## The Annals of Probability

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

Thomas M. Liggett

#### 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

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