On D_infty the elephant random walk's signed location follows simple random walk on Z to leading orders, with memory neutralized by the involutive generators and appearing only in an explicit functional correction from the Z case.
Functional limit theorems for the multi-dimensional elephant random walk
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Elephant random walks on d-regular infinite trees have asymptotic speed (d-2)/d independent of memory parameter p, with p-dependent upper bounds on convergence rate that exhibit a phase transition at p_d = (d+1)/(2d).
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Elephant random walk on the infinite dihedral group $\mathbb{Z}_2 * \mathbb{Z}_2$
On D_infty the elephant random walk's signed location follows simple random walk on Z to leading orders, with memory neutralized by the involutive generators and appearing only in an explicit functional correction from the Z case.
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Elephant random walks on infinite Cayley trees
Elephant random walks on d-regular infinite trees have asymptotic speed (d-2)/d independent of memory parameter p, with p-dependent upper bounds on convergence rate that exhibit a phase transition at p_d = (d+1)/(2d).