## The Annals of Probability

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

### Attractive Nearest Particle Systems

#### 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

**Permanent link to this document**

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**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

**Keywords**

Interacting particle systems ergodicity reversible Markov processes free energy

#### Citation

Liggett, Thomas M. Attractive Nearest Particle Systems. Ann. Probab. 11 (1983), no. 1, 16--33. doi:10.1214/aop/1176993656. https://projecteuclid.org/euclid.aop/1176993656