Ergodic theorems for the asymmetric simple exclusion process

Abstract:

Consider the infinite particle system on the integers with the simple exclusion interaction and one-particle motion determined by $p(x,x + 1) = p$ and $p(x,x - 1) = q$ for $x \in Z$, where $p + q = 1$ and $p > q$. If $\mu$ is the initial distribution of the system, let ${\mu _t}$ be the distribution at time t. The main results determine the limiting behavior of ${\mu _t}$ as $t \to \infty$ for simple choices of $\mu$. For example, it is shown that if $\mu$ is the pointmass on the configuration in which all sites to the left of the origin are occupied, while those to the right are vacant, then the system converges as $t \to \infty$ to the product measure on ${\{ 0,1\} ^Z}$ with density $1/2$. For the proof, an auxiliary process is introduced which is of interest in its own right. It is a process on the positive integers in which particles move according to the simple exclusion process, but with the additional feature that there can be creation and destruction of particles at one. Ergodic theorems are proved for this process also.