The Annals of Probability

Two Critical Exponents for Finite Reversible Nearest Particle Systems

Thomas M. Liggett

Abstract

Finite nearest particle systems are certain one parameter families of continuous time Markov chains $A_t$ whose state space is the collection of all finite subsets of the integers. Points are added to or taken away from $A_t$ at rates which have a particular form. The empty set is absorbing for these chains. In the reversible case, the parameter $\lambda$ is normalized so that extinction at the empty set is certain if and only if $\lambda \leq 1$. Let $\sigma(\lambda)$ be the probability of nonextinction starting from a singleton. In a recent paper, Griffeath and Liggett obtained the bounds $\lambda^{-1}(\lambda - 1) \leq \sigma(\lambda) \leq |\log \lambda^{-1}(\lambda -1)|^{-1}$ for $\lambda > 1$, and raised the question of determining the correct asymptotics of $\sigma(\lambda)$ as $\lambda \downarrow 1$. In the present paper, this question is largely answered by showing under a moment assumption that for $\lambda > 1, \sigma(\lambda)$ is bounded above by a constant multiple of $\lambda - 1$. In the critical case $\lambda = 1$, a similar improvement is made on the known bounds on the asymptotics as $n \rightarrow \infty$ of the probability that $A_t$ is of cardinality at least $n$ sometime before extinction. Similar results have been conjectured, but remain open problems in nonreversible situations--for example, for the basic one-dimensional contact process.

Article information

Source
Ann. Probab., Volume 11, Number 3 (1983), 714-725.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993516

Digital Object Identifier
doi:10.1214/aop/1176993516

Mathematical Reviews number (MathSciNet)
MR704558

Zentralblatt MATH identifier
0527.60093

JSTOR