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Abstract

In this article, we initiate the study of elephant random walks on finitely generated infinite groups whose Cayley graphs are homogeneous trees of degree $d \ge 3$ (e.g., groups of the form $\mathbb{Z}^{* d_1} * \mathbb{Z}_2^{*d_2}$ with $2d_1 + d_2 \ge 3$). 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. The rate of convergence to the limiting speed turns out to depend on the memory parameter $p$ and one encounters phase transitions akin to elephant random walks on $\mathbb{Z}^d$.