• DOI:10.1017/CBO9780511616808
• Corpus ID: 115617071

Statistical Mechanics of Disordered Systems

@inproceedings{Bovier2006StatisticalMO,
  title={Statistical Mechanics of Disordered Systems},
  author={Anton Bovier},
  year={2006}
}

• A. Bovier
• Published 1 June 2006
• Physics

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 some basic knowledge of classical physics; on the mathematics side, the reader should have a good working knowledge of graduate-level probability theory. Part I gives a concise introduction to thermodynamics and statistical mechanics, which provides the tools and concepts needed later. The main topics treated here are the classical ensembles of statistical mechanics, lattice gases and spin systems, the rigorous setting of the theory of infinite-volume Gibbs states (DLR theory), and cluster expansions for highand low-temperature phases. Part II proceeds to disordered lattice models. It presents the general theory of random Gibbs states and metastates in the spirit of Newman–Stein and the random-field Ising model. Part III is devoted to disordered mean-field models. It begins with the random energy model as a toy example and then explains in depth the geometric structures arising in the description of the infinite-volume limit of the Gibbs states in the generalized random energy models. Finally, it presents 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. The last two chapters treat applications to non-physical systems: the Hopfield neural network model and the number partitioning problem.