Our mathematical understanding of the statistical mechanics of disordered systems is going through a period of stunning progress. This self-contained book is a graduate-level introduction for mathematicians and for physicists interested in the mathematical foundations of the field, and can be used as a textbook for a two-semester course on mathematical statistical mechanics. It assumes only basic knowledge of classical physics and, on the mathematics side, a good working knowledge of graduate-level probability theory. The book starts with a concise introduction to statistical mechanics, proceeds to disordered lattice spin systems, and concludes with a presentation of the latest developments in the mathematical understanding of mean-field spin glass models. In particular, recent progress towards a rigorous understanding of the replica symmetry-breaking solutions of the Sherrington-Kirkpatrick spin glass models, due to Guerra, Aizenman-Sims-Starr and Talagrand, is reviewed in some detail.
Contents
Preface; Part I. Statistical Mechanics: 1. Introduction; 2. Principles of statistical mechanics; 3. Lattice gases and spin systems; 4. Gibbsian formalism; 5. Cluster expansions; Part II. Disordered Systems: Lattice Models: 6. Gibbsian formalism and metastates; 7. The random field Ising model; Part III: Disordered Systems: Mean Field Models; 8. Disordered mean field models; 9. The random energy model; 10. Derrida’s generalised random energy models; 11. The SK models and the Parisi solution; 12. Hopfield models; 13. The number partitioning problem; Bibliography; Index of notation; Index.
Review
"This book grew out of lecture notes and courses, and was prepared and polished over the last six years. Long-awaited, it is carefully prepared in a pedagogical style, with selected themes and progressive difficulty. It presents a broad overview of the field with modern tools and elaborate techniques, culminating with deep results and fascinating pictures. However, the author succeeds in also making it nice to read and easy to handle, keeping the style as direct as possible..."
Francis Comets, Mathematical Reviews
Printer friendly version
Email a colleague