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Institute of Mathematical Statistics Textbooks : Probability on Graphs: Random Processes on Graphs and Lattices Series Number 1 - Geoffrey Grimmett

Institute of Mathematical Statistics Textbooks

Probability on Graphs: Random Processes on Graphs and Lattices Series Number 1

Hardcover Published: 16th August 2010
ISBN: 9780521197984
Number Of Pages: 260

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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-Loewner 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.

'The book under review serves admirably for this 'getting started' purpose. It provides a rigorous introduction to a broad range of topics centered on the percolation-IPS field discussed above ... This book, like a typical Part III course, requires only undergraduate background knowledge but assumes a higher level of general mathematical sophistication. It also requires active engagement by the reader. As I often tell students, 'Mathematics is not a spectator sport - you learn by actually doing the exercises!' For the reader who is willing to engage the material and is not fazed by the fact that some proofs are only outlined or are omitted, this style enables the author to cover a lot of ground in 247 pages.' David Aldous, Bulletin of the American Mathematical Society
'It is written in a condensed style with only the briefest of introductions or motivations, but it is a mine of information for those who are well prepared and know how to use it. It formed the basis for a Probability reading group at the University of Warwick last term and was well received, and parts of it are being used by a colleague for an undergraduate module this term on Probability and Discrete Mathematics.' R.S. MacKay, Contemporary Physics
'This is clearly a successful advanced textbook.' Fernando Q. Gouvea, MAA Reviews
"The book under review serves admirably for this "getting started" purpose. It provides a rigorous introduction to a broad range of topics centered on the percolation-IPS field discussed above... This book, like a typical Part III course, requires only undergraduate background knowledge but assumes a higher level of general mathematical sophistication. It also requires active engagement by the reader. As I often tell students, "Mathematics is not a spectator sport - you learn by actually doing the exercises!" For the reader who is willing to engage the material and is not fazed by the fact that some proofs are only outlined or are omitted, this style enables the author to cover a lot of ground in 247 pages." David Aldous, Bulletin of the American Mathematical Society
"It is written in a condensed style with only the briefest of introductions or motivations, but it is a mine of information for those who are well prepared and know how to use it. It formed the basis for a Probability reading group at the University of Warwick last term and was well received, and parts of it are being used by a colleague for an undergraduate module this term on Probability and Discrete Mathematics." R.S. MacKay, Contemporary Physics
'This is clearly a successful advanced textbook.' Fernando Q. Gouvea, MAA Reviews

Prefacep. ix
Random walks on graphsp. 1
Random walks and reversible Markov chainsp. 1
Electrical networksp. 3
Flows and energyp. 8
Recurrence and resistancep. 11
Pólya's theoremp. 14
Graph theoryp. 16
Exercisesp. 18
Uniform spanning treep. 21
Definitionp. 21
Wilson's algorithmp. 23
Weak limits on latticesp. 28
Uniform forestp. 31
Schramm-Löwner evolutionsp. 32
Exercisesp. 37
Percolation and self-avoiding walkp. 39
Percolation and phase transitionp. 39
Self-avoiding walksp. 42
Coupled percolationp. 45
Oriented percolationp. 45
Exercisesp. 48
Association and influencep. 50
Holley inequalityp. 50
FKG inequalityp. 53
BK inequalityp. 54
Hoeffding inequalityp. 56
Influence for product measuresp. 58
Proofs of influence theoremsp. 63
Russo's formula and sharp thresholdsp. 75
Exercisesp. 78
Further percolationp. 81
Subcritical phasep. 81
Supercritical phasep. 86
Uniqueness of the infinite clusterp. 92
Phase transitionp. 95
Open paths in annulip. 99
The critical probability in two dimensionsp. 103
Cardy's formulap. 110
The critical probability via the sharp-threshold theoremp. 121
Exercisesp. 125
Contact processp. 127
Stochastic epidemicsp. 127
Coupling and dualityp. 128
Invariant measures and percolationp. 131
The critical valuep. 133
The contact model on a treep. 135
Space-time percolationp. 138
Exercisesp. 141
Gibbs statesp. 142
Dependency graphsp. 142
Markov fields and Gibbs statesp. 144
Ising and Potts modelsp. 148
Exercisesp. 150
Random-cluster modelp. 152
The random-cluster and Ising/Potts modelsp. 152
Basic propertiesp. 155
Infinite-volume limits and phase transitionp. 156
Open problemsp. 160
In two dimensionsp. 163
Random even graphsp. 168
Exercisesp. 171
Quantum Ising modelp. 175
The modelp. 175
Continuum random-cluster modelp. 176
Quantum Ising via random-clusterp. 179
Long-range orderp. 184
Entanglement in one dimensionp. 185
Exercisesp. 189
Interacting particle systemsp. 190
Introductory remarksp. 190
Contact modelp. 192
Voter modelp. 193
Exclusion modelp. 196
Stochastic Ising modelp. 200
Exercisesp. 203
Random graphsp. 205
Erdos-Rényi graphsp. 205
Giant componentp. 206
Independence and colouringp. 211
Exercisesp. 217
Lorentz gasp. 219
Lorentz modelp. 219
The square Lorentz gasp. 220
In the planep. 223
Exercisesp. 224
Referencesp. 226
Indexp. 243
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780521197984
ISBN-10: 0521197988
Series: Institute of Mathematical Statistics Textbooks
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 260
Published: 16th August 2010
Publisher: Cambridge University Press
Country of Publication: GB
Dimensions (cm): 22.8 x 15.2  x 2.0
Weight (kg): 0.54

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