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Let $S=\Gamma\backslash \mathbb{H}$ be a hyperbolic surface of finitetopological type, such that the Fuchsian group $\Gamma \le\operatorname{PSL}_2(\mathbb{R})$ is non-elementary, and consider anygenerating set $\mathfrak S$ of $\Gamma$. When sampling by an $n$-step randomwalk 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 elementsapproaches full measure as $n\to \infty$, and for this subset, the distributionof geometric lengths obeys a Law of Large Numbers, Central Limit Theorem, LargeDeviations Principle, and Local Limit Theorem. We give a proof of this knowntheorem using Gromov's theorem on translation lengths of Gromov-hyperbolicgroups.

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