Abstract

The balanced excited random walk, introduced by Benjamini, Kozma and Schapira in 2011, is defined as a discrete time stochastic process in Zd, depending on two integer parameters 1<=d1, d2 <=d, which whenever it is at a site x " Zd at time n, it jumps to x ± ei with uniform probability, where e1, . . . , ed are the canonical vectors, for 1<= i <=d1, if the site x was visited for the first time at time n, while it jumps to x ± ei with uniform probability, for 1 + d − d2 <= i !<=d, if the site x was already visited before time n. Herewe give an overview of this model when d1+d2 = d and introduce and study the cases when d1+d2 > d. In particular, we prove that for all the cases d >=5 and most cases d = 4, the balanced excited random walk is transient.