Spin Glasses: A Challenge for Mathematicians: Cavity and Mean Field Models

by Springer Berlin Heidelberg

Overview

In the eighties, a group of theoretical physicists introduced several models for certain disordered systems, called "spin glasses". These models are simple and rather canonical random structures, that physicists studied by non-rigorous methods. They predicted spectacular behaviors, previously unknown in probability theory. They believe these behaviors occur in many models of considerable interest for several branches of science (statistical physics, neural networks and computer science).

This book introduces in a rigorous manner this exciting new area to the mathematically minded reader. It requires no knowledge whatsoever of any physics, and contains proofs in complete detail of much of what is rigorously known on spin glasses at the time of writing.

Product Details

ISBN-13:	9783642055683
Publisher:	Springer Berlin Heidelberg
Publication date:	12/07/2010
Series:	Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics , #46
Edition description:	Softcover reprint of hardcover 1st ed. 2003
Pages:	586
Product dimensions:	6.10(w) x 9.25(h) x 0.05(d)

Table of Contents

0. Introduction.- 1. A Toy Model, the REM.- 2. The Sherrington-Kirkpatrick Model.- 3. The Capacity of the Perceptron: The Ising Case.- 4. Capacity of the Perceptron: The Gaussian and the Spherical Case.- 5. The Hopfield Model.- 6. The p-Spin Interaction Model at Low Temperature.- 7. The Diluted SK Model and the K-Sat Problems.- 8. An Assignment Problem.- A. Appendix.- Elements of Probability Theory.- References.- Index.

Editorial Reviews

From the reviews:

"The book presents most of the technical tools available today for a rigorous analysis of random mean field models. It will be indispensable for anyone working in the field." (Anton Bovier, Combinatorics, Probability & Computing, Issue 13, 2004)

"This book seeks to start from basics … to develop the rigorous theory, in almost complete detail. … this is a book of ideas and calculations which build upon each other to create a rich and informative theory." (David Aldous, SIAM Reviews, Vol. 47 (1), 2005)

"Each section of the book ends with some comments on the literature. … The author has … put a great effort into rendering the exposition as clear as possible. … The book will, certainly, find a favorite place on the desk of anyone working in the field." (Anton Bovier, Zentralblatt MATH, Vol. 1033 (8), 2004)

"This is a book on structures, which are very interesting for physics and – at the same time – rather fundamental for mathematics. … In conclusion, this is a very important, impressive book, written by a prominent mathematician and leading expert in the field … . the book remains indispensable as a basic introduction and reference on this extremely difficulty and interesting subject." (EMS Newsletter, March, 2006)

From the Publisher