A special set of exceptional times for dynamical random walk on Z²

Benjamini, Haggstrom, Peres and Steif introduced the model of dynamical random walk on Z^d [2]. This is a continuum of random walks indexed by a parameter t. They proved that for d = 3,4 there almost surely exist t such that the random walk at time t visits the origin infinitely often, but for d . 5 there almost surely do not exist such t. Hoffman showed that for d ≥ 2 there almost surely exists t such that the random walk at time t visits the origin only finitely many times [5]. We refine the results of [5] for dynamical random walk on Z², showing that with probability one the are times when the origin is visited only a finite number of times while other points are visited infinitely often.