The Legendre transform of two replicas of the Sherrington-Kirkpatrick spin glass model
type: Item
instance of: scholarly article
title: The Legendre transform of two replicas of the Sherrington-Kirkpatrick spin glass model
zbMATH Open document ID: 0847.60080
DOI: 10.1007/BF01192213
author: Olivier Catoni
publication date: 1996-09-24T00:00:00Z
review text: We prove a variational inequality linking the values of the free energy per site at different temperatures. This inequality is based on the Legendre transform of the free energy of two replicas of the system. We prove that equality holds when \(\beta \leq 1/ \sqrt 2\) and fails when \(1/ \sqrt 2 < \beta \leq 1\). We deduce from this that the mean entropy per site of the uniform distribution with respect to the distribution of the coupling \(\sigma^1_i \sigma^2_i = \psi_i\) between two replicas is null when \(0 \leq \beta \leq 1/ \sqrt 2\) and strictly positive when \(1/ \sqrt 2 < \beta \leq 1\). We exhibit thus a new secondary critical phenomenon within the high temperature region \(0 \leq \beta \leq 1\). We give an interpretation of this phenomenon showing that the fluctuations of the law of the coupling with the interactions remain strong in the thermodynamic limit when \(\beta > 1/ \sqrt 2\). We also use our inequality numerically within the low temperature region to improve (slightly) the best previously known lower bounds for the free energy and the ground state energy per site.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B44
zbMATH DE Number: 902419
zbMATH Keywords: Sherrington-Kirkpatrick model
zbMATH Keywords: variational inequality
zbMATH Keywords: mean entropy
zbMATH Keywords: thermodynamic limit
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01192213
OpenAlex ID: W1552622233
published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 902419