Spin Glasses: A Challenge for Mathematicians. Cavity and Mean Field Model
This is a book on structures, which are very useful for physics and, at the same time, rather important for mathematics. The subject of spin glasses has played a very special role in theoretical and mathematical physics for more than twenty years. The formulations of basic objects and the hypothesis of the theory are typically very simple and natural, requiring almost no mathematical and physical prerequisites and are easily understandable even to a novice in the field. However, there are almost no easy and simple results in this theory and exact answers even to the most basic questions are extremely difficult to obtain. Many different opinions concerning the behavior of these objects still exist after decades of study.
The theory of spin glasses remained for long time purely in the domain of theoretical physics. The great achievements in the field, due to Parisi and others, created a challenge for mathematical physicists a long time ago, due to the fact that these results were very non-rigorous mathematically. Considerable progress in understanding these mathematical ideas was achieved in recent years by several researchers, most notably Guerra. Thus, the field became more and more interesting not only for mathematical physicists but also for mathematicians. The author of this book is a prominent example of a pure mathematician finally attracted by these physical models, discovering that they describe objects of a fundamental, abstract mathematical nature.
The book describes most of this recent progress, achieved in the mathematical theory of spin glasses by the author and other mathematicians and mathematical physicists. It is directed to the mathematical community, thus requiring almost no prerequisites outside probability theory. It starts with the investigation of a ‘trivial’ random energy model of Derrida. Even in this introductory chapter we find delicate and detailed material in the description of low temperature behaviour of the REM model. Chapter 2 describes the Sherrington Kirkpatrick model (the short range spin glass models are even more complicated so that even among physicists there is no real consensus about their low temperature behaviour), where any two spins interact randomly. Firstly, the basic work for high temperatures is reviewed. Then the famous cavity (induction) method is described in detail, incorporating the recent important results of Guerra, the author and others. Chapter 3 deals with the ”capacity of a perceptron” and chapter 4 treats its special cases: the gaussian and spherical models. Chapter 5 describes the Hopfield model, a popular “model of memory” (which is somehow more accessible mathematically than the true spin glass model). Chapter 6 treats the p-spin interaction model (replacing the two spin interactions of usual models), chapter 7 discusses the “diluted” SK model and chapter 8 covers the “assignment problem” for permutations. A short appendix comprises prerequisites from probability theory, namely some basics on tail estimates, nets, random matrices, Poisson point process, etc.
In conclusion, this is a very important, impressive book, written by a prominent mathematician and leading expert in the field, on some fundamental mathematical questions and deep technical estimates in “high dimensional” probability theory. Even though remarkable new progress in the field of spin glasses was recently (after publication of the book) made in a new seminal paper by its author, this book remains indispensable as a basic introduction and reference on this extremely difficult and interesting subject.