The Sherrington-Kirkpatrick Model

verfasst von: Dmitry Panchenko

Verlag: Springer New York

Buchreihe : Springer Monographs in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

The celebrated Parisi solution of the Sherrington-Kirkpatrick model for spin glasses is one of the most important achievements in the field of disordered systems. Over the last three decades, through the efforts of theoretical physicists and mathematicians, the essential aspects of the Parisi solution were clarified and proved mathematically. The core ideas of the theory that emerged are the subject of this book, including the recent solution of the Parisi ultrametricity conjecture and a conceptually simple proof of the Parisi formula for the free energy. The treatment is self-contained and should be accessible to graduate students with a background in probability theory, with no prior knowledge of spin glasses. The methods involved in the analysis of the Sherrington-Kirkpatrick model also serve as a good illustration of such classical topics in probability as the Gaussian interpolation and concentration of measure, Poisson processes, and representation results for exchangeable arrays.

Inhaltsverzeichnis

Frontmatter

Chapter 1. The Free Energy and Gibbs Measure

Abstract

In Sect.1.1, we will introduce the Sherrington–Kirkpatrick model and a family of closely related mixed p-spin models and give some motivation for the problem of computing the free energy in these models. A solution of this problem in Chap.?? will be based on a description of the structure of the Gibbs measure in the thermodynamic limit and in this chapter we will outline several connections between the free energy and Gibbs measure. At the same time, we will introduce various ideas and techniques, such as the Gaussian integration by parts, Gaussian interpolation, and Gaussian concentration, that will play essential roles in the key results of this chapter and throughout the book. In the last section, we will prove the Dovbysh–Sudakov representation for Gram-de Finetti arrays, which will allow us to define a certain analogue of the Gibbs measure in the thermodynamic limit. As a first step, we will prove the Aldous–Hoover representation for exchangeable and weakly exchangeable arrays. In Sect.1.4, we will give a classic probabilistic proof of this result for weakly exchangeable arrays and, for a change, in the Appendix we will prove the representation for exchangeable arrays using a different approach, based on more recent ideas of Lovász and Szegedy in the framework of limits of dense graph sequences. We will describe another application of the Aldous–Hoover representations for exchangeable arrays in Chap.??.

Dmitry Panchenko

Chapter 2. The Ruelle Probability Cascades

Abstract

In this chapter we will describe a remarkable family of random measures on a Hilbert space, called the Ruelle probability cascades, that play a central role in the Sherrington–Kirkpatrick model, and the first three sections will be devoted to the construction of these measures and study of their properties. We will see that they satisfy certain special invariance properties, one of which, called the Ghirlanda–Guerra identities, will serve as a key link between the Ruelle probability cascades and the Gibbs measure in the Sherrington–Kirkpatrick model. This connection will be explained in the last two sections, where it will be shown that, in a certain sense, the Ghirlanda–Guerra identities completely determine a random measure up to a functional order parameter. We will see in the next chapter that, as a consequence, the asymptotic Gibbs measures in the Sherrington–Kirkpatrick and mixed p-spin models can be approximated by the Ruelle probability cascades.

Dmitry Panchenko

Chapter 3. The Parisi Formula

Abstract

The main goal of this chapter is to prove the celebrated Parisi formula for the free energy in the mixed p-spin models. The Ruelle probability cascades studied in the previous chapter will be used in a number of different ways, but their most important role will be as an approximation of the asymptotic Gibbs measures that generate the overlap matrix in the thermodynamic limit. As we explained in Sect.2.4, a link between the Gibbs measure and the Ruelle probability cascades can be established using the Ghirlanda–Guerra identities and in this chapter we will show how these identities arise in the setting of the mixed p-spin models. In addition, the proof of the Parisi formula will be based on several other essential ideas, such as the Talagrand positivity principle, the Guerra replica symmetry breaking interpolation, and the Aizenman–Sims–Starr scheme.

Dmitry Panchenko

Chapter 4. Toward a Generalized Parisi Ansatz

Abstract

In the analysis of the Sherrington–Kirkpatrick and mixed p-spin models, a key role is played by the fact that the Hamiltonian of these models is a Gaussian process with the covariance given by a function of the overlap of spin configurations in {−1,+1}^N.The distribution of such processes is invariant under orthogonal transformations and, as a result, the computation of the free energy can be reduced to the description of the asymptotic distributions of the overlaps, which, in some sense, encode the Gibbs measure up to orthogonal transformations. However, for other random Hamiltonians on {−1,+1}^N, understanding the distribution of the overlaps is not sufficient and one would like to study the asymptotic distributions of all coordinates, or spins, of the configurations sampled from the Gibbs measure. In certain models, it is expected that the structure of these asymptotic distributions can be described by some particular realizations of the Ruelle probability cascades on a separable Hilbert space, but, in most cases, these predictions remain an open problem. In this chapter, we will describe an approach that, in some sense, proves these predictions in the setting of the mixed p-spin models. Unfortunately, again, the special Gaussian nature of the Hamiltonian will play a crucial role but at least, we will obtain new information beyond the distribution of the overlaps.

Dmitry Panchenko

Backmatter

Titel: The Sherrington-Kirkpatrick Model
verfasst von: Dmitry Panchenko
Verlag: Springer New York
Electronic ISBN: 978-1-4614-6289-7
Print ISBN: 978-1-4614-6288-0
DOI: https://doi.org/10.1007/978-1-4614-6289-7