Probability on Graphs: Random Processes on

This introduction to some of the principal models in the theory of disordered systems leads the reader through the basics, to the very edge of contemporary research, with the minimum of technical fuss. Topics covered include random walk, percolation, self-avoiding walk, interacting particle systems, uniform spanning tree, random graphs, as well as the Ising, Potts, and random-cluster models for ferromagnetism, and the Lorentz model for motion in a random medium. Schramm-L_wner evolutions (SLE) arise in various contexts. The choice of topics is strongly motivated by modern applications and focuses on areas that merit further research. Special features include a simple account of Smirnov's proof of Cardy's formula for critical percolation, and a fairly full account of the theory of influence and sharp-thresholds. Accessible to a wide audience of mathematicians and physicists, this book can be used as a graduate course text. Each chapter ends with a range of exercises.

Preface	p. ix
Random walks on graphs	p. 1
Random walks and reversible Markov chains	p. 1
Electrical networks	p. 3
Flows and energy	p. 8
Recurrence and resistance	p. 11
Pólya's theorem	p. 14
Graph theory	p. 16
Exercises	p. 18
Uniform spanning tree	p. 21
Definition	p. 21
Wilson's algorithm	p. 23
Weak limits on lattices	p. 28
Uniform forest	p. 31
Schramm-Löwner evolutions	p. 32
Exercises	p. 37
Percolation and self-avoiding walk	p. 39
Percolation and phase transition	p. 39
Self-avoiding walks	p. 42
Coupled percolation	p. 45
Oriented percolation	p. 45
Exercises	p. 48
Association and influence	p. 50
Holley inequality	p. 50
FKG inequality	p. 53
BK inequality	p. 54
Hoeffding inequality	p. 56
Influence for product measures	p. 58
Proofs of influence theorems	p. 63
Russo's formula and sharp thresholds	p. 75
Exercises	p. 78
Further percolation	p. 81
Subcritical phase	p. 81
Supercritical phase	p. 86
Uniqueness of the infinite cluster	p. 92
Phase transition	p. 95
Open paths in annuli	p. 99
The critical probability in two dimensions	p. 103
Cardy's formula	p. 110
The critical probability via the sharp-threshold theorem	p. 121
Exercises	p. 125
Contact process	p. 127
Stochastic epidemics	p. 127
Coupling and duality	p. 128
Invariant measures and percolation	p. 131
The critical value	p. 133
The contact model on a tree	p. 135
Space-time percolation	p. 138
Exercises	p. 141
Gibbs states	p. 142
Dependency graphs	p. 142
Markov fields and Gibbs states	p. 144
Ising and Potts models	p. 148
Exercises	p. 150
Random-cluster model	p. 152
The random-cluster and Ising/Potts models	p. 152
Basic properties	p. 155
Infinite-volume limits and phase transition	p. 156
Open problems	p. 160
In two dimensions	p. 163
Random even graphs	p. 168
Exercises	p. 171
Quantum Ising model	p. 175
The model	p. 175
Continuum random-cluster model	p. 176
Quantum Ising via random-cluster	p. 179
Long-range order	p. 184
Entanglement in one dimension	p. 185
Exercises	p. 189
Interacting particle systems	p. 190
Introductory remarks	p. 190
Contact model	p. 192
Voter model	p. 193
Exclusion model	p. 196
Stochastic Ising model	p. 200
Exercises	p. 203
Random graphs	p. 205
Erdos-Rényi graphs	p. 205
Giant component	p. 206
Independence and colouring	p. 211
Exercises	p. 217
Lorentz gas	p. 219
Lorentz model	p. 219
The square Lorentz gas	p. 220
In the plane	p. 223
Exercises	p. 224
References	p. 226
Index	p. 243
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Probability on Graphs: Random Processes on Graphs and Lattices

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