system speed is at least

(1 +

1 K m −1 (2p − 1)N + K −1 K ) = (1 + ) = (2p + )−1 ≥(2p>1) − . N N N 2p N

But by Theorem 3.1 system speed is at most K N



1 2p ,

thus system speed is exactly

− for some 0 ≤ K ≤ K ≤ C = O(1), thus K = O(1) proving the first part of the theorem. By (5) we get total number of segments in the system is at most N m + 1, applying m ≥ (2p − 1)N we get the number of segments is bounded by: N 1 2p +1≤ +1= = O(1). m 2p − 1 2p − 1 Thus the second part proven.  1 2p

3.4. p < 0.5 revisited The characterization in 3.10 can be also proven useful to handle p < 0.5. Actually the main result for p < 0.5, Theorem 3.5, can be shown using similar technique, and even sharpened.3 Corollary2 3.14.3 For p < 0.5 the junction reaches speed of at least 1 − C(p) N , for p 1 C(p) = 1−2p . In particular, for p < 3 the junction reaches speed 1, for any initial configuration. Proof. Let R, B be any initial configuration. Looking at the configuration after it reached the stable state, if the system reached speed 1 we have nothing to 3 To Theorem 3.5 defence, we should note that key components of its proof, such as Lemma 3.3, were also used in the proof of 3.10.

The Biham-Middleton-Levine Traffic Model for a Single Junction

113

prove. Assume the speed is less than 1. Since in this case violations still occur, at some time the stable configuration will satisfy Theorem 3.10. As before, letting m = min(sR , sB ) at this time, by claim (4) in Corollary 3.11 we have: 1 2m N + m ≤ 2pN ⇒ m ≤ (2p − 1 + )N. 2m + 1 2m + 1 p 1 In particular 2p − 1 + 2m+1 > 0, rearranging we get m < 1−2p , and since m ∈ Z, 2 3 3 2 p p 1 m ≤ 1−2p = C(p). m must be positive, thus for p < 3 , having C(p) = 1−2p = 0 we get a contradiction, thus the assumption (speed < 1) cannot hold. For p ≥ 13 −1 by claim (2) in Corollary 3.11 the system speed is at least (1 + C(p) ≥ 1− N ) C(p) .  N 3.5. The critical p = 0.5 Gathering the results so far we get an almost-complete description of the behaviour of a junction. For junction of size N and density p: • If p < 0.5 the junction will reach speed 1 − o(1) (asymptotically optimal), and contain linearly many different segments in the stable state. 1 • If p > 0.5 the junction will reach speed 2p − o(1) (asymptotically optimal), and contain constant many segments in the stable state. From the description above one sees that the junction system goes through a sharp phase transition at p = 0.5, as the number of segments of cars as the system stabilizes drops from being linear to merely constant. The last curiosity, is what happens at p = 0.5. Once again by using the powerful Theorem 3.10 we can deduce: Theorem 3.15. A junction of size N with p = 0.5 reaches speed of at least 1 − √1N √ and contains at most N different segments. Proof. For p = 0.5 we have exactly N cars in the system. As we reach stable state, violations must still occur (since a system with exactly N cars must contain at least one violation), thus at some time Theorem 3.10 is satisfied. For m = min(sR , sB ) at this time, by claim (4) in Corollary 3.11 we have: 2m N + m ≤ N ⇒ m(2m + 1) ≤ N. 2m + 1 √ √ Thus m < N . From here by claim (2) the system speed is at least (1 + NN )−1 ≥ 1 − √1N . If S is the number of segments, then by Theorem 3.10 we can deduce that the total number of cars, N , is: 2m cars in places [N − m, N − 1], and N − m − S cars in places [0, N − M − 1] (since each place contains one car, except transitions between segments that are empty). √ N = (N − m − S) + 2m = N + m − S ⇒ S = m ≤ N . √ Thus the configuration contains at most N segments. 

114

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky

Simulation results show that these bounds are not only tight, but typical, meaning √ that a junction with a random initial configuration with density p indeed has O( N) segments in the stable state, with the largest segment of size near n1/2 . This suggests that the system undergoes a second order phase transition.

4. Simulation results Following are computer simulation results for the junction for critical and near critical p, demonstrating the phase transition. The columns below consist of N , the average asymptotic speed, the average number of car segments in the stable state, the average longest segment in the stable state, and the average number of √ segments divided by N and N . 4.1. p = 0.48 For large N , system reaches speed 1 and the average number of segments is linear (approx. 0.037N ). √ N Speed No. segs Longest No. segs/N No. segs/ N 1000 5000 10000 50000

0.99970 1.00000 1.00000 1.00000

38.7 186.4 369.8 1850.6

6.8 8.5 7.5 8.2

4.2. p = 0.52 For large N , system reaches speed 0.961 = is about constant. N

Speed

1000 5000 10000 50000

0.95703 0.96041 0.96091 0.96142

0.0387 0.0373 0.0370 0.0370

1 2p

and the average number of segments

No. segs Longest No. segs/N 5.7 6.9 7.3 7.2

76.7 330.0 416.1 3207.1

0.0057 0.0014 0.0007 0.0001

4.3. p = 0.5 At criticality, the speed is approaching 1 like 1 − √ segments is around 0.43 · N. N

Speed

1000 5000 10000 50000

0.98741 0.99414 0.99570 0.99812

1.2238 2.6361 3.6980 8.2761

√C N

0.1802 0.0976 0.0730 0.0322

and the average number of

No. segs Longest No. segs/N 13.4 30.0 43.8 95.0

38.4 82.8 142.1 248.4

0.0134 0.0060 0.0044 0.0019

√ No. segs/ N

√ No. segs/ N 0.4237 0.4243 0.4375 0.4251

The Biham-Middleton-Levine Traffic Model for a Single Junction

115

Figure 9. K-Lanes junction in a time-space normalized view, before and after organization. Junction is marked in yellow and advances left each turn

5. Summary The fascinating phenomena observed in the BML traffic model are still far from being completely understood. In this paper we showed a very simplified version of this model, which, despite its relative simplicity, displayed very similar phenomena of phase transition at some critical density and of self-organization, which in our case both can be proven and well understood. We used two approaches in this paper: The first one was to use some sort of “independence” in the way the system evolves, as in the way we handle p < 0.5. We showed that the system self-organizes “locally” and with a bounded affect on the rest of the configuration, thus it will eventually organize globally. The second approach is the notion of the stable configuration, i.e., we characterize the combinatorial structure of any state that the system can “preserve” and use it to show it is optimal (in a way saying, that as long as we are not optimal the system must continue to evolve). Can these results be extended to handle more complicated settings than the junction? Possibly yes. For example, considering a k-lanes junction (i.e., k consecutive red lines meeting k consecutive blue rows), one can look at a timespace-normalized version of the system, as shown in Figure 9, with the junction now being a k × k parallelogram travelling along the red and blue lines, and “propagating violations” (which now have a more complicated structure depending on the k × k configuration within the junction). Stable states of this configurations seem to have the same characteristics of a single junction, with a red (blue) car equivalent to some red (blue) car in any of the k red (blue) lines. Thus a zero1 effort corollary for a k-lanes junction is that it reaches speed 1 for p < 2k , but for k nearing O(N ) this bound is clearly non-significant. It is not surprising though, since combinatorics alone cannot bring us too far, at least for the complete BML model – even as little as 2N cars in an N 2 size torus can be put in a stuck configuration – i.e., reach speed 0.

116

I. Benjamini, O. Gurel-Gurevich and R. Izkovsky

References [1] T. Austin, I. Benjamini, For what number of cars must self organization occur in the Biham-Middleton-Levin traffic model from any possible starting configuration?, 2006. [2] O. Angel, A.E. Holroyd, J.B. Martin, The jammed phase of the Biham-MiddletonLevine traffic model, Elect. Comm. in Probability 10 (2005), 167–178. [3] O. Biham, A.A. Middleton, D. Levine, Self organization and a dynamical transition in traffic flow models, Phys. Rev. A 46 (1992), R6124. [4] R.M. D’Souza, Geometric structure of coexisting phases found in the BihamMiddleton-Levine traffic model, Phys. Rev. E 71 (2005), 0066112. Itai Benjamini Department of Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel e-mail: [email protected] Ori Gurel-Gurevich Department of Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel e-mail: [email protected] Roey Izkovsky Department of Mathematics, Weizmann Institute of Science, Rehovot, 76100, Israel e-mail: [email protected]

Progress in Probability, Vol. 60, 117–135 c 2008 Birkhauser  ¨ Verlag Basel/Switzerland

Box-Crossings and Continuity Results for Self-Destructive Percolation in the Plane J. van den Berg, R. Brouwer and B. V´´agvolgyi ¨ Abstract. A few years ago (see [1]) two of us introduced, motivated by the study of certain forest-fire processes, the self-destructive percolation model (abbreviated as sdp model). A typical configuration for the sdp model with parameters p and δ is generated in three steps: First we generate a typical configuration for the ordinary site percolation model with parameter p. Next, we make all sites in the infinite occupied cluster vacant. Finally, each site that was already vacant in the beginning or made vacant by the above action, becomes occupied with probability δ (independent of the other sites). Let θ(p, δ) be the probability that some specified vertex belongs, in the final configuration, to an infinite occupied cluster. In our earlier paper we stated the conjecture that, for the square lattice and other planar lattices, the function θ(·, ·) has a discontinuity at points of the form (pc , δ), with δ sufficiently small. We also showed (see [2]) remarkable consequences for the forest-fire models. The conjecture naturally raises the question whether the function θ(·, ·) is continuous outside some region of the above-mentioned form. We prove that this is indeed the case. An important ingredient in our proof is a somewhat modified (improved) form of a recent RSW-like (box-crossing) result of Bollobas ´ and Riordan ([4]). We believe that this modification is also useful for many other percolation models. Mathematics Subject Classification (2000). 60K35, 82B43. Keywords. Percolation, RSW results, box crossings.

1. Introduction and outline of results 1.1. Background and motivation The self-destructive percolation model (abbreviated as sdp model) on the square lattice is described as follows: First we perform independent site percolation on this lattice: we declare each site occupied with probability p, and vacant with Part of J. van den Berg’s research has been funded by the Dutch BSIK/BRICKS project.

118

J. van den Berg, R. Brouwer and B. V´´agvolgyi ¨

probability 1−p, independent of the other sites. We will use the notation {V ↔ W } for the event that there is an occupied path from the set of sites V to the set of sites W . We write {V ↔ ∞} for the event that there is an infinite occupied path starting at V . Let, as usual, θ(p) denote the probability that a given site, say O = (0, 0), belongs to an infinite occupied cluster. It is known that there is a critical value 0 < pc < 1 such that θ(p) > 0 for all p > pc , and θ(p) = 0 for all p ≤ pc . Now suppose that, by some catastrophe, the infinite occupied cluster (if present) is destroyed; that is, each site in this cluster becomes vacant. Further suppose that after this catastrophe we give the sites independent ‘enhancements’, as follows: Each site that was already vacant in the beginning, or was made vacant by the catastrophe, becomes occupied with probability δ, independent of the others. Let Pp,δ be the distribution of the final configuration. A more formal, and often very convenient description of the model is as follows: Let Xi , i ∈ Z2 be independent 0 − 1-valued random variables, each Xi being 1 with probability p and 0 with probability 1 − p. Further, let Yi , i ∈ Z2 , be independent 0 − 1-valued random variables, each Yi being 1 with probability δ and 0 with probability 1 − δ. Moreover, we take the collection of Yi ’s independent of that of the Xi ’s. Let Xi∗ , i ∈ Z2 be defined by  1 if Xi = 1 and there is no X-occupied path from i to ∞ Xi∗ = (1) 0 otherwise, where by ‘X-occupied path’ we mean a path on which each site j has Xj = 1. Finally, define Zi = Xi∗ ∨Y Yi . This collection (Z Zi , i ∈ Z2 ) is (with 0 meaning ‘vacant’ and 1 ‘occupied’) what we called ‘the final configuration’, and the above-mentioned Pp,δ is its distribution. We use the notation θ(p, δ) for the probability that, in the final configuration, O is in an infinite occupied cluster: θ(p, δ) := Pp,δ (O ↔ ∞). Note that O is occupied in the final configuration if and only if the abovementioned enhancement was successful, or O belonged initially (before the catastrophe) to a non-empty but finite occupied cluster. This gives Pp,δ (O is occupied ) = δ + (1 − δ)(p − θ(p)). Also note that, in the case that p ≤ pc , nothing happens in the above catastrophe, so that in the final configuration the sites are independently occupied with probability p + (1 − p)δ. Formally, if p ≤ pc , then Pp,δ = Pp+(1−p)δ ,

(2)

where we use the notation Pp for the product measure with parameter p. In particular, θ(pc , δ) = θ(pc + (1 − pc )δ) > 0, (3) for each δ > 0.

Crossings and Continuity Results for Self-Destructive Percolation

119

Remark 1.1. Most of what we said above has straightforward analogs for arbitrary countable graphs, but there are subtle differences. For instance, on the cubic lattice it has not yet been proved that θ(pc ) = 0 (although this is generally believed to be true). So, for that lattice, (2) with p = pc , and hence (3), are not rigorously known. It is also clear from the construction that Pp,δ stochastically dominates Pδ . Hence, if δ > pc then θ(p, δ) ≥ θ(δ) > 0 for all p. It turns out (see Proposition 3.1 of [1]) that, if p > pc , a ‘non-negligible’ enhancement is needed after the catastrophe to create again an infinite occupied cluster. More precisely, for each p > pc there is a δ > 0 with θ(p, δ) = 0. A much more difficult question is whether the needed enhancement goes to 0 as p ↓ pc . By (3) one might be tempted to reason intuitively that this is indeed the case. In [1] it was shown that for the analogous model on the binary tree this is correct. However, in [1] a conjecture is presented which says, in particular, that for the square lattice (and other planar lattices) there is a δ > 0 for which θ(p, δ) = 0 for all p > pc . In Section 4 of [1] and in [2] we showed remarkable consequences for certain forest-fire models. Note that, since θ(pc , δ) > 0, the above conjecture says that the function θ(·, ·) has discontinuities at points of the form (pc , δ) with δ sufficiently small. This naturally raises the question whether this function is continuous in the complement of a region of such form: is there a δ > 0 such that θ(·, ·) is continuous outside the set {pc } × [0, δ]? In the next subsection we state that this is indeed the case, and give a summary of the methods and intermediate results used in the proof. At the end of Section 6 we point out why our proof does not work at points (pc , δ) with small δ. We hope our arguments provide a better understanding of the earlier mentioned conjecture and will trigger new attempts to prove (or disprove) it. 1.2. Outline of results The conjecture mentioned in the previous subsection raises the natural question whether θ(·, ·) is continuous outside the indicated ‘suspected’ region. The following theorem states that this is indeed the case. Theorem 1.2. There is a δ ∈ (0, 1) such that the function θ(·, ·) is continuous outside the segment {pc } × (0, δ). As could be expected, the proof widely uses tools and results from ordinary percolation. However, the dependencies introduced by the self-destructive mechanism cause complications. Until recently, a serious obstacle was the absence of a suitable RSW-like theorem. This obstacle could be removed by the use of (a modified and somewhat stronger form of) a recent theorem of Bollob´ a´s and Riordan ([4]). A rough outline of the proof of Theorem 1.2, and the needed intermediate results that are interesting in themselves, is as follows: In Section 2 we list some basic properties of our model, which will be used later. The results in Section 3,

120

J. van den Berg, R. Brouwer and B. V´´agvolgyi ¨

which are also contained in the recent PhD thesis [6] of one of us, show that if θ(·, ·) is strictly positive in some open region, then it is continuous on this region. It is also shown that if θ(p, δ) = 0, then θ(·, ·) is continuous at (p, δ). These two results reduce the proof of Theorem 1.2 to showing that if θ(p, δ) > 0 and p = pc , then θ(p, δ) > 0 in an open neighborhood of (p, δ). This in turn requires a suitable finite-size criterion (see below) for sdp. In Section 4 we give the modified form of the Bollob´ a´s-Riordan theorem. This is used in Section 5 to obtain the abovementioned finite-size criterion. Finally, in Section 6 we combine these results and prove the main theorem. We end this section with the following remark: When we say that a function f is ‘increasing’ (‘decreasing’) this should, unless this is preceded by the word ‘strictly’, be interpreted in the weak sense: x < y implies f (x) ≤ f (y).

2. Basic properties In this section we state some basic properties which will be used later. First some more terminology and notation: If v = (v1 , v2 ) and w = (w1 , w2 ) are two vertices, we let |v − w| denote their (graph) distance |v1 − w1 | + |v2 − w2 |. By B(v, k) and ∂B(v, k) we denote the set of vertices w for which |v − w| is at most k, respectively equal to k. For V, W ⊂ Z2 , we define the distance between V and W as min{|v − w| : v ∈ V, w ∈ W }. Recall that Pp,δ denotes the sdp distribution (that is, the distribution of the collection (Z Zi , i ∈ Z2 ) defined in Subsection 1.1). This is a distribution on Z2 Ω := {0, 1} (with the usual σ-field). Elements of Ω are typically denoted by ω(= (ωi , i ∈ Z2 )), σ etc. We write ω ≤ σ if ωi ≤ σi for all i. Let V be a set of vertices and A an event. We say that A lives on V if ω ∈ A and σi = ωi for all i ∈ V , implies σ ∈ A. And we say that A is a cylinder event if A lives on some finite set of vertices. As usual, we say that A is increasing if ω ∈ A and ωi ≤ σi for all i, implies σ ∈ A. The first two lemma’s below come from Section 2.2 and 2.4 respectively in [1]. Lemma 2.1. Let A and B be two increasing cylinder events. We have Pp,δ (A ∩ B) ≥ Pp,δ (A)P Pp,δ (B). As to monotonicity, it is obvious that the sdp model has monotonicity in δ: If δ1 ≥ δ2 , then Pp,δ1 stochastically dominates Pp,δ2 . Although there seems to be no ‘nice’ monotonicity in p we have the following property. Lemma 2.2. If p2 ≥ p1 and p2 + (1 − p2 )δ2 ≤ p1 + (1 − p1 )δ1 , then Pp1 ,δ1 dominates Pp2 ,δ2 . The next result is about ‘almost independence’ of cylinder events which live on widely separated sets. As usual, the lattice which has the same vertices as the square lattice but where each vertex has, besides the four edges to its nearest neighbours, also four ‘diagonal edges’ is called the matching lattice (of the square

Crossings and Continuity Results for Self-Destructive Percolation

121

lattice). To distinguish paths and circuits in the matching lattice from those in the square lattice, we use the terminology *-paths and *-circuits. Lemma 2.3. Let k be a positive integer and let V and W be subsets of Z2 that have distance larger than 2k. Further, let A and B be events which live on V and W respectively. Then |P Pp,δ (A ∩ B) − Pp,δ (A)P Pp,δ (B)| ≤ 2(|V | + |W |) Pp (∃ vacant *-circuit surrounding O and some vertex in ∂B(O, k)) .

(4)

Proof. Recall how we formally defined the sdp model in terms of random variables X, Y and Z. We use a modification of those variables: Let X and Y be as before, but in addition to X ∗ and Z we now define X ∗(f ) and Z (f ) by

1 if Xi = 1 and  X-occupied path from i to ∂B(i, k) ∗(f ) Xi = 0 otherwise; (f )

Zi

∗(f )

= Xi

∨ Yi .

(5)

(f )

Let Pp,δ denote the distribution of Z (f ) . It is clear that the random variables (f )

(f )

Zi , i ∈ V are independent of the random variables Zi , i ∈ W , and hence (f )

(f )

(f )

Pp,δ (B). Pp,δ (A ∩ B) = Pp,δ (A)P

(6)

(f )

Also note that if Zi = Zi , then the X-occupied cluster of i intersects ∂B(i, k) but is finite. Hence there is an X-vacant circuit in the matching lattice that surrounds i and some site in ∂B(i, k). Hence, since the X-variables are Bernoulli random variables with parameter p, we have for any finite set K of vertices and any event E living on K, (f )

(f )

|P Pp,δ (E) − Pp,δ (E)| ≤ P (ZK = ZK ) ≤

(7)

|K|P Pp (∃ a vacant *-circuit surrounding O and some vertex in ∂B(O, k)) . The lemma now follows easily from (6) and (7)



Our last result in this section is on the uniqueness of the infinite cluster. Lemma 2.4. If θ(p, δ) > 0, then r = 1. Pp,δ (∃ a unique infinite occupied cluster) Proof. From the earlier construction of the sdp model in terms of the X- and Y -variables, it is clear that Pp,δ is stationary and ergodic. It is also clear that in the sdp model the conditional probability that a given site is occupied given the configuration at all other sites, is at least δ. So this model has the so-called positive finite energy property. The result now follows from an extension in [9] of the well-known Burton-Keane ([7]) uniqueness result. 

122

J. van den Berg, R. Brouwer and B. V´´agvolgyi ¨

3. Partial continuity results In this section we first prove that in the sdp model the probabilities of cylinder events are continuous functions of (p, δ). Next we prove that the function θ(·, ·) is continuous at (p, δ) if θ(p, δ) = 0 or there is an open neighborhood of (p, δ) on which θ is strictly positive. Note that, once we have this, the proof of Theorem 1.2 is basically reduced to showing that if p = pc and θ(pc , δ) > 0, then θ(·, ·) is strictly positive on an open neighborhood of (p, δ). Lemma 3.1. Let A be a cylinder event. The function (p, δ) → Pp,δ (A) is continuous on [0, 1]2 . Remark 3.2. The proof (see below) uses the well-known fact that θ(pc ) = 0. For many lattices (e.g., the cubic lattice) this fact has not been proved. For those lattices the arguments below show that the function in the statement of 3.1 is continuous on [0, 1]2 \ ({pc } × [0, 1]). Related to this, Proposition 3.3 below would also need some modification for those lattices. Proof. Let A be an event which lives on some finite set V . Recall the construction of the sdp model in terms of random variables X, Y and Z. Let, for σ ∈ Ω, σV denote the tuple (σi , i ∈ V ). It is clear that the distribution of XV∗ is a function of p only, and that, conditioned on XV∗ , the probability that ZV ∈ A is a polynomial (of degree |V |) in δ. Therefore it is sufficient to prove that, for each α ∈ {0, 1}V , the function f : p → P(XV∗ = α) is continuous. Recall that the X-variables are Bernoulli random variables (with parameter p). Now let 0 < p1 < p2 . In a standard way, by introducing independent, uniformly on the interval (0, 1) distributed random variables Ui , i ∈ Z2 , we can suitably couple two collections of Bernoulli random variables with parameters p1 , respectively p2 . Such argument easily gives that |f (p2 )−f (p1 )| is less than or equal to the sum over i ∈ V of P(U Ui ∈ (p1 , p2 )) + P(i is in an infinite p2 -open but not in an infinite p1 -open cluster), which equals |V |(p2 − p1 + θ(p2 ) − θ(p1 )). The lemma now follows from the continuity of θ(.).



Proposition 3.3. Let (p, δ) ∈ [0, 1]2 . If (a) or (b) below holds, the function θ(·, ·) is continuous at (p, δ). (a) θ(·, ·) > 0 on an open neighborhood of (p, δ). (b) θ(p, δ) = 0, Proof. For this (and some other) results it is convenient to describe the sdp model in terms of Poisson processes: Assign to each site, independently of the other sites, a Poisson clock with rate 1. These clocks govern the following time evolution: Initially each site is vacant. Whenever the clock of a site rings, the site becomes occupied. (If it was already occupied, the ring is ignored.) Note that if occupied sites would always remain occupied, then for each time t, the configuration at time t would be a collection of independent Bernoulli random variables with parameter 1−exp(−t). In particular, before and at time tc , defined by the relation pc = 1 − exp(−tc ),

Crossings and Continuity Results for Self-Destructive Percolation

123

there would be no infinite occupied cluster, but after tc there would be a (unique) infinite cluster. However, we do allow occupied sites to become vacant, although only once, as follows: Fix a time τ , a parameter of the time evolution. At time τ all sites in the infinite occupied cluster become vacant. (If there is no infinite occupied cluster, which is a.s. the case if τ ≤ tc , nothing happens.) After time τ we let the evolution behave as before; that is, each vacant site becomes occupied when its Poisson clock rings. Let, for this time evolution with parameter τ , Pˆτ,t denote the distribution of the configuration at time t , and let ˆ t) = Pˆτ,t (O is in an infinite occupied cluster). θ(τ,

(8)

It is easy to see that Pˆτ,t = Pp,δ ,

(9)

where p = 1 − exp(−τ ) and δ = 1 − exp(−(t − τ )). It is also easy to see that Pˆτ,t is stochastically decreasing in τ and stochastically increasing in t. In fact this is the key behind Lemma 2.2. Now we come back to the proof of Proposition 3.3. From (8) and (9) we get (since the map between pairs (p, δ) and (τ, t) in (9) is continuous) that this ˆ t) = 0 or θˆ is strictly positive on an proposition is equivalent to saying that if θ(τ, open neighborhood of (τ, tc ), then θˆ is continuous at (τ, t). To prove this equivalent form of Proposition 3.3 we use ideas from [3]. The introduction of pairs (τ, t) as replacement of (p, δ) not only has the advantage that, as we already saw, we now have a more suitable form of monotonicity, but, more importantly, that we now have a more ‘detailed’ structure (the Poisson processes) in the background which gives the appropriate ‘room’ needed to get a suitable modification of the arguments in [3]. Let (τ, t) be as above. Divide the parameter space in four ‘quadrants’, numbered I to IV : I := [0, τ ] × [t, ∞), II := [τ, ∞) × [t, ∞), III := [τ, ∞) × [0, t], IV := [0, τ ] × [0, t]. Note that it is sufficient to prove that for each monotone sequence (ττi , ti )i≥0 that lies in one of the above quadrants and converges to (τ, t), one has ˆ τi , ti ) = θ(τ, ˆ t). lim θ(τ

i→∞

We handle each of the quadrants separately. Quadrant I) This is easy and corresponds to the (easy) proof of right continuity of ordinary percolation: Let (ττi ) be a monotone sequence which converges from below to τ and let (ti ) be a monotone sequence which converges from above to t. Let An denote the event that there is an occupied path from O to ∂B(O, n). By

124

J. van den Berg, R. Brouwer and B. V´´agvolgyi ¨

monotonicity and Lemma 3.1 we have that ˆ τi , ti ) as n → ∞; For each i, Pˆτi ,ti (An ) ↓ θ(τ

(10)

ˆ t) as n → ∞; Pˆτ,t (An ) ↓ θ(τ,

(11)

For each n, Pˆτi ,ti (An ) ↓ Pˆτ,t (An ) as i → ∞,

(12)

ˆ τi , ti ) tends to θ(τ, ˆ t) as i → ∞. From these three statements it is easy to see that θ(τ Quadrant III) Let (ττi ) be a monotone sequence which converges from above to τ and (ti ) a monotone sequence which converges from below to t. By the earlier ˆ τi , ti ) is increasing in i, and has a limit monotonicity arguments, the sequence θ(τ ˆ ˆ t) = 0, the proof smaller than or equal to θ(τ, t). So for the situation where θ(τ, ˆ is done. Now we handle the other situation: we assume θ is positive in an open neighborhood of (τ, t). For this situation considerable work has to be done. Note that in the dynamic description given earlier in this section, the underlying Poisson processes were the same for each choice of the model parameter τ . This allows us (and we already used this to derive some monotonicity properties) to couple the models with the different τi ’s and τ . Let, for s < u, Cs,u denote the occupied cluster of site O at time u in the process with parameter s (that is, under the time evolution where the infinite occupied cluster is destroyed at time s). Further, we use the notation ω(s, u) for the configuration at time u in that model. It is also convenient to consider ω(u), the configuration at time u in the model where no destruction takes place. (So, ωv (u), v ∈ Z2 , are independent 0 − 1-valued random variables, each being 1 with probability 1 − exp(−u)). Again we emphasize that all these models are defined in terms of the same Poisson processes. From monotonicity (note that Cτi ,ti ⊂ Cτi+1 ,ti+1 for all i) it is clear that ˆ τi , ti ) = P(∃ i |C Cτi ,ti | = ∞), lim θ(τ

i→∞

and ˆ t) − lim θ(τ ˆ τi , ti ) = P(|C θ(τ, Cτ,t | = ∞, ∀i |C Cτi ,ti | < ∞). i→∞

(13)

So we have to show that the r.h.s. of (13) is 0. Fix a j with the property that ˆ τj , tj ) > 0. Such j exists by the condition we assumed for (τ, t). To show that θ(τ the r.h.s. of (13) is 0, it is sufficient (and necessary) to prove the following claim: Claim Apart from an event of probability 0, the event {|C Cτ,t | = ∞} is contained in the event that there is a k > j for which |C Cτk ,tk | = ∞. So suppose |C Cτ,t | = ∞. By our choice of j we may assume that ω(ττj , tj ) has an infinite occupied cluster, and by Lemma 2.4 that this cluster is unique. We denote it by Ij . If O ∈ Ij we are done. From monotonicity and the uniqueness of the infinite cluster (see Lemma 2.4), we have Ij ⊂ Cτ,t . Hence there is a finite path π from O to some site in Ij such that ω(τ, t) ≡ 1 on π. Since, a.s. there are no

Crossings and Continuity Results for Self-Destructive Percolation

125

vertices whose clock rings exactly at time t or τ , we may assume that for every site v on π, (a) or (b) below holds: (a) The clock of v rings in the interval (τ, t). (b) ωv (τ ) = 1 but the occupied cluster of v in ω(τ ) is finite. If (a) occurs we define: iv := min{i : i ≥ j and the clock of v rings in (ττi , ti )}. Note that then, by the monotonicity of the sequence (ττi , ti ), the clock of v rings in the interval (ττl , tl ) for all l ≥ iv . If (a) does not occur, (b) occurs, and hence there is a finite set Kv of sites on which ω(τ ) = 0 and which separates v from ∞. Then we use the following alternative definition of iv : iv := min{i : i ≥ j and ω(ττi ) ≡ 0 on Kv }. This minimum exists since Kv is finite and (again) we assume that no Poisson clock rings exactly at time τ . Now let k := max iv , v∈π

which exists since π is finite. From the above procedure it is clear that ωv (ττk , tk ) = 1 for all v on π. Further, since k ≥ j and by monotonicity, also ωv (ττk , tk ) = 1 for all v ∈ Ij . Since π is a path from O to Ij this implies that Ij is contained in Cτk ,tk and hence that |C Cτk ,tk | = ∞. This proves the Claim above. Quadrants II) and IV) The required results for these quadrants follow very easily from monotonicity and the above results for quadrants I and III: Let (ττi , ti ) be a sequence in quadrant II that converges to (τ, t). We have, by earlier stated monotonicity properties, ˆ τi , t) ≤ θ(τ ˆ τi , ti ) ≤ θ(τ, ˆ ti ). θ(τ Since the sequence (ττi , t)) lies in quadrant III and the sequence (τ, ti ) lies in ˆ t). This completes the quadrant I, the upper and lower bound both converge to θ(τ, treatment of quadrant II. Quadrant IV is treated in the same way. This completes  the proof of Proposition 3.3

4. An RSW-type result For our main result we need to prove that if the crossing probability of an n by n square goes to 1 as n → ∞, then also the crossing probability of a (say) 3n × n rectangle in the ‘difficult direction’ goes to 1 as n → ∞. Such (and stronger) results were proved for ordinary percolation in the late nineteen seventies by Russo, and by Seymour and Welsh, and therefore became known as RSW theorems. Their proofs used careful conditioning on the lowest horizontal crossing in a rectangle, after which the area above that crossing was treated, and a new, vertical crossing in that area was ‘constructed’. Such arguments work for ordinary percolation because

126

J. van den Berg, R. Brouwer and B. V´´agvolgyi ¨

there the above-mentioned area can be treated as ‘fresh’ territory. However, they usually break down in situations where we have dependencies, as in the sdp model. Recently, Bollobas ´ and Riordan ([4]) made significant progress on these matters. For the so-called Voronoi percolation model they proved an RSW type result. That result is one of the main ingredients in their proof that the critical probability for Voronoi percolation equals 1/2 (which had been conjectured but stayed open for a long time). Although they explicitly proved their RSW type result only for the Voronoi model, their proof works (as they remark in their paper) for a large class of models. The result we needed is a little stronger than that of [4]. We think that this improvement is useful for many other models as well. The rest of this section is organised as follows. First we give a short introduction to Voronoi percolation. Then we state the above-mentioned RSW-like theorem of [4], and point out where and how its proof needs to be modified to obtain the stronger version. Finally we state the analog for the sdp model and explain why the proof for the Voronoi model works for this model as well. 4.1. The Voronoi percolation model We start with a brief description of the Voronoi percolation model. The (random) Voronoi percolation model is as follows: Let Z denote the (random) set of points in a Poisson point process with density 1 in the plane. This set gives rise to a random Voronoi tessellation of the plane: Assign to each z ∈ Z the set of all x ∈ R2 for which z is the nearest point in Z. The closure of this set is called the (Voronoi) cell of z. It is known that (with probability 1) each Voronoi cell is a convex polygon, and that two cells are either disjoint or share an entire edge. In the latter case the two cells are said to be neighbours or adjacent. This notion of adjacency gives, in a natural way, rise to the notion of paths, clusters etc. Now consider the percolation model where each cell, independently of everything else, is coloured black with probability p and white with probability 1 − p. Based on analogies with ordinary percolation (in particular with the self-matching property of the usual triangular lattice) it has been conjectured for a long time that the critical value for this percolation model is 1/2: for p < 1/2 there is (a.s.) no infinite black cluster, but for p > 1/2 there is an infinite black cluster (a.s.). As we said before, this was recently proved rigorously by Bollobas ´ and Riordan ([4]), and a key ingredient in their proof is an ingenious RSW-like result. 4.2. The RSW-like result for Voronoi percolation As in [4] we define, for the Voronoi percolation model with parameter p, fp (ρ, s) as the probability that there is a horizontal black crossing of the rectangle [0, ρs] × [0, s]. The following is Theorem 4.1 in [4] Theorem 4.1. (Bollob´ a ´s and Riordan) Let 0 < p < 1 be fixed. If then

lim inf fp (1, s) > 0, s→∞

lim sup fp (ρ, s) > 0 for all ρ > 0. s→∞

(14)

Crossings and Continuity Results for Self-Destructive Percolation

127

Studying the proof we realised that the condition can be weakened, so that the following theorem is obtained: Theorem 4.2. Let 0 < p < 1 be fixed. If

lim sup fp (ρ, s) > 0 for some ρ > 0, s→∞

then

lim sup fp (ρ, s) > 0 for all ρ > 0. s→∞

As we shall point out, this somewhat stronger Theorem 4.2 can be proved in almost the same way as Theorem 4.1. But see Remark 4.4 about the global structure of the proof. First note that Theorem 4.2 is (trivially) equivalent to the following: Theorem 4.3. Let 0 < p < 1 be fixed. If l