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

TL;DR

This paper proves that the geometric lengths of hyperbolic elements sampled by a random walk in a Fuchsian group follow classical probabilistic laws, using Gromov's theorem on hyperbolic groups.

Contribution

It provides a new proof of known distribution laws for geometric lengths in hyperbolic groups via Gromov's theorem, connecting geometric group theory with probabilistic results.

Findings

Hyperbolic elements dominate the sampled set as steps increase.

Geometric lengths satisfy LLN, CLT, LDP, and Local Limit Theorem.

Distribution laws hold for large random walks in Fuchsian groups.

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.