Large deviations for random walks on Gromov-hyperbolic spaces

TL;DR

This paper establishes large deviation principles for the distance and translation length of random walks on Gromov-hyperbolic spaces, connecting geometric group actions with probabilistic spectral radius behaviors.

Contribution

It proves large deviations results for random walks on Gromov-hyperbolic spaces under finite exponential moment conditions, advancing understanding of geometric and spectral properties.

Findings

Large deviations for distance and translation length established

Results imply a special case of a spectral radius conjecture

Connects probabilistic behavior with geometric group theory

Abstract

Let $\Gamma$ be a countable group acting on a geodesic Gromov-hyperbolic metric space $X$ and $\mu$ a probability measure on $\Gamma$ whose support generates a non-elementary subsemigroup. Under the assumption that $\mu$ has a finite exponential moment, we establish large deviations results for the distance and the translation length of a random walk with driving measure $\mu$. From our results, we deduce a special case of a conjecture regarding large deviations of spectral radii of random matrix products.