2025/09/03 by Soumendu Sundar Mukherjee, Mukherjee, Soumendu Sundar · 1 citation
Mathematics · #60G42 #60G50 #60K99 #82C41 #FOS: Mathematics #FOS: Physical sciences #Geometric and Algebraic Topology #Limits and Structures in Graph Theory #Markov Chains and Monte Carlo Methods #Mathematical Physics (math-ph) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.2509.03048
openalex publication_date 2025/09/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We introduce a generalisation of Schütz 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 ≥ 3. We show that the asymptotic speed of the walk does not depend on the memory parameter p ∈ [0, 1) and equals (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 pd = (d + 1)/(2d). Numerical experiments suggest that these upper bounds are tight. Along the way, we also obtain estimates on the return probability.