Asymptotic Renyi Entropies of Random Walks on Groups

TL;DR

This paper introduces asymptotic Rnyi entropies as a new family of invariants for random walks on groups, connecting growth, entropy, and spectral properties, with applications to large deviations.

Contribution

It defines and studies the properties of asymptotic Rnyi entropies, providing a unified framework linking various random walk invariants on groups.

Findings

Asymptotic Rnyi entropies interpolate between known invariants.

Basic properties hold for all groups, including positivity and analyticity in specific cases.

Applications include large deviation principles for random walks.

Abstract

We introduce asymptotic R\'enyi 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 R\'enyi entropies that apply to all groups, and discuss their analyticity and positivity for the free group and lamplighter groups.