Phase Transitions in the Theory of Lattice Gases

A comprehensive account of methods used in the rigorous study of Ising models and classical and quantum Heisenberg models.

Barry Simon (Author)

9781108491853, Cambridge University Press

Hardback, published 13 November 2025

644 pages
25.4 x 17.8 x 3.5 cm, 1.47 kg

'Lattice gases are the simplest systems in which to study a most important topic in statistical mechanics, namely phase transitions. Barry Simon has been a major contributor to that field as well as one of the very best expositors of mathematical physics. This book results from the combination of these two qualities.' Jean Bricmont, Université Catholique de Louvain

The intersection of statistical mechanics and mathematical analysis has proved a fertile ground for mathematical physics and probability, and in the decades since lattice gases were first proposed as a model for describing physical systems at the atomic level, our understanding of them has grown tremendously. A book that provides a comprehensive account of the methods used in the study of phase transitions for Ising models and classical and quantum Heisenberg models has been long overdue. This book, written by one of the masters of the subject, is just that. Topics covered include correlation inequalities, Lee-Yang theorems, the Peierls method, the Hohenberg-Mermin-Wagner method, infrared bounds, random cluster methods, random current methods and BKT transition. The final section outlines major open problems to inspire future work. This is a must-have reference for researchers in mathematical physics and probability and serves as an entry point, albeit advanced, for students entering this active area.

Notation
1. A framework for lattice gases
2. Correlation inequalities
3. The Lee-Yang theorem
4. Peierls argument
5. No continuous symmetry breaking in 2D (and other situations)
6. Infrared bounds
7. Stochastic geometric representations
8. Berezinskii-Kosterlitz-Thouless transition
9. Final thoughts
References
Author index
Subject index.

Subject Areas: Maths for scientists [PDE]