Long Range Exclusion Processes

Abstract

Let $S$ be a countable set and $p(x, y)$ be the transition probabilities for a discrete time Markov chain on $S$. Consider the motion of particles on $S$ which obey the following rules: (a) there is always at most one particle at each site in $S$, (b) particles wait independent exponential times with mean one before moving, and (c) when a particle at $x$ is to move, it moves to $X_\tau$, where $\{X_n\}$ is the Markov chain starting at $x$ with transition probabilities $p(x, y)$ and $\tau$ is the first time that $X_n = x$ or $X_n$ is an unoccupied site. This process was introduced by Spitzer, and will be called a long range exclusion process because particles may travel long distances in short times. The process is well defined for finite configurations, and we will show how to use monotonicity arguments to define it for arbitrary configurations. It is shown that the configuration in which all sites are occupied may or may not be absorbing for the process. It always is if $p(x, y)$ is translation invariant on $S = Z^d$, but if $p(x, y)$ is a birth and death chain on $S = \{0,1,2,\cdots\}$, it is absorbing if and only if $p(x, y)$ is recurrent. For each positive function $\pi(x)$ on $S$ such that $\pi P = \pi$, there is a product measure $\nu_\pi$ on $\{0, 1\}^S$ which is a natural candidate for an invariant measure for the process. When $p(x, y)$ is translation invariant on $Z^d$, it is probably the case that $\nu_\pi$ is in fact invariant if and only if $\pi$ is constant. This will be verified under a mild regularity assumption, which is automatically satisfied if $d = 1$ or 2 or if the Laplace transform of $p(o, x)$ is finite in a neighborhood of the origin.