## The Annals of Probability

- Ann. Probab.
- Volume 2, Number 6 (1974), 989-998.

### Convergence to Total Occupancy in an Infinite Particle System with Interactions

#### Abstract

Let $p(x, y)$ be the transition function for an irreducible, positive recurrent, reversible Markov chain on the countable set $S$. Let $\eta_t$ be the infinite particle system on $S$ with the simple exclusion interaction and one-particle motion determined by $p$. The principal result is that there are no nontrivial invariant measures for $\eta_t$ which concentrate on infinite configurations of particles on $S$. Furthermore, it is proved that if the system begins with an arbitrary infinite configuration, then it converges in probability to the configuration in which all sites are occupied.

#### Article information

**Source**

Ann. Probab., Volume 2, Number 6 (1974), 989-998.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176996494

**Digital Object Identifier**

doi:10.1214/aop/1176996494

**Mathematical Reviews number (MathSciNet)**

MR362564

**Zentralblatt MATH identifier**

0295.60086

**JSTOR**

links.jstor.org

**Subjects**

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

Secondary: 47A35: Ergodic theory [See also 28Dxx, 37Axx]

**Keywords**

Infinite particle systems simple exclusion model ergodic theorems

#### Citation

Liggett, Thomas M. Convergence to Total Occupancy in an Infinite Particle System with Interactions. Ann. Probab. 2 (1974), no. 6, 989--998. doi:10.1214/aop/1176996494. https://projecteuclid.org/euclid.aop/1176996494