Differentiating the entropy of random walks on hyperbolic groups

TL;DR

This paper proves that the asymptotic entropy and rate of escape for symmetric random walks on hyperbolic groups are differentiable functions, with derivatives characterized as correlations, advancing understanding of their probabilistic behavior.

Contribution

It establishes the differentiability of entropy and escape rate in hyperbolic groups and identifies their derivatives as correlation functions, a novel theoretical insight.

Findings

Entropy and escape rate are differentiable functions.

Derivatives are identified as correlation functions.

Results apply to symmetric, bounded increment random walks.

Abstract

We show that the asymptotic entropy of a random walk on a nonelementary hyperbolic group, with symmetric and bounded increments, is differentiable and we identify its derivative as a correlation. We also prove similar results for the rate of escape.