Preface .................................................... ix
1 Random walks on graphs ...................................... 1
1.1 Random walks and reversible Markov chains .............. 1
1.2 Electrical networks .................................... 3
1.3 Flows and energy ....................................... 8
1.4 Recurrence and resistance ............................. 11
1.5 Pólya's theorem ....................................... 14
1.6 Graph theory .......................................... 16
1.7 Exercises ............................................. 18
2 Uniform spanning tree ...................................... 21
2.1 Definition ............................................ 21
2.2 Wilson's algorithm .................................... 23
2.3 Weak limits on lattices ............................... 28
2.4 Uniform forest ........................................ 31
2.5 Schramm-Löwner evolutions ............................. 32
2.6 Exercises ............................................. 37
3 Percolation and self-avoiding walk ......................... 39
3.1 Percolation and phase transition ...................... 39
3.2 Self-avoiding walks ................................... 42
3.3 Coupled percolation ................................... 45
3.4 Oriented percolation .................................. 45
3.5 Exercises ............................................. 48
4 Association and influence .................................. 50
4.1 Holley inequality ..................................... 50
4.2 FKG inequality ........................................ 53
4.3 BK inequality ......................................... 54
4.4 Hoeffding inequality .................................. 56
4.5 Influence for product measures ........................ 58
4.6 Proofs of influence theorems .......................... 63
4.7 Russo's formula and sharp thresholds .................. 75
4.8 Exercises ............................................. 78
5 Further percolation ........................................ 81
5.1 Subcritical phase ..................................... 81
5.2 Supercritical phase ................................... 86
5.3 Uniqueness of the infinite cluster .................... 92
5.4 Phase transition ...................................... 95
5.5 Open paths in annuli .................................. 99
5.6 The critical probability in two dimensions ........... 103
5.7 Cardy's formula ...................................... 110
5.8 The critical probability via the sharp-threshold
theorem .............................................. 121
5.9 Exercises ............................................ 125
6 Contact process ........................................... 127
6.1 Stochastic epidemics ................................. 127
6.2 Coupling and duality ................................. 128
6.3 Invariant measures and percolation ................... 131
6.4 The critical value ................................... 133
6.5 The contact model on a tree .......................... 135
6.6 Space-time percolation ............................... 138
6.7 Exercises ............................................ 141
7 Gibbs states .............................................. 142
7.1 Dependency graphs .................................... 142
7.2 Markov fields and Gibbs states ....................... 144
7.3 Ising and Potts models ............................... 148
7.4 Exercises ............................................ 150
8 Random-cluster model ...................................... 152
8.1 The random-cluster and Ising/Potts models ............ 152
8.2 Basic properties ..................................... 155
8.3 Infmile-volume limits and phase transition ........... 156
8.4 Open problems ........................................ 160
8.5 In two dimensions .................................... 163
8.6 Random even graphs ................................... 168
8.7 Exercises ............................................ 171
9 Quantum Ising model ....................................... 175
9.1 The model ............................................ 175
9.2 Continuum random-cluster model ....................... 176
9.3 Quantum Ising via random-cluster ..................... 179
9.4 Long-range order ..................................... 184
9.5 Entanglement in one dimension ........................ 185
9.6 Exercises ............................................ 189
10 Interacting particle systems .............................. 190
10.1 Introductory remarks ................................. 190
10.2 Contact model ........................................ 192
10.3 Voter model .......................................... 193
10.4 Exclusion model ...................................... 196
10.5 Stochastic Ising model ............................... 200
10.6 Exercises ............................................ 203
11 Random graphs ............................................. 205
11.1 Erdős-Rényi graphs ................................... 205
11.2 Giant component ...................................... 206
11.3 Independence and colouring ........................... 211
11.4 Exercises ............................................ 217
12 Lorentz gas ............................................... 219
12.1 Lorentz model ........................................ 219
12.2 The square Lorentz gas ............................... 220
12.3 In the plane ......................................... 223
12.4 Exercises ............................................ 224
References ................................................ 226
Index ..................................................... 243
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