Infinite reversible nearest particle systems in inhomogeneous and random environments
type: Item
instance of: scholarly article
title: Infinite reversible nearest particle systems in inhomogeneous and random environments
zbMATH Open document ID: 0732.60115
DOI: 10.1016/0304-4149(91)90096-U
author: Xijian Liu
published in: Stochastic Processes and their Applications
publication date: 1991-01-01T00:00:00Z
review text: Let \(\eta\) : \({\mathbb{Z}}\to \{0,1\}\). Consider a particle system with state space \(\{\eta:\;\sum_{x<0}\eta (x)=\sum_{x>0}\eta (x)=\infty \}.\) The system evolves in the following way: a particle at site x dies out with rate one. If \(\eta (x)=0\), then a flip occurs at x with rate \(\beta\) (\(\ell,r)\) depending only on the nearest sites to the right and the left which have value one. A particular interesting case is that \[ \beta (\ell,r)=\lambda \beta (\ell)\beta (r)/\beta (\ell +r), \] where \(\lambda,\beta >0\), \(\sum_{n}n\beta (n)<\infty\) and \(\beta (n+1)/\beta (n)\uparrow 1.\) In this case, the Markov process is reversible and attractive. This model attracts a lot of attention since it exhibits phase transition with explicit critical value \(\lambda =1\). The author studies the system in a random environment. That is, \(\lambda\) is i.i.d. random variable on \({\mathbb{Z}}\). The main result of the paper is summarized as follows: Assume that \({\mathbb{E}} \lambda (0)\) and \({\mathbb{E}} \log \lambda (0)\) exist. Then the system survives almost surely if \({\mathbb{E}} \log \lambda (0)>0\); dies out almost surely if \({\mathbb{E}} \lambda (0)<1\). For the remainder case, both phenomena can be happened either the system survives or dies out, depending on \(\beta\) (n). It should be pointed out that altough the conclusion is the same as in the finite-dimension case studied by Chen and Liggett the techniques used in the paper are complete different.
Mathematics Subject Classification ID: 60K35
zbMATH DE Number: 4211246
zbMATH Keywords: particle system
zbMATH Keywords: reversible
zbMATH Keywords: random environment
reviewed by: Mu-Fa Chen
MaRDI profile type: Publication
cites work: Controlled spin-flip systems
cites work: Critical phenomena for Spitzer's reversible nearest particle systems
cites work: Subadditive ergodic theory
cites work: Attractive nearest particle systems
cites work: Q5185817
cites work: Random walks in a random environment
cites work: Stochastic time evolution of one dimensional infinite particle systems
description: scientific article; zbMATH DE number 4211246