Two Critical Exponents for Finite Reversible Nearest Particle Systems

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.