Enlargement of Obstacles for the Simple Random Walk

Abstract

We consider a continuous time simple random walk moving among obstacles, which are sites (resp., bonds) of the lattice $Z^d$. We derive in this context a version of the technique of enlargement of obstacles developed by Sznitman in the Brownian case. This method gives controls on exponential moments of certain death times as well as good lower bounds for certain principal eigenvalues. We give an application to recover an asymptotic result of Donsker and Varadhan on the number of sites visited by the random walk and another application to the number of bonds visited by the random walk.