Mathematical Aspects of Spin Glasses and Neural Networks PDF
Preview Mathematical Aspects of Spin Glasses and Neural Networks
Progress in Probability Volume 41 Series Editors Thomas Liggett Charles Newman Loren Pitt Mathematical Aspects of Spin Glasses and Neural Networks Anton Bovier Pierre Picco Editors 1998 Birkhauser Boston • Basel • Berlin Anton Bovier Pierre Picco Weierstrass - Institut fUr Angewandle CNRS-Luminy Analysis und Siochaslik Centre de Physique Theorique Berlin 0-10117 Marseille 13288 Germany France Library of Congress Cataloging-in-Publication Data Mathematical aspects of spin glasses and neural networks , Anion Bovier, Pierre Picco, editors. p. cm. -- (Progress in probability ; v. 41) Includes bibliographical references. ISBN-13: 978-1-4612-8653-0 c-ISBN-13: 978-1-4612-4102-7 DOl: 10.1007/978-1-4612-4102-7 1. Spin glasses--Mathematics. 2. Neural networks (Computer scicnce)--Mathematics. I. Bovier, Anton 1957- II. Picco, Pierre, 1953-. III. Series: Progress in probability ; 41. QCI76.8.s68M38 1997 538'.4--dc21 97-20693 CIP AMS Classification Codes: 6OXX, 82CXX, 92BXX, 94CXX Printed on acid-free paper ~ © 1998 Birkhauser Boston Birkhiiuser Copyright is not claimed for works of U.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any fonn or by any means, electronic, mechanical, photocopy ing, recording, or otherwise, without prior pennission of the copyright owner. Pennission to photocopy for internal or personal use of specific clients is gramed by Birkh!iuser Boston for libraries and other users registered with the Copyright Clearance Center (Ccq, provided that the base fee of $6.00 per copy, plus $0.20 pcr page is paid directly to CCC, 222 Rosewood Drive, Danvers, MA 01923, U.S.A. Special requests should be addressed directly to Birkhauser Boston, 675 Massachusetts Avenue, Cam bridge, MA 02139, U.S.A. Refonnatted from authors' disks by Te:'lniques, Inc. in TEX. Printed and bound by Quinn-Woodbine, Woodbine, NJ. 9 8 7 654 3 2 I Table of Contents Prologue ........................................................ vii PART 1: STATICS 1.1 Mean Field Models Hopfield Models as Generalized Random Mean Field Models A. Bovier and V. Gayrard ......................................... 3 The Martingale Method for Mean-Field Disordered Systems at High Temperature F. Comets ........................................................ 91 On the Central Limit Theorem for the Overlap in the Hopfield Model B. Gentz ........................................................ 115 Limiting Behavior of Random Gibbs Measures: Metastates in Some Disordered Mean Field Models C. Kiilske ....................................................... 151 On the Storage Capacity of the Hopfield Model M. Lowe ........................................................ 161 1.2 Lattice Models Typical Profiles of the Kac-Hopfield Model A. Bovier, V. Gayrard, and P. Picco ............................ 187 Thermodynamic Chaos and the Structure of Short-Range Spin Glasses C.M. Newman and D.L. Stein ................................... 243 Random Spin Systems with Long-Range Interactions B. Zegarlinski ................................................... 289 vi Table of Contents PART 2: Dynamics Langevin Dynamics for Sherrington-Kirkpatrick Spin Glasses G. Ben Arous and A. Guionnet .................................. 323 Sherrington-Kirkpatrick Spin-Glass Dynamics Part II: The Discrete Setting M. Grunwald ..................................................... 355 Prologue Spin glass theory has been an extremely active field of research in both experimental and theoretical physics for more than twenty years and is still producing papers at a prodigious rate. Soon after the in troduction of the first spin glass models, the close relationship between spin glasses and certain aspects of the theory of neural networks was discovered, and "spin glasses and neural networks" is now a well estab lished subfield of theoretical physics of enormous promise. Virtually all the important theoretical results - besides simulation based on Monte Carlo techniques - rely on one great analytic tool, the replica trick and Parisi's theory of replica symmetry breaking. While this is without doubt one of the most impressive achievements of modern theoretical physics, there is no mathematical justification, and not even under standing, of this theoretical tool. A mathematically rigorous analysis of spin glass models and possibly a rigorous derivation of the results of replica theory presents therefore one of the great challenges for math ematics and for probability theory in particular. Over the last years an increasing number of probabilists and mathematical physicists has indeed begun to work on this field, and results have been produced at an impressive rate, even though many of the most interesting heuristic results are still beyond the reach of rigour. In this situation, the ques tion "Where can I find a good introduction to the subject?" was put to us more and more frequently by colleagues seeking to inform them selves or give some graduate student some useful reading to get started on a research project. There was no really satisfactory answer to this question: Excellent reviews exist on the theoretical physics aspects of the subject, but they cannot serve as an introduction to the mathe matical problems and are usually not easy to read for mathematicians. Two recent review papers by Petritis [Pl,P2] give a guided tour to the literature and some first ideas, but are hardly self-contained. To get to the subject through the original literature is an admittedly hard task. So the idea of a book suggested itself: but, to write a comprehensive monograph on the subject that would be sufficiently deep and suffi ciently broad would be a tremendous task, and almost futile in view of the speed at which the field evolves. The idea to invite comprehen sive reviews from some of the main experts in the field and to collect them into a book appeared to us the most promising solution. Our enthusiasm for this idea, which was born in the spring of 1995, was luckily shared by the series editors of "Progress in Probability" and by the Publisher, Birkhiiuser, and, most importantly, by the authors viii Prologue that contributed to this volume. The timing proved well chosen. The reviews were written between the fall of 95 and the winter 96, and a number of remarkable breakthrough results were produced during that period that could be included in the texts. Thus, more than reviews, most of the contributions present original results for the first time. Of course, within the space of one volume, no such project can claim encyclopedic comprehensiveness. The subject area was chosen deliberately narrow to focus on intrinsically "spin glass" aspects. Thus results on random fields, percolation, etc. are absent. Also, high tem perature results which are not specific to spin glasses (although they apply to them) were largely excluded. Within that frame, the choice of the contributions is still effected by randomness and by personal prejudice on the part of the editors. But we hope, in spite of this, to have produced a volume that may serve that need for a first and basic reference to the mathematics of spin glasses and neural networks, and we hope that it will help to promote this challenging and fascinating field. We thank all contributors, the series editors, and the publisher for their help and for the work they invested. A. Bovier and P. Picco Berlin and Marseille, March 1997 PART 1: Statics 1.1 Mean Field Models Hopfield Models as Generalized Random Mean Field Models # Anton Bovier and Veronique Gayrard L'intuition ne peut nous donnerla rigueur, ni meme la certitude, on s'en est apen;u de plus en plus. Henri Poincare, "La Valeur de La Science" 1. Introduction Twenty years ago, Pastur and Figotin [FP1,FP2] first introduced and studied what has become known as the Hopfield model and which turned out, over the years, to be one of the more successful and im portant models of a disordered system. This is also reflected in the fact that several contributions in this book are devoted to it. The Hopfield model is quite versatile and models various situations. Pastur and Figotin introduced it as a simple model for a spin glass, and, in 1982, Hopfield independently considered it as a model for associative memory. The first viewpoint naturally put it in the context of equilibrium statistical mechanics, whereas Hopfield's main interest was its dynam ics. But the great success of what became known as the Hopfield model came from the realization, mainly in the work of Amit, Gutfreund, and Sompolinsky [AGS] , that a more complicated version of this model is reminiscent of a spin glass, and that the (then) recently developed methods of spin-glass theory, in particular, the replica trick and Parisi's replica symmetry breaking scheme could be adapted to this model and allowed a "complete" analysis of the equilibrium statistical mechanics of the model and the recovery of some of the most prominent "experimen tally" observed features of the model, such as the "storage capacity" and "loss of memory" , in a precise analytical way. This observation sparked a surge of interest by theoretical physi cists into neural network theory in general and has led to considerable progress in the field (the literature on the subject is extremely rich, and there are a great number of good review papers. See, for example # Work partially supported by the Commission of the European Union under contract CHRX-CT93-0411
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