Speed of excited random walks with long backward steps

TL;DR

This paper investigates the speed of multi-excited random walks with long backward jumps on the integer lattice, establishing conditions for positive speed based on the expected total drift, and extends previous results to more general jump lengths.

Contribution

It introduces a multi-type branching structure for the walk, proves a limit theorem for a related Galton-Watson process, and extends existing results to models with longer backward jumps.

Findings

Positive speed occurs if and only if expected total drift > 2

Extended previous results to jumps of length greater than 1

Confirmed a special case of a conjecture by Davis and Peterson

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.