Probability on Graphs
Random Processes on Graphs and Lattices

A user-friendly introduction for mathematicians to some of the principal stochastic models near the interface of probability and physics.

Geoffrey Grimmett (Author)

9781108438179, Cambridge University Press

Paperback / softback, published 25 January 2018

276 pages
22.8 x 15.1 x 1.5 cm, 0.4 kg

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. This new edition features accounts of major recent progress, including the exact value of the connective constant of the hexagonal lattice, and the critical point of the random-cluster model on the square lattice. The choice of topics is strongly motivated by modern applications, and focuses on areas that merit further research. 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
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.

Subject Areas: Mathematical physics [PHU], Statistical physics [PHS], Stochastics [PBWL], Probability & statistics [PBT], Calculus & mathematical analysis [PBK]