Hamiltonians on random walk trajectories
type: Item
instance of: scholarly article
title: Hamiltonians on random walk trajectories
zbMATH Open document ID: 0927.60081
DOI: 10.1016/S0304-4149(98)00052-0
author: Pablo A. Ferrari
author: Servet Martínez
published in: Stochastic Processes and their Applications
publication date: 1999-11-18T00:00:00Z
review text: The Gibbs states on the set of those elements \(w\in \mathbb Z_+^{\mathbb Z}\) are studied for which \(| w_i-w_{i+1}| =1\). The considered Hamiltonians are characterized by pair interaction potentials \(\Phi ^\alpha_{\{i,i+1\}}(w)=\alpha (w_{i+1})\) and \(\Phi ^{b,c}_{\{i,i+1\}}(w)= b(w_i)(w_{i+1}-w_i)^++c(w_{i+1})(w_{i+1}-w_i)^-\) for fixed sequences \(\alpha , b, c\) on \(\mathbb Z_+\). The main results give necessary and sufficient conditions on the sequences \(\alpha \), or \(b\) and \(c\), respectively, such that there is a translation invariant Gibbs state corresponding to the interactions \(\Phi ^\alpha \) or \(\Phi ^{b,c}\) provided that \(\alpha , b, c\) are ultimately constant sequences. The key tool is an eigenvalue problem for special infinite matrices corresponding to sequences \(b\) and \(c\). A comparison with symmetric models on \(\mathbb Z^{\mathbb Z}\) goes to an explanation of the phenomenon of entropic repulsion for the studied ``solid on solid models''. The relation to stationary states of an infinite particle system is also discussed.
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 82B41
Mathematics Subject Classification ID: 82C22
zbMATH DE Number: 1364514
zbMATH Keywords: interface
zbMATH Keywords: solid on solid models
zbMATH Keywords: entropic repulsion
zbMATH Keywords: random walk
reviewed by: Petr Holický
MaRDI profile type: Publication
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full work available at URL: s0304-4149(98)00052-0
OpenAlex ID: W2025564155
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description: scientific article; zbMATH DE number 1364514