Statistical Mechanics of Lattice Systems A Concrete Mathematical Introduction

This motivating textbook gives a friendly, rigorous introduction to fundamental concepts in equilibrium statistical mechanics, covering a selection of specific models, including the Curie–Weiss and Ising models, the Gaussian free field, O(n) models, and models with Kac interactions. Using classical concepts such as Gibbs measures, pressure, free energy, and entropy, the book exposes the main features of the classical description of large systems in equilibrium, in particular the central problem of phase transitions. It treats such important topics as the Peierls argument, the Dobrushin uniqueness, Mermin–Wagner and Lee–Yang theorems, and develops from scratch such workhorses as correlation inequalities, the cluster expansion, Pirogov–Sinai Theory, and reflection positivity. Written as a self-contained course for advanced undergraduate or beginning graduate students, the detailed explanations, large collection of exercises (with solutions), and appendix of mathematical results and concepts also make it a handy reference for researchers in related areas.

Preface; Convention; 1. Introduction; 2. The Curie–Weiss model; 3. The Ising model; 4. Liquid-vapor equilibrium; 5. Cluster expansion; 6. Infinite-volume Gibbs measures; 7. Pirogov–Sinai theory; 8. The Gaussian free field on Zd; 9. Models with continuous symmetry; 10. Reflection positivity; A. Notes; B. Mathematical appendices; C. Solutions to exercises; Bibliography; Index.

Sacha Friedli is Associate Professor of Mathematics at the Universidade Federal de Minas Gerais, Brazil. His current research interests are in statistical mechanics, mathematical physics, and Markov processes.
Yvan Velenik is Professor of Mathematics at the Université de Genève. His current work focuses on applications of probability theory to the study of classical statistical mechanics, especially lattice random fields and random walks.