Probability on Graphs

Probability on Graphs : Random Processes on Graphs and Lattices

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Description

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-Lowner 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.show more

Product details

  • Electronic book text | 260 pages
  • CAMBRIDGE UNIVERSITY PRESS
  • Cambridge University Press (Virtual Publishing)
  • Cambridge, United Kingdom
  • English
  • 45 b/w illus. 90 exercises
  • 1139036602
  • 9781139036603

Review quote

"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 Physicsshow more

About Geoffrey Grimmett

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

Table of contents

Preface; 1. Random walks on graphs; 2. Uniform spanning tree; 3. Percolation and self-avoiding walk; 4. Association and influence; 5. Further percolation; 6. Contact process; 7. Gibbs states; 8. Random-cluster model; 9. Quantum Ising model; 10. Interacting particle systems; 11. Random graphs; 12. Lorentz gas; References; Index.show more