Exponential \(L_ 2\) convergence of attractive reversible nearest particle systems
type: Item
instance of: scholarly article
title: Exponential \(L_ 2\) convergence of attractive reversible nearest particle systems
zbMATH Open document ID: 0679.60093
author: Thomas M. Liggett
published in: The Annals of Probability
publication date: 1989-01-01T00:00:00Z
review text: Nearest particle systems on \({\mathbb{Z}}\) which are attractive and reversible are studied. The main result shows that the convergence of the system to the equilibrium \(\nu\) in \(L_ 2(\nu)\) is exponential (even if phase transition occurs), and the upper and lower bounds for the exponent are given (up to the critical value of parameter). The method used for the bound which proves the exponential convergence is a comparison of the corresponding exponent to the exponent for a Markov chain on \(({\mathbb{Z}}^+)^{{\mathbb{Z}}}\) which is the family of independent birth and death chains in \({\mathbb{Z}}^+\). The comparison is based on a representation of the invariant measure \(\nu\), which is the distribution of a renewal process, as an image of a distribution of a sequence of independent identically distributed random variables in \({\mathbb{Z}}^+\).
Mathematics Subject Classification ID: 60K35
Mathematics Subject Classification ID: 60K05
Mathematics Subject Classification ID: 60J27
zbMATH DE Number: 4113710
zbMATH Keywords: Nearest particle systems
zbMATH Keywords: exponential convergence
zbMATH Keywords: birth and death chains
reviewed by: Petr Holický
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1214/aop/1176991408
OpenAlex ID: W2018430793
DOI: 10.1214/AOP/1176991408
description: scientific article; zbMATH DE number 4113710