Mean field models for spin glasses: a first course

Michel Talagrand

Equipe d’Analyse, E.S.A. au C.N.R.S. n^{∘}7064, Université Paris 6, 4 Place Jussieu, 75252 Paris Cedex 05, France, www.proba.jussieu.fr

Contents

Chapter 1. Introduction 1 Chapter 2. What this is all about: the REM 5 Chapter 3. The Sherrington-Kirkpatrick model at high temperature 19 Chapter 4. The p-spin interaction model 31 Chapter 5. External field and the replica-symmetric solution 39 Chapter 6. Exponential inequalities 59 Chapter 7. Central limit theorems and the Almeida-Thouless line 73 Chapter 8. Emergence and separation of the lumps in the p-spin

interaction model 89

Bibliography 105

Chapter 1

Introduction

Consider a large collection of random variables (H_{s})_{s≤M}. What is the value of the largest of them? More generally, what is the structure of the “few largest” values? Certainly the question is too general. Even if the variables are identically distributed, the answer depends upon both the distribution and the correlation structure of the variables. When the variables are independent, then of course, everything can be computed. To move beyond that case, one should ask what correlation structures are of interest. A collection of random variables is, in other words, a stochastic process. The examples that first come to mind are indexed by the real line. The correlation structure one will consider will naturally take advantage of the structure of the real line, a “one-dimensional” object.

The direction purposed here will be completely different and probably completely new to the reader. The correlation structure considered will be high dimensional, in a sense that will soon be clear. This type of structures appear naturally in problems of combinatorial optimization in a random setting. One can dream that there exists an entire theory in the above direction; the present monograph presents only a few steps of the study of a typical example.

The motivation behind the present research is the work of physicists on mean field models for spin glasses, and in particular the book “Spin glass theory and beyond” by Mézard, Parisi, Virasoro. The physicists have built a truly remarkable theory, which they believe applies not only to the models they have introduced, but also to a number of problems of fundamental interest in probability. Due to space (and energy) limitation, these models will not be discussed here, and we refer the reader to [T6] for a non-technical discussion.

The predictions of the physicists are most likely true. A reader of this volume is however probably well aware that studying a problem by the methods of theoretical physics (in the present case heuristic arguments backed by numerical simulation) and providing mathematical arguments are two distinct endeavors. We make a definite choice for the later approach.

The author has devoted a major effort since 1994 to the task of providing proofs to some of the physicists’ statements. This has resulted in a number of very long and very technical papers. The author’s concern is that they will remain forever hardly accessible to others. The main goal of the present notes is to address this concern. Our overwhelming objective is readability. No knowledge of physics, or statistical mechanics is required, or even probably useful to read these notes. There are no prerequisite of any kind, beside a familiarity with the basic concepts of probability theory. In fact, the most advanced tool that will not be proved is Hölder’s inequality. With the exception of a few results that are probably still far from their final form, every proof is given in complete detail. The usual dreadful “it is easy to see … ” have not been permitted to creep in. In order to improve readability we will discuss only the SK model and its cousin, the p-spin interaction model.

A regrettable consequence of this (necessary) choice is that it will not be possible to introduce the reader to what is possibly the greatest charm of the topic: the existence of several models, which yield to the same overall approach, but seem to require different technical tools. But of course the reader having mastered the present notes should be well armed to explore this aspect by reading the relevant research papers.

Let us now briefly describe the content of each chapter.

Chapter 2 is devoted to a toy model, Derrida’s Random Energy Model, that helps understanding what this is all about. This model is not treated in the simplest possible manner, but is also an opportunity to introduce some tools.

Chapter 3 studies the high temperature case of the SK model without external field. Some special symmetries make this case particularly simple and well understood.

The main motivation is that the methods developed there also apply to the p-spin interaction model, considered in Chapter 4. In this chapter, we prove the basic a priori estimates that will be the starting point of our study of this model below the critical temperature.

In Chapter 5, we return to the SK model, this time with external field. This is a much more interesting situation. We give a simple and complete proof of the mysterious “replica-symmetric” formula and we describe in great detail the structure behind them. This chapter is essentially self contained.

In Chapter 6, we investigate in much greater detail the structure behind the replica-symmetric formula. We replace crude moment inequalities by sharp exponential inequalities. Most importantly, the proofs there, rather than taking advantage of specific features of the SK model, rely on tools of much wider applicability.

In Chapter 7 we compute the inter-spin correlation. We explain the occurrence of the AT line, the conjectured boundary between the high and low temperature region. We also argue that the validity of the replicasymmetric solution on the entire high-temperature region is not self-evident (as the physicists think) but rather is a difficult problem.

In Chapter 8 we return to the p-spin interaction model. We study it below the critical temperature. We prove that the configuration space spontaneously decomposes in small pieces that we call the lumps. We prove that these are as much separated from each other as they can be.

The structure of this model below (but not too far below) the critical temperature is now (January 2000) rigorously understood in much greater detail than is presented here [T7]. Having decided at the onset that this monograph would contain only proofs that were being written a second time, we have not included these recent results here. For the same reason, the reader is referred to [T8] for another major progress (April 2000) on the region of validity of the replica-symmetric solution. The reader having penetrated the present monograph should however be well prepared to read [T7, T8].

Acknowledgment. The author expresses his deepest gratitude to Professor Anne Boutet de Monvel who made the publication of this monograph possible by accepting the unrewarding labor of typing it. He also expresses his thanks to the many participants of the school that helped to correct countless typos and inaccuracies.