The Annals of Probability

Attractive Nearest Particle Systems

Thomas M. Liggett

Abstract

We consider certain Markov processes with state space $\{0, 1\}^z$ which were introduced and first studied by Spitzer. In these systems, deaths occur at rate one independently of the configuration, and births occur at rate $\beta(\ell, r)$ where $\ell$ and $r$ are the distances to the nearest particles to the left and right, respectively. In his paper, Spitzer gave a necessary and sufficient condition for this process to have a reversible invariant measure, and showed that such a measure must be a stationary renewal process. It was that fact which motivated the study of these systems. Assuming that the process is attractive in the sense of Holley, we give conditions under which (a) the pointmass on the configuration "all zeros" is invariant, and (b) the reversible renewal process is the only nontrivial invariant measure which is translation invariant. As an application, these results allow us to determine exactly the values of $\lambda > 0$ and $p > 0$ for which the process with $\beta(\ell, r) = \lambda(1/\ell + 1/r)^p$ is ergodic.

Article information

Source
Ann. Probab., Volume 11, Number 1 (1983), 16-33.

Dates
First available in Project Euclid: 19 April 2007

https://projecteuclid.org/euclid.aop/1176993656

Digital Object Identifier
doi:10.1214/aop/1176993656

Mathematical Reviews number (MathSciNet)
MR682797

Zentralblatt MATH identifier
0508.60081

JSTOR