The Random-Cluster Model (eBook)

PhD (Oxford 1974) under supervision of John Hammersley and Dominic Welsh. Member of the Mathematics Department of Bristol University (1976-1992), and subsequently appointed to the Professorship of Mathematics Statistics at Cambridge University. Author of around 100 articles and five books in probability and related fields, including Percolation (Springer 1999), Probability and Random Processes (with David Stirzaker, Oxford University Press 2001). Managing Editor of "Probability Theory and Related Fields", 2001-2005.

Preface 6
Contents 10
Chapter 1 Random-Cluster Measures 13
1.1 Introduction 13
1.2 Random-cluster model 16
1.3 Ising and Potts models 18
1.4 Random-cluster and Ising/Potts models coupled 20
1.5 The limit as q . 0 25
1.6 Basic notation 27
Chapter 2 Monotonic Measures 31
2.1 Stochastic ordering of measures 31
2.2 Positive association 37
2.3 Influence for monotonic measures 42
2.4 Sharp thresholds for increasing events 45
2.5 Exponential steepness 47
Chapter 3 Fundamental Properties 49
3.1 Conditional probabilities 49
3.2 Positive association 51
3.3 Differential formulae and sharp thresholds 52
3.4 Comparison inequalities 55
3.5 Exponential steepness 61
3.6 Partition functions 65
3.7 Domination by the Ising model 69
3.8 Series and parallel laws 73
3.9 Negative association 75
Chapter 4 Infinite-Volume Measures 79
4.1 Infinite graphs 79
4.2 Boundary conditions 82
4.3 Infinite-volume weak limits 84
4.4 Infinite-volume random-cluster measures 90
4.5 Uniqueness via convexity of pressure 97
4.6 Potts and random-cluster models on infinite graphs 107
Chapter 5 Phase Transition 110
5.1 The critical point 110
5.2 Percolation probabilities 114
5.3 Uniqueness of random-cluster measures 119
5.4 The subcritical phase 122
5.6 Exponential decay of volume 131
5.7 The supercritical phase and the Wulff crystal 134
5.8 Uniqueness when q < 1
Chapter 6 In Two Dimensions 145
6.1 Planar duality 145
6.2 The value of the critical point 150
6.3 Exponential decay 155
6.4 First-order phase transition 156
6.5 General lattices in two dimensions 164
6.6 Square, triangular, and hexagonal lattices 166
6.7 Stochastic L wner evolutions 176
Chapter 7 Duality in Higher Dimensions 179
7.1 Surfaces and plaquettes 179
7.2 Basic properties of surfaces 181
7.3 A contour representation 185
7.4 Polymer models 191
7.5 Discontinuous phase transition for large q 194
7.6 Dobrushin interfaces 207
7.7 Probabilistic and geometric preliminaries 211
7.8 The law of the interface 214
7.9 Geometry of interfaces 220
7.10 Exponential bounds for group probabilities 227
7.11 Localization of interface 230
Chapter 8 Dynamics of Random-Cluster Models 234
8.1 Time-evolution of the random-cluster mode 234
8.2 Glauber dynamics 236
8.3 Gibbs sampler 237
8.4 Coupling from the past 239
8.5 Swendsen–Wang dynamics 242
8.6 Coupled dynamics on a finite graph 244
8.7 Box dynamics with boundary conditions 249
8.8 Coupled dynamics on the infinite lattice 252
Chapter 9 Flows in Poisson Graphs 269
9.1 Potts models and flows 269
9.2 Flows for the Ising model 274
9.3 Exponential decay for the Ising model 285
9.4 The Ising model in four and more dimensions 286
Chapter 10 On Other Graphs 288
10.1 Mean-field theory 288
10.2 On complete graphs 289
10.3 Main results for the complete graph 293
10.4 The fundamental proposition 296
10.5 The size of the largest component 298
10.6 Proofs of main results for complete graphs 301
10.7 The nature of the singularity 307
10.8 Large deviations 308
10.9 On a tree 311
10.10 The critical point for a tree 317
10.11 (Non-)uniqueness of measures on trees 325
10.12 On non-amenable graphs 327
Chapter 11 Graphical Methods for Spin Systems 332
11.1 Random-cluster representations 332
11.2 The Potts model 333
11.3 The Ashkin–Teller model 338
11.4 The disordered Potts ferromagnet 342
11.5 The Edwards–Anderson spin-glass model 345
11.6 The Widom–Rowlinson lattice gas 349
Appendix The Origins of FK(G) 353
List of Notation 362
References 365
Subject Index 386