Recurrence of Simple Random Walk on $Z^2$ is Dynamically Sensitive

Details Recurrence of Simple Random Walk on $Z^2$ is Dynamically Sensitive Recurrence of Simple Random Walk on $Z^2$ is Dynamically Sensitive

2005-03-03

arxiv.org/abs/math/0503065v1

Mathematics

Probability

Scientific paper

Benjamini, Haggstrom, Peres and Steif introduced the concept of a dynamical random walk. This is a continuous family of random walks, {S_n(t)}. Benjamini et. al. proved that if d=3 or d=4 then there is an exceptional set of t such that {S_n(t)} returns to the origin infinitely often. In this paper we consider a dynamical random walk on Z^2. We show that with probability one there exists t such that {S_n(t)} never returns to the origin. This exceptional set of times has dimension one. This proves a conjecture of Benjamini et. al.

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