One-dimensional random Ising systems with interaction decay \(r^{- (1+\epsilon)}:\) A convergent cluster expansion
type: Item
instance of: scholarly article
title: One-dimensional random Ising systems with interaction decay \(r^{- (1+\epsilon)}:\) A convergent cluster expansion
zbMATH Open document ID: 0642.60103
DOI: 10.1007/BF01219074
author: E. Olivieri
author: Massimo Campanino
published in: Communications in Mathematical Physics
publication date: 1987-01-01T00:00:00Z
review text: We consider a one-dimensional random Ising model with Hamiltonian \[ H\quad =\sum_{i\neq j}\frac{J_{ij}}{| i-j|^{1+\epsilon}}s_ is_ j\quad +\quad h\sum_{i}s_ i, \] where \(\epsilon >0\) and \(J_{ij}\) are independent, identically distributed random variables with distribution dF(x) such that i) \(\int x dF(x)=0\), ii) \(\int e^{tx} dF(x)<\infty\forall t\in {\mathbb{R}}\). We construct a cluster expansion for the free energy and the Gibbs expectations of local observables. This expansion is convergent almost surely at every temperature. In this way we obtain that the free energy and the Gibbs expectations of local observables are \(C^{\infty}\) functions of the temperature and of the magnetic field h. Moreover we can estimate the decay of truncated correlation functions. In particular, for every \(\epsilon '>0\) there exists a random variable c(\(\omega)\), finite almost everywhere, such that \[ | <s_ 0s_ j>_ H-<s_ 0>_ H<s_ j>_ H| \quad \leq \frac{c(\omega)}{| j|^{1+\epsilon -\epsilon '}}, \] where \(< >_ H\) denotes the Gibbs average with respect to the Hamiltonian H.
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 4046867
zbMATH Keywords: one-dimensional random Ising model
zbMATH Keywords: cluster expansion for the free energy
zbMATH Keywords: Gibbs expectations
zbMATH Keywords: decay of truncated correlation functions
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01219074
OpenAlex ID: W2047979049
description: scientific article; zbMATH DE number 4046867