{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:X7UGV457M6LYHBOQNZKBQ3EGMN","short_pith_number":"pith:X7UGV457","canonical_record":{"source":{"id":"1708.02506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-08-08T14:39:53Z","cross_cats_sorted":[],"title_canon_sha256":"c29bcbdb8f940110a9d45f1d6f85f6f81c83d6c7c77e64c5300b05858ec62cb9","abstract_canon_sha256":"f1b8634153a86fe343534f48fd730173604010418a1c3a64f0b13497aea5292b"},"schema_version":"1.0"},"canonical_sha256":"bfe86af3bf67978385d06e54186c86634228ad49a0395f86f5fa08ab3b617dae","source":{"kind":"arxiv","id":"1708.02506","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.02506","created_at":"2026-05-18T00:38:24Z"},{"alias_kind":"arxiv_version","alias_value":"1708.02506v1","created_at":"2026-05-18T00:38:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.02506","created_at":"2026-05-18T00:38:24Z"},{"alias_kind":"pith_short_12","alias_value":"X7UGV457M6LY","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"X7UGV457M6LYHBOQ","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"X7UGV457","created_at":"2026-05-18T12:31:53Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:X7UGV457M6LYHBOQNZKBQ3EGMN","target":"record","payload":{"canonical_record":{"source":{"id":"1708.02506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-08-08T14:39:53Z","cross_cats_sorted":[],"title_canon_sha256":"c29bcbdb8f940110a9d45f1d6f85f6f81c83d6c7c77e64c5300b05858ec62cb9","abstract_canon_sha256":"f1b8634153a86fe343534f48fd730173604010418a1c3a64f0b13497aea5292b"},"schema_version":"1.0"},"canonical_sha256":"bfe86af3bf67978385d06e54186c86634228ad49a0395f86f5fa08ab3b617dae","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:38:24.063456Z","signature_b64":"w9AGzZkl79Es7GID8kYPoZmKjWzVh31I8T2D4nYmNQFN/TSJX9XNkmYfqEWkCil/2Hc0JTTJTPxZXIyO+GOFBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bfe86af3bf67978385d06e54186c86634228ad49a0395f86f5fa08ab3b617dae","last_reissued_at":"2026-05-18T00:38:24.062788Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:38:24.062788Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1708.02506","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:38:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4l1XLOiCSe932eZ1ov7Nhk4lFPnvwWvSykoMSuK8DI99M+k1WDDA7f5ms5cPOsmgMPnXWM8C4Wqf9ge3wXTZAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T00:23:22.006189Z"},"content_sha256":"c1f776e0223c3c83c97e5ef3228ff7203840e19e7cd9023ffa7d841bfa915abb","schema_version":"1.0","event_id":"sha256:c1f776e0223c3c83c97e5ef3228ff7203840e19e7cd9023ffa7d841bfa915abb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:X7UGV457M6LYHBOQNZKBQ3EGMN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Random walks in the hyperbolic plane and the question mark function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Gerard Letac, Mauro Piccioni","submitted_at":"2017-08-08T14:39:53Z","abstract_excerpt":"Consider $G=SL_2(\\mathbb{Z})/\\{\\pm I\\}$ acting on the complex upper half plane $H$ by $h_M(z)=\\frac{az+b}{cz+d},$ for $M \\in G$. Let $D=\\{z \\in H: |z|\\geq 1, |\\Re(z)|\\leq 1/2\\}$. We consider the set $\\mathcal{E} \\subset G$ with the $9$ elements $M$, different from the identity, such that $(MM^T)\\leq 3$. We equip the tiling of $H$ defined by $\\mathbb{D}=\\{h_M(D), M \\in G\\}$ with a graph structure where the neighbours are defined by $h_M(D) \\cap h_{M'}(D) \\neq \\emptyset$, equivalently $M^{-1}M' \\in \\mathcal{E}$.\n  The present paper studies several Markov chains related to the above structure. We"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02506","kind":"arxiv","version":1},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:38:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cLCdQj8OWZfmWw2hXhBG3+0eN3d1wNi4PkopxvOZ4GzoKmmFgUXBJ9fmgyhRvlqHyn+3JWxiX9ItfM3k/xZ9AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-08T00:23:22.006951Z"},"content_sha256":"9263214351ef038914eeff63cff30b6492e8fb8ca201f5257651a9f681b9d69f","schema_version":"1.0","event_id":"sha256:9263214351ef038914eeff63cff30b6492e8fb8ca201f5257651a9f681b9d69f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/X7UGV457M6LYHBOQNZKBQ3EGMN/bundle.json","state_url":"https://pith.science/pith/X7UGV457M6LYHBOQNZKBQ3EGMN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/X7UGV457M6LYHBOQNZKBQ3EGMN/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-08T00:23:22Z","links":{"resolver":"https://pith.science/pith/X7UGV457M6LYHBOQNZKBQ3EGMN","bundle":"https://pith.science/pith/X7UGV457M6LYHBOQNZKBQ3EGMN/bundle.json","state":"https://pith.science/pith/X7UGV457M6LYHBOQNZKBQ3EGMN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/X7UGV457M6LYHBOQNZKBQ3EGMN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:X7UGV457M6LYHBOQNZKBQ3EGMN","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":"f1b8634153a86fe343534f48fd730173604010418a1c3a64f0b13497aea5292b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-08-08T14:39:53Z","title_canon_sha256":"c29bcbdb8f940110a9d45f1d6f85f6f81c83d6c7c77e64c5300b05858ec62cb9"},"schema_version":"1.0","source":{"id":"1708.02506","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1708.02506","created_at":"2026-05-18T00:38:24Z"},{"alias_kind":"arxiv_version","alias_value":"1708.02506v1","created_at":"2026-05-18T00:38:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.02506","created_at":"2026-05-18T00:38:24Z"},{"alias_kind":"pith_short_12","alias_value":"X7UGV457M6LY","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_16","alias_value":"X7UGV457M6LYHBOQ","created_at":"2026-05-18T12:31:53Z"},{"alias_kind":"pith_short_8","alias_value":"X7UGV457","created_at":"2026-05-18T12:31:53Z"}],"graph_snapshots":[{"event_id":"sha256:9263214351ef038914eeff63cff30b6492e8fb8ca201f5257651a9f681b9d69f","target":"graph","created_at":"2026-05-18T00:38:24Z","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":"Consider $G=SL_2(\\mathbb{Z})/\\{\\pm I\\}$ acting on the complex upper half plane $H$ by $h_M(z)=\\frac{az+b}{cz+d},$ for $M \\in G$. Let $D=\\{z \\in H: |z|\\geq 1, |\\Re(z)|\\leq 1/2\\}$. We consider the set $\\mathcal{E} \\subset G$ with the $9$ elements $M$, different from the identity, such that $(MM^T)\\leq 3$. We equip the tiling of $H$ defined by $\\mathbb{D}=\\{h_M(D), M \\in G\\}$ with a graph structure where the neighbours are defined by $h_M(D) \\cap h_{M'}(D) \\neq \\emptyset$, equivalently $M^{-1}M' \\in \\mathcal{E}$.\n  The present paper studies several Markov chains related to the above structure. We","authors_text":"Gerard Letac, Mauro Piccioni","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-08-08T14:39:53Z","title":"Random walks in the hyperbolic plane and the question mark function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.02506","kind":"arxiv","version":1},"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:c1f776e0223c3c83c97e5ef3228ff7203840e19e7cd9023ffa7d841bfa915abb","target":"record","created_at":"2026-05-18T00:38:24Z","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":"f1b8634153a86fe343534f48fd730173604010418a1c3a64f0b13497aea5292b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-08-08T14:39:53Z","title_canon_sha256":"c29bcbdb8f940110a9d45f1d6f85f6f81c83d6c7c77e64c5300b05858ec62cb9"},"schema_version":"1.0","source":{"id":"1708.02506","kind":"arxiv","version":1}},"canonical_sha256":"bfe86af3bf67978385d06e54186c86634228ad49a0395f86f5fa08ab3b617dae","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bfe86af3bf67978385d06e54186c86634228ad49a0395f86f5fa08ab3b617dae","first_computed_at":"2026-05-18T00:38:24.062788Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:38:24.062788Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"w9AGzZkl79Es7GID8kYPoZmKjWzVh31I8T2D4nYmNQFN/TSJX9XNkmYfqEWkCil/2Hc0JTTJTPxZXIyO+GOFBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:38:24.063456Z","signed_message":"canonical_sha256_bytes"},"source_id":"1708.02506","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c1f776e0223c3c83c97e5ef3228ff7203840e19e7cd9023ffa7d841bfa915abb","sha256:9263214351ef038914eeff63cff30b6492e8fb8ca201f5257651a9f681b9d69f"],"state_sha256":"9aaf73ae07875602a27d143e4058c478d4c141895ebc0fc9f128c6fe21eda51d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"zQ0Jx/YkTRvcHx1UzwcsYIGpVZ0J2anEUUp2xGKftEcQdg6Z0iuP8OgET30iqmAK9mNK/bFVUSVTBuiqu5ENCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-08T00:23:22.011130Z","bundle_sha256":"0821d9892fb67a74eb4826639b45b87cf23f19041d1dda45a0b1d5687cc4d10c"}}