Moderate deviations for the overlap parameter in the Hopfield model
type: Item
instance of: scholarly article
title: Moderate deviations for the overlap parameter in the Hopfield model
zbMATH Open document ID: 1062.60021
author: Peter Eichelsbacher
author: M. Löwe
publication date: 2005-02-22T00:00:00Z
review text: The authors consider the following model. Given \(N, M\in \mathbb{N}\), the model Hamiltonian is set to be \[ H_N (\sigma) = - \frac{1}{2N}\sum_{i, j = 1}^N J_{ij}( M) \sigma_i \sigma_j, \quad J_{ij} (M) = \sum_{\mu=1}^M \xi_i^{\mu} \xi_j^\mu , \] where all \(\sigma_i\), \(\xi^\mu_i\), \(i = 1, \dots , N\), \(\mu = 1, \dots , M\), are random variables taking values \(\pm 1\). The \(\xi\)'s are symmetric i.i.d.; the a priori distribution of the \(\sigma\)'s is the same as the one for the \(\xi\)'s. The Hopfield model at temperature \(\beta^{-1} >0\) is identified with the Gibbs measure \[ \varrho_{N, \beta} (\sigma) = 2^{-N} \exp ( - \beta H_N (\sigma)] /Z_{N, \beta}. \] The vectors \(m_N (\sigma) = ( m_N^\mu (\sigma))_{\mu = 1}^N\), with \(m_N^\mu (\sigma) = \frac{1}{N} \sum_{i = 1}^N \xi_i^\mu \sigma_i, \) are called overlaps. Then a number of statements describing thermodynamic limits, as \(N \rightarrow + \infty\), \(\alpha (N) = M/N \rightarrow 0\), of the distributions of the re-scaled overlaps \(N^\gamma m_N (\sigma)\), \(\gamma >0\), obtained from the measures \(\varrho_{N, \beta}\), are proven. The paper may be viewed as a continuation of the research of the same authors [Markov Process. Relat. Fields 10, No. 2, 345--366 (2004; Zbl 1059.60031)].
Mathematics Subject Classification ID: 60F10
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82C32
Mathematics Subject Classification ID: 82D30
zbMATH DE Number: 2137174
zbMATH Keywords: large deviations
zbMATH Keywords: moderate deviations
zbMATH Keywords: neural network
zbMATH Keywords: critical temperature
zbMATH Keywords: disorder
reviewed by: Yuri Kozitsky
MaRDI profile type: Publication
full work available at URL: s00440-004-0349-8
OpenAlex ID: W2073504893
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DOI: 10.1007/S00440-004-0349-8
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 2137174