Analyticity of the entropy and the escape rate of random walks in hyperbolic groups

TL;DR

This paper proves that the entropy and escape rate of random walks on hyperbolic groups vary analytically with the probability measure, providing new proofs and insights into their statistical properties.

Contribution

It establishes the analyticity of entropy and escape rate in hyperbolic groups and introduces spectral methods for new proofs of the central limit theorem.

Findings

Entropy and escape rate depend analytically on the probability measure.

Spectral techniques yield a new proof of the central limit theorem.

Variance associated with the CLT is also analytic.

Abstract

We consider random walks on a non-elementary hyperbolic group endowed with a word distance. To a probability measure on the group are associated two numerical quantities, the rate of escape and the entropy. On the set of admissible probability measures whose support is contained in a given finite set, we show that both quantities depend in an analytic way on the probability measure. Our spectral techniques also give a new proof of the central limit theorem, and imply that the corresponding variance is analytic.