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PERCOLATION and DISORDERED SYSTEMS Geoffrey GRIMMETT PERCOLATION AND DISORDERED SYSTEMS Geoffrey GRIMMETT Percolation and Disordered Systems 143 PREFACE This course aims to be a (nearly) self-contained account of part of the math- ematical theory of percolation and related topics. The first nine chapters summarise rigorous results in percolation theory, with special emphasis on results obtained since the publication of my book [155] entitled ‘Percolation’, and sometimes referred to simply as [G] in these notes. Following this core material are chapters on random walks in random labyrinths, and fractal percolation. The final two chapters include material on Ising, Potts, and random-cluster models, and concentrate on a ‘percolative’ approach to the associated phase transitions. The first target of this course is to draw a picture of the mathematics of percolation, together with its immediate mathematical relations. Another target is to present and summarise recent progress. There is a considerable overlap between the first nine chapters and the contents of the principal reference [G]. On the other hand, the current notes are more concise than [G], and include some important extensions, such as material concerning strict inequalities for critical probabilities, the uniqueness of the infinite cluster, the triangle condition and lace expansion in high dimensions, together with material concerning percolation in slabs, and conformal invariance in two dimensions. The present account differs from that of [G] in numerous minor ways also. It does not claim to be comprehensive. A second edition of [G] is planned, containing further material based in part on the current notes. A special feature is the bibliography, which is a fairly full list of papers published in recent years on percolation and related mathematical phenom- ena. The compilation of the list was greatly facilitated by the kind responses of many individuals to my request for lists of publications. Many people have commented on versions of these notes, the bulk of which have been typed so superbly by Sarah Shea-Simonds. I thank all those who have contributed, and acknowledge particularly the suggestions of Ken Alexander, Carol Bezuidenhout, Philipp Hiemer, Anthony Quas, and Alan Stacey, some of whom are mentioned at appropriate points in the text. In addition, these notes have benefited from the critical observations of various members of the audience at St Flour. Members of the 1996 summer school were treated to a guided tour of the library of the former seminary of St Flour. We were pleased to find there a copy of the Encyclop´edie, ou Dictionnaire Raisonn´edes Sciences, des Arts et des M´etiers, compiled by Diderot and D’Alembert, and published in Geneva around 1778. Of the many illuminating entries in this substantial work, the following definition of a probabilist was not overlooked. 144 Geoffrey Grimmett ÈÊÇBABÁÄÁËÌE,s.m.(Gram. Th´eol.) celui qui tient pour la doctrine abom- inable des opinions rendues probables par la d´ecision d’un casuiste, & qui assure l’innocence de l’action faite en cons´equence. Pascal a foudroy´ece systˆeme, qui ouvroit la porte au crime en accordant `al’autorit´eles pr´erogatives de la certitude, `al’opinion & la s´ecurit´equi n’appartient qu’`ala bonne conscience. This work was aided by partial financial support from the European Union under contract CHRX–CT93–0411, and from the Engineering and Physical Sciences Research Council under grant GR/L15425. Geoffrey R. Grimmett Statistical Laboratory University of Cambridge 16 Mill Lane Cambridge CB2 1SB United Kingdom. October 1996 Note added at reprinting. GRG thanks Roberto Schonmann for pointing out an error in the proof of equation (8.20), corrected in this reprint. June 2012 Percolation and Disordered Systems 145 CONTENTS 1. Introductory Remarks 147 1.1 Percolation 147 1.2 Some possible questions 148 1.3 History 149 2. Notation and Definitions 151 2.1 Graph terminology 151 2.2 Probability 152 2.3 Geometry 153 2.4 A partial order 154 2.5 Site percolation 155 3. Phase Transition 156 3.1 Percolation probability 156 3.2 Existence of phase transition 156 3.3 A question 158 4. Inequalities for Critical Probabilities 159 4.1 Russo’s formula 159 4.2 Strict inequalities for critical probabilities 161 4.3 The square and triangular lattices 162 4.4 Enhancements 167 5. Correlation Inequalities 169 5.1 FKG inequality 169 5.2 Disjoint occurrence 172 5.3 Site and bond percolation 174 6. Subcritical Percolation 176 6.1 Using subadditivity 176 6.2 Exponential decay 179 6.3 Ornstein–Zernike decay 190 7. Supercritical Percolation 191 7.1 Uniqueness of the infinite cluster 191 7.2 Percolation in slabs 194 7.3 Limit of slab critical points 195 7.4 Percolation in half-spaces 209 7.5 Percolation probability 209 7.6 Cluster-size distribution 211 146 Geoffrey Grimmett 8. Critical Percolation 212 8.1 Percolation probability 212 8.2 Critical exponents 212 8.3 Scaling theory 212 8.4 Rigorous results 214 8.5 Mean-field theory 214 9. PercolationinTwoDimensions 224 1 9.1 The critical probability is 2 224 9.2 RSW technology 225 9.3 Conformal invariance 228 10. Random Walks in Random Labyrinths 232 10.1 Random walk on the infinite percolation cluster 232 10.2 Random walks in two-dimensional labyrinths 236 10.3 General labyrinths 245 11. Fractal Percolation 251 11.1 Random fractals 251 11.2 Percolation 253 11.3 A morphology 255 11.4 Relationship to Brownian Motion 257 12. Ising and Potts Models 259 12.1 Ising model for ferromagnets 259 12.2 Potts models 260 12.3 Random-cluster models 261 13. Random-Cluster Models 263 13.1 Basic properties 263 13.2 Weak limits and phase transitions 264 13.3Firstandsecondordertransitions 266 13.4 Exponential decay in the subcritical phase 267 13.5 The case of two dimensions 272 References 280 Percolation and Disordered Systems 147 1. INTRODUCTORY REMARKS 1.1. Percolation We will focus our ideas on a specific percolation process, namely ‘bond per- colation on the cubic lattice’, defined as follows. Let Ld = (Zd, Ed) be the hypercubic lattice in d dimensions, where d 2. Each edge of Ld is de- clared open with probability p, and closed otherwise.≥ Different edges are given independent designations. We think of an open edge as being open to the transmission of disease, or to the passage of water. Now concentrate on the set of open edges, a random set. Percolation theory is concerned with ascertaining properties of this set. The following question is considered central. If water is supplied at the origin, and flows along the open edges only, can it reach infinitely many ver- tices with strictly positive probability? It turns out that the answer is no for small p, and yes for large p. There is a critical probability pc dividing these two phases. Percolation theory is particularly concerned with understanding the geometry of open edges in the subcritical phase (when p<pc), the super- critical phase (when p>pc), and when p is near or equal to pc (the critical case). As an illustration of the concrete problems of percolation, consider the function θ(p), defined as the probability that the origin lies in an infinite cluster of open edges (this is the probability referred to above, in the dis- cussion of pc). It is believed that θ has the general appearance sketched in Figure 1.1. θ should be smooth on (pc, 1). It is known to be infinitely differentiable, • but there is no proof known that it is real analytic for all d. Presumably θ is continuous at pc. No proof is known which is valid for • all d. Perhaps θ is concave on (pc, 1], or at least on (pc,pc + δ) for some • positive δ. β As p pc, perhaps θ(p) a(p pc) for some constant a and some • ‘critical↓ exponent’ β. ∼ − We stress that, although each of the points raised above is unproved in gen- eral, there are special arguments which answer some of them when either d = 2 or d is sufficiently large. The case d = 3 is a good one on which to concentrate. 148 Geoffrey Grimmett θ(p) 1 pc 1 p Fig. 1.1. It is generally believed that the percolation probability θ(p) behaves roughly as indicated here. It is known, for example, that θ is infinitely differen- tiable except at the critical point pc. The possibility of a jump discontinuity at pc has not been ruled out when d ≥ 3 but d is not too large. 1.2 Some Possible Questions Here are some apparently reasonable questions, some of which turn out to be feasible. What is the value of pc? • What are the structures of the subcritical and supercritical phases? • What happens when p is near to pc? • Are there other points of phase transition? • What are the properties of other ‘macroscopic’ quantities, such as the • mean size of the open cluster containing the origin? What is the relevance of the choice of dimension or lattice? • In what ways are the large-scale properties different if the states of • nearby edges are allowed to be dependent rather than independent? There is a variety of reasons for the explosion of interest in the percolation model, and we mention next a few of these. The problems are simple and elegant to state, and apparently hard to • solve. Their solutions require a mixture of new ideas, from analysis, geometry, • and discrete mathematics. Physical intuition has provided a bunch of beautiful conjectures. • Techniques developed for percolation have applications to other more • complicated spatial random processes, such as epidemic models. Percolation gives insight and method for understanding other physical • models of spatial interaction, such as Ising and Potts models.
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