Many visits to a single site by a transient random walk in random environment

Abstract

We consider a transient random walk on Z in random environment, and study the almost sure asymptotics of the supremum of its local time. Our main result states that if the random walk has zero speed, there is a (random) sequence of sites and a (random) sequence of times such that the walk spends a positive fraction of the times at these sites. This was known for a recurrent random walk in random environment (Random Walk in Random and Non-Random Environments, World Scientific, Singapore, 1990; Stochastic Process. Appl. 76 (1998) 231). Our method of proof is different and relies on the connection of random walk in random environment with branching processes in random environment used in Kesten et al. (Compositio Math. 30 (1975) 145).