Asymptotic Rényi entropies of random walks on groups

Abstract

We introduce asymptotic Rényi entropies as a parameterized family of invariants for random walks on groups. These invariants interpolate between various well-studied properties of the random walk, including the growth rate of the group, the Shannon entropy, and the spectral radius. They furthermore offer large deviation counterparts of the Shannon-McMillan-Breiman Theorem. We prove some basic properties of asymptotic Rényi entropies that apply to all groups, and discuss their analyticity and positivity for the free group and lamplighter groups.