Groups, Geometry, and Dynamics

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Volume 7, Issue 4, 2013, pp. 791–820
DOI: 10.4171/GGD/206

Published online: 2013-11-20

Spectral properties of a class of random walks on locally finite groups

Alexander Bendikov[1], Barbara Bobikau[2] and Christophe Pittet[3]

(1) Uniwersytet Wrocławski, Poland
(2) Uniwersytet Wrocławski, Poland
(3) Aix-Marseille Université, France

We study some spectral properties of random walks on infinite countable amenable groups with an emphasis on locally finite groups, e.g. the infinite symmetric group $S_{\infty}$. On locally finite groups, the random walks under consideration are driven by infinite divisible distributions. This allows us to embed our random walks into continuous time Lévy processes. We obtain examples of fast/slow decays of return probabilities, a recurrence criterion, exact values and estimates of isospectral profiles and spectral distributions.

Keywords: Random walk, locally finite group, ultra-metric space, infinite divisible distribution, Laplace transform, Köhlbecker transform, Legendre transform, return probability, spectral distribution, isospectral profile

Bendikov Alexander, Bobikau Barbara, Pittet Christophe: Spectral properties of a class of random walks on locally finite groups. Groups Geom. Dyn. 7 (2013), 791-820. doi: 10.4171/GGD/206