Probability laws for the distribution of geometric lengths when sampling by a random walk in a Fuchsian fundamental group

Abstract

Let $S=\Gamma\backslash \mathbb{H}$ be a hyperbolic surface of finite topological type, such that the Fuchsian group $\Gamma \le \operatorname{PSL}_2(\mathbb{R})$ is non-elementary, and consider any generating set $\mathfrak S$ of $\Gamma$. When sampling by an $n$-step random walk in $\pi_1(S) \cong \Gamma$ with each step given by an element in $\mathfrak S$, the subset of this sampled set comprised of hyperbolic elements approaches full measure as $n\to \infty$, and for this subset, the distribution of geometric lengths obeys a Law of Large Numbers, Central Limit Theorem, Large Deviations Principle, and Local Limit Theorem. We give a proof of this known theorem using Gromov's theorem on translation lengths of Gromov-hyperbolic groups.