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February, 1983 Attractive Nearest Particle Systems
Thomas M. Liggett
Ann. Probab. 11(1): 16-33 (February, 1983). DOI: 10.1214/aop/1176993656

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.

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Thomas M. Liggett. "Attractive Nearest Particle Systems." Ann. Probab. 11 (1) 16 - 33, February, 1983. https://doi.org/10.1214/aop/1176993656

Information

Published: February, 1983
First available in Project Euclid: 19 April 2007

zbMATH: 0508.60081
MathSciNet: MR682797
Digital Object Identifier: 10.1214/aop/1176993656

Subjects:
Primary: 60K35

Rights: Copyright © 1983 Institute of Mathematical Statistics

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Vol.11 • No. 1 • February, 1983
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