Martingale defocusing and transience of a self-interacting random walk

Suppose that (X;Y;Z) is a random walk in Z 3 that moves in the following way: on the rst visit to a vertex only Z changes by 1 equally likely, while on later visits to the same vertex (X;Y ) performs a two-dimensional random walk step. We show that this walk is transient thus answering a question of Benjamini, Kozma and Schapira. One important ingredient of the proof is a dispersion result for martingales.