Probability on Graphs: Random Processes on Graphs and Lattices

Probability on Graphs: Random Processes on Graphs and Lattices

by Geoffrey Grimmett
ISBN-10:
0521197988
ISBN-13:
9780521197984
Pub. Date:
08/16/2010
Publisher:
Cambridge University Press

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Overview

Probability on Graphs: Random Processes on Graphs and Lattices

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–Löwner 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.

Product Details

ISBN-13: 9780521197984
Publisher: Cambridge University Press
Publication date: 08/16/2010
Series: Institute of Mathematical Statistics Textbooks Series , #1
Edition description: Reissue
Pages: 260
Product dimensions: 6.00(w) x 9.10(h) x 0.80(d)

About the Author

Geoffrey Grimmett is Professor of Mathematical Statistics in the Statistical Laboratory at the University of Cambridge.

Table of Contents

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 Infinite-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 Erdos-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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