The Sherrington-Kirkpatrick model: A challenge for mathematicians
type: Item
instance of: scholarly article
title: The Sherrington-Kirkpatrick model: A challenge for mathematicians
zbMATH Open document ID: 0909.60083
author: Michel Talagrand
publication date: 1999-03-21T00:00:00Z
review text: This is the first in a series of papers by the author in which he attacks a variety of models for disordered systems from statistical mechanics and/or combinatorial optimisation by rigourous probabilistic methods. Such models are `solved' in the physics literature by powerful methods, before all the so-called ``replica trick'', but no mathematical justification of these techniques and their results has so far been found. Rigorous results on these systems are sparse [see the collection of review papers in \textit{A. Bovier, P. Picco} (eds.), ``Mathematical aspects of spin glasses and neural networks'' (Progress in Probability 41, Boston, 1998)[, and the present paper contains some remarkable new contributions in this directions. The model considered here is the most classical model of a mean field spin glass, the Sherrington-Kirkpatrick model. This model is defined through the Hamiltonian function \(H_N\) on the set \(\{-1\}^N\) given by \[ H_N(\sigma)=-\sum_{1\leq i<j\leq N} J_{ij}\sigma_i\sigma_j +h\sum_{1\leq i\leq N} \sigma_i, \] where \(J_{ij}\) are a family of independent standard normal variables. An object of particular concern is the partition function \[ Z_N(\overline b,h)\equiv \sum_{\sigma\in\{-1\}^N}\exp(-\beta H_N(\sigma)). \] The parameters \(\overline b\) and \(h\) are called the inverse temperature and the magnetic field, respectively. While completely understood on the heuristic level [see the book by \textit{M. Mézar, G. Parisi} and \textit{M. A. Virasoro}, ``Spin glass theory and beyond'' (Singapore, 1987)] prior rigorous results were only available for the case \(h=0\) and \(\overline b\leq 1\), where \textit{M. Aizenman, J. L. Lebowitz} and \textit{D. Ruelle} [Commun. Math. Phys. 112, 3-20 (1987)] proved that \[ \lim_{N\uparrow\infty} -\tfrac 1 N\ln Z_N(\overline b,0) = \overline b^2/2-\ln 2 \] and that \(Z_N(\overline b,0)/E Z_N(\overline b,0)\) converges to a log-Gaussian random variable. In the present paper a number of additional results is …
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B44
zbMATH DE Number: 1129463
zbMATH Keywords: spin glass
zbMATH Keywords: Sherrington-Kirkpatrick model
zbMATH Keywords: cavity method
zbMATH Keywords: replica symmetric solution
reviewed by: Anton Bovier
MaRDI profile type: Publication
full work available at URL: s004400050147
OpenAlex ID: W1976736848
DOI: 10.1007/S004400050147
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1129463