{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:BG2HNFAGQOPIGK3KILDPET3RBW","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"65585c1e847026d12c6f202f6b16834f5814098be4d161bc83a5cd2d75bf1303","cross_cats_sorted":["math.DS","math.GR","math.NT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-07-10T17:54:03Z","title_canon_sha256":"9ff2f104beb932cfcbfddd2c2b9bf84b81b9714db0836d4f03b9a4a37cd1b997"},"schema_version":"1.0","source":{"id":"1807.03775","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.03775","created_at":"2026-05-17T23:54:28Z"},{"alias_kind":"arxiv_version","alias_value":"1807.03775v3","created_at":"2026-05-17T23:54:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.03775","created_at":"2026-05-17T23:54:28Z"},{"alias_kind":"pith_short_12","alias_value":"BG2HNFAGQOPI","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_16","alias_value":"BG2HNFAGQOPIGK3K","created_at":"2026-05-18T12:32:16Z"},{"alias_kind":"pith_short_8","alias_value":"BG2HNFAG","created_at":"2026-05-18T12:32:16Z"}],"graph_snapshots":[{"event_id":"sha256:0b565bf4ef31d3b29ed915db386fdeac0bc0861b92f4032ae910138a0cd473d2","target":"graph","created_at":"2026-05-17T23:54:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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","authors_text":"Peter S. Park","cross_cats":["math.DS","math.GR","math.NT","math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-07-10T17:54:03Z","title":"Probability laws for the distribution of geometric lengths when sampling by a random walk in a Fuchsian fundamental group"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03775","kind":"arxiv","version":3},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:cdf4515dd36c8902a3aea3211f48d9666bee26ef8d72a0b33a0faa84d521d938","target":"record","created_at":"2026-05-17T23:54:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"65585c1e847026d12c6f202f6b16834f5814098be4d161bc83a5cd2d75bf1303","cross_cats_sorted":["math.DS","math.GR","math.NT","math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GT","submitted_at":"2018-07-10T17:54:03Z","title_canon_sha256":"9ff2f104beb932cfcbfddd2c2b9bf84b81b9714db0836d4f03b9a4a37cd1b997"},"schema_version":"1.0","source":{"id":"1807.03775","kind":"arxiv","version":3}},"canonical_sha256":"09b4769406839e832b6a42c6f24f710d84ba2b72512d0cb7dda7c5e9e11c22cf","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"09b4769406839e832b6a42c6f24f710d84ba2b72512d0cb7dda7c5e9e11c22cf","first_computed_at":"2026-05-17T23:54:28.239910Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:54:28.239910Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tt0VuJ9JRzMMoxm9Ua+7sdK/5dAGNkO1T0N4CylZBhBkKTDjvC1W5+xr1lekxYTZwUUvZckkl0tPvf9OG4sEDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:54:28.240512Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.03775","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cdf4515dd36c8902a3aea3211f48d9666bee26ef8d72a0b33a0faa84d521d938","sha256:0b565bf4ef31d3b29ed915db386fdeac0bc0861b92f4032ae910138a0cd473d2"],"state_sha256":"6944aea289529ba2097e274ea299105adf70407fbe6569a0f7fd1de066815b04"}