Mixing Properties for Random Walk in Random Scenery

Abstract

Consider the lattice $Z^d, d \geq 1$, together with a stochastic black-white coloring of its points and on it a random walk that is independent of the coloring. A local scenery perceived at a given time is a pattern of colors seen by the walker in a finite box around his current position. Under weak assumptions on the probability distributions governing walk and coloring, we prove asymptotic independence of local sceneries perceived at times 0 and $n$, in the limit as $n\rightarrow\infty$, and at times 0 and $T_k$, in the limit as $k \rightarrow \infty$, where $T_k$ is the random $k$th hitting time of a black point. An immediate corollary of the latter result is the convergence in distribution of the interarrival times between successive black hits, i.e., of $T_{k+1} - T_k$ as $k\rightarrow\infty$. The limit distribution is expressed in terms of the distribution of the first hitting time $T_1$. The proof uses coupling arguments and ergodic theory.