Elephant random walks on infinite Cayley trees

Abstract

We introduce a generalisation of Sch\"{u}tz and Trimper's elephant random walk to finitely generated groups. We focus on the simplest non-abelian setting, i.e. groups whose Cayley graphs are homogeneous trees of degree $d \ge 3$. We show that the asymptotic speed of the walk does not depend on the memory parameter $p \in [0, 1)$ and equals $\frac{d - 2}{d}$, the asymptotic speed of simple random walk on these graphs. We also establish upper bounds on the rate of convergence to the limiting speed. These upper bounds depend on $p$ and exhibit a phase transition at the critical value $p_d = \frac{d + 1}{2d}$. Numerical experiments suggest that these upper bounds are tight. Along the way, we also obtain estimates on the return probability.