The Random-Cluster Model

3540328904......Page 1
Contents......Page 10
1.1 Introduction......Page 13
1.2 Random-cluster model......Page 16
1.3 Ising and Potts models......Page 18
1.4 Random-cluster and Ising/Potts models coupled......Page 20
1.5 The limit as q ↓ 0......Page 25
1.6 Basic notation......Page 27
2.1 Stochastic ordering of measures......Page 31
2.2 Positive association......Page 37
2.3 Influence for monotonic measures......Page 42
2.4 Sharp thresholds for increasing events......Page 45
2.5 Exponential steepness......Page 47
3.1 Conditional probabilities......Page 49
3.2 Positive association......Page 51
3.3 Differential formulae and sharp thresholds......Page 52
3.4 Comparison inequalities......Page 55
3.5 Exponential steepness......Page 61
3.6 Partition functions......Page 65
3.7 Domination by the Ising model......Page 69
3.8 Series and parallel laws......Page 73
3.9 Negative association......Page 75
4.1 Infinite graphs......Page 79
4.2 Boundary conditions......Page 82
4.3 Infinite-volume weak limits......Page 84
4.4 Infinite-volume random-cluster measures......Page 90
4.5 Uniqueness via convexity of pressure......Page 97
4.6 Potts and random-cluster models on infinite graphs......Page 107
5.1 The critical point......Page 110
5.2 Percolation probabilities......Page 114
5.3 Uniqueness of random-cluster measures......Page 119
5.4 The subcritical phase......Page 122
5.5 Exponential decay of radius......Page 125
5.6 Exponential decay of volume......Page 131
5.7 The supercritical phase and the Wulff crystal......Page 134
5.8 Uniqueness when q < 1......Page 143
6.1 Planar duality......Page 145
6.2 The value of the critical point......Page 150
6.3 Exponential decay of radius......Page 155
6.4 First-order phase transition......Page 156
6.5 General lattices in two dimensions......Page 164
6.6 Square, triangular, and hexagonal lattices......Page 166
6.7 Stochastic Löwner evolutions......Page 176
7.1 Surfaces and plaquettes......Page 179
7.2 Basic properties of surfaces......Page 181
7.3 A contour representation......Page 185
7.4 Polymer models......Page 191
7.5 Discontinuous phase transition for large q......Page 194
7.6 Dobrushin interfaces......Page 207
7.7 Probabilistic and geometric preliminaries......Page 211
7.8 The law of the interface......Page 214
7.9 Geometry of interfaces......Page 220
7.10 Exponential bounds for group probabilities......Page 227
7.11 Localization of interface......Page 230
8.1 Time-evolution of the random-cluster model......Page 234
8.2 Glauber dynamics......Page 236
8.3 Gibbs sampler......Page 237
8.4 Coupling from the past......Page 239
8.5 Swendsen–Wang dynamics......Page 242
8.6 Coupled dynamics on a finite graph......Page 244
8.7 Box dynamics with boundary conditions......Page 249
8.8 Coupled dynamics on the infinite lattice......Page 252
8.9 Simultaneous uniqueness......Page 267
9.1 Potts models and flows......Page 269
9.2 Flows in the Ising model......Page 274
9.3 Exponential decay for the Ising model......Page 285
9.4 The Ising model in four and more dimensions......Page 286
10.1 Mean-field theory......Page 288
10.2 On complete graphs......Page 289
10.3 Main results for the complete graph......Page 293
10.4 The fundamental proposition......Page 296
10.5 The size of the largest component......Page 298
10.6 Proofs of main results for complete graphs......Page 301
10.7 The nature of the singularity......Page 307
10.8 Large deviations......Page 308
10.9 On a tree......Page 311
10.10 The critical point for a tree......Page 317
10.11 (Non-)uniqueness of measures on trees......Page 325
10.12 On non-amenable graphs......Page 327
11.1 Random-cluster representations......Page 332
11.2 The Potts model......Page 333
11.3 The Ashkin–Teller model......Page 338
11.4 The disordered Potts ferromagnet......Page 342
11.5 The Edwards–Anderson spin-glass model......Page 345
11.6 The Widom–Rowlinson lattice gas......Page 349
Appendix. The Origins of FK(G)......Page 353
List of Notation......Page 362
References......Page 365
C......Page 385
G......Page 386
P......Page 387
R......Page 388
Z......Page 389