No abstract

The balanced excited random walk, introduced by Benjamini, Kozma and Schapira in 2011, is defined as a discrete time stochastic process in Z d , depending on two integer parameters 1 ≤ d 1 , d 2 ≤ d with d 1 , d 2 ∈ {1, . . . , d}, which whenever it is at a site x ∈ Z d at time n, it jumps to x ± e i , where e 1 , . . . , e d are the canonical vectors, for 1 ≤ i ≤ d 1 , if the site x was visited for the first time at time n, while it jumps to x ± e i , for d − d 2 ≤ d, if the site x was already visited before time n. Here we give an overview of this model when d 1 + d 2 = d and introduce and study the cases when d 2 + d 2 > d. In particular, we prove that for all the cases d ≥ 5 and most cases d = 4, the balanced excited random walk is transient.