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

Abstract: 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. 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 L\'evy processes whose heat kernels have shapes similar to the ones of alpha-stable processes. We obtain examples of fast/slow decays of return probabilities, a recurrence criterion, exact values and estimates of isospectral profiles and spectral distributions, formulae and estimates for the escape rates and for heat kernels.