{"paper":{"title":"Probability laws for the distribution of geometric lengths when sampling by a random walk in a Fuchsian fundamental group","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS","math.GR","math.NT","math.PR"],"primary_cat":"math.GT","authors_text":"Peter S. Park","submitted_at":"2018-07-10T17:54:03Z","abstract_excerpt":"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 Pri"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03775","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}