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.

‘This remarkable near-encyclopaedic volume will become a classic text. Barry Simon’s vision and knowledge are evident on every page, and the result is a must-have for experts and students alike. The material proceeds from the classical to the ultra-modern, including recent probabilistic methodology and a fascinating list of open problems.’

Geoffrey Grimmett - University of Cambridge

‘My old joke was that to study or research phase transitions in lattice models, install Barry Simon in your office to answer all questions. This book obviates the need for a live Barry, takes up less room and is not subject to AI hallucinations – Highly Recommended.’

Chuck Newman - Courant Institute of Mathematical Sciences

‘This volume on the theory of Lattice Gases covers most of the recent developments of statistical mechanics. Here, the scholar finds what the modern trends are, and what open problems one might work on. Together with Simon’s previous book on Lattice Gases, it is an indispensable encyclopedia both for students and scientists interested in rigorous mathematical physics.’

Senya Shlosman - Skolkovo Institute of Science and Technology

‘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