Speed of excited random walks with long backward steps

Abstract

We study a model of multi-excited random walk with non-nearest neighbour steps on $\mathbb Z$, in which the walk can jump from a vertex $x$ to either $x+1$ or $x-i$ with $i\in \{1,2,\dots,L\}$, $L\ge 1$. We first point out the multi-type branching structure of this random walk and then prove a limit theorem for a related multi-type Galton-Watson process with emigration, which is of independent interest. Combining this result and the method introduced by Basdevant and Singh [Probab. Theory Related Fields (2008), 141 (3-4)], we extend their result (w.r.t the case $L=1$) to our model. More specifically, we show that in the regime of transience to the right, the walk has positive speed if and only if the expected total drift $\delta>2$. This confirms a special case of a conjecture proposed by Davis and Peterson.