Regularity of the entropy for random walks on hyperbolic groups

TL;DR

This paper proves that for random walks on hyperbolic groups, the entropy and escape rate vary in a Lipschitz continuous manner with respect to the probability distribution, provided the support is fixed.

Contribution

It establishes the Lipschitz regularity of entropy and escape rate functions for finitely supported random walks on hyperbolic groups.

Findings

Entropy and escape rate are Lipschitz functions of the probability measure.

The results hold for nondegenerate, finitely supported random walks on hyperbolic groups.

Support constancy is essential for the Lipschitz property.

Abstract

We consider nondegenerate, finitely supported random walks on a finitely generated Gromov hyperbolic group. We show that the entropy and the escape rate are Lipschitz functions of the probability if the support remains constant.