Central limit theorem on CAT(0) spaces with contracting isometries

TL;DR

This paper proves a Central Limit Theorem for random walks on groups acting on CAT(0) spaces with contracting isometries, showing convergence to a Gaussian distribution under certain conditions.

Contribution

It establishes a CLT for random walks on CAT(0) spaces with contracting isometries, using hyperbolic models and demonstrating the prevalence of contracting actions.

Findings

Random walks satisfy a CLT with Gaussian limit.

Probability of contracting isometries approaches 1 over time.

Hyperbolic models facilitate analysis of CAT(0) space actions.

Abstract

Let $G$ be a group with a non-elementary action on a proper CAT(0) space $X$, and let $\mu$ be a measure on $G$ such that the random walk $(Z_n)_n$ generated by $\mu$ has finite second moment on $X$. Let $o$ be a basepoint in $X$, and assume that there exists a rank one isometry in $G$. We prove that in this context, $(Z_n o )_n$ satisfies a Central Limit Theorem, namely that the random variables $\frac{1}{\sqrt{n}}(d(Z_n o, o) - n \lambda) $ converge in law to a non-degenerate Gaussian distribution $N_\mu$, for $\lambda$ the (positive) drift of the random walk. The strategy relies on the use of hyperbolic models introduced by H. Petyt, A. Zalloum and D. Spriano, which are analogues of curve graphs and cubical hyperplanes for the class of CAT(0) spaces. As a side result, we prove that the probability that the nth-step $Z_n$ acts on $X$ as a contracting isometry goes to 1 as $n$ goes to infinity.