Recurrence of Simple Random Walk on Z² is Dynamically Sensitive

الأصل · EN

Benjamini, Haggstrom, Peres and Steif introduced the concept of a dynamical random walk. This is a continuous family of random walks, Sₙ(t). Benjamini et. al. proved that if d=3 or d=4 then there is an exceptional set of t such that Sₙ(t) returns to the origin infinitely often. In this paper we consider a dynamical random walk on Z². We show that with probability one there exists t such that Sₙ(t) never returns to the origin. This exceptional set of times has dimension one. This proves a conjecture of Benjamini et. al.