{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:6V5ZLRBFVDTS36P5ZWVAKEAULY","short_pith_number":"pith:6V5ZLRBF","canonical_record":{"source":{"id":"0903.4697","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-27T17:23:31Z","cross_cats_sorted":[],"title_canon_sha256":"bf279ed44c8729855ebc03ea3354324658b4019f133784d77763938ed8a37d9c","abstract_canon_sha256":"0dd67f0a045cdf8e47a4a1bc810fa44ef4c1a293cc337724ff914b2e40ba5a74"},"schema_version":"1.0"},"canonical_sha256":"f57b95c425a8e72df9fdcdaa0510145e3e5a46e5f3520fc4489bd9f699b539f8","source":{"kind":"arxiv","id":"0903.4697","version":5},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.4697","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"arxiv_version","alias_value":"0903.4697v5","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.4697","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"pith_short_12","alias_value":"6V5ZLRBFVDTS","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"6V5ZLRBFVDTS36P5","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"6V5ZLRBF","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:6V5ZLRBFVDTS36P5ZWVAKEAULY","target":"record","payload":{"canonical_record":{"source":{"id":"0903.4697","kind":"arxiv","version":5},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-27T17:23:31Z","cross_cats_sorted":[],"title_canon_sha256":"bf279ed44c8729855ebc03ea3354324658b4019f133784d77763938ed8a37d9c","abstract_canon_sha256":"0dd67f0a045cdf8e47a4a1bc810fa44ef4c1a293cc337724ff914b2e40ba5a74"},"schema_version":"1.0"},"canonical_sha256":"f57b95c425a8e72df9fdcdaa0510145e3e5a46e5f3520fc4489bd9f699b539f8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:43:53.163242Z","signature_b64":"/ohCJOYQ/hxhrWDQakL7IHjV/k+bNg2/zEhif7tkq1v3ZfJSqZNZLxXqedJR9xSCqzvuQP8nwOg/DN1cnoxKCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f57b95c425a8e72df9fdcdaa0510145e3e5a46e5f3520fc4489bd9f699b539f8","last_reissued_at":"2026-05-18T03:43:53.162820Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:43:53.162820Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0903.4697","source_version":5,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:43:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"l/oRimV7ivEmPNlDH4HDWcZSSP1I8x+eRKFKcsrVOo2TYICrDTFH/sD6lOSsGLeOglMYTQeomZGV5ZQtKP32Aw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:09:38.915702Z"},"content_sha256":"8fcee62d3a5513046086336b7b9b7f527827b184e20fea656bd3cfccc2d78dde","schema_version":"1.0","event_id":"sha256:8fcee62d3a5513046086336b7b9b7f527827b184e20fea656bd3cfccc2d78dde"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:6V5ZLRBFVDTS36P5ZWVAKEAULY","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the moments of the meeting time of independent random walks in random environment","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Christophe Gallesco","submitted_at":"2009-03-27T17:23:31Z","abstract_excerpt":"We consider, in the continuous time version, $\\gamma$ independent random walks on $\\mathbb{Z_+}$ in random environment in the Sinai's regime. Let $T_\\gam$ be the first meeting time of one pair of the $\\gamma$ random walks starting at different positions. We first show that the tail of the quenched distribution of $T_\\gamma$, after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being $\\Eo$ the quenched expectation, we show that, for almost"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4697","kind":"arxiv","version":5},"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":"2026-05-18T03:43:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Q1k9nNCN+iQYdM1S0mU1A5L4kkK/AGDmDdVTgnXzYsQbjW/sgFEYCwneibjft7RiWK1WH42pYQrVIuVj1rZbDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:09:38.916353Z"},"content_sha256":"efec2a253df4158395d4695f00a583fff15f7c87a82ce3340a54659eea6bbd74","schema_version":"1.0","event_id":"sha256:efec2a253df4158395d4695f00a583fff15f7c87a82ce3340a54659eea6bbd74"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6V5ZLRBFVDTS36P5ZWVAKEAULY/bundle.json","state_url":"https://pith.science/pith/6V5ZLRBFVDTS36P5ZWVAKEAULY/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6V5ZLRBFVDTS36P5ZWVAKEAULY/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":"2026-05-27T14:09:38Z","links":{"resolver":"https://pith.science/pith/6V5ZLRBFVDTS36P5ZWVAKEAULY","bundle":"https://pith.science/pith/6V5ZLRBFVDTS36P5ZWVAKEAULY/bundle.json","state":"https://pith.science/pith/6V5ZLRBFVDTS36P5ZWVAKEAULY/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6V5ZLRBFVDTS36P5ZWVAKEAULY/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:6V5ZLRBFVDTS36P5ZWVAKEAULY","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":"0dd67f0a045cdf8e47a4a1bc810fa44ef4c1a293cc337724ff914b2e40ba5a74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-27T17:23:31Z","title_canon_sha256":"bf279ed44c8729855ebc03ea3354324658b4019f133784d77763938ed8a37d9c"},"schema_version":"1.0","source":{"id":"0903.4697","kind":"arxiv","version":5}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0903.4697","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"arxiv_version","alias_value":"0903.4697v5","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0903.4697","created_at":"2026-05-18T03:43:53Z"},{"alias_kind":"pith_short_12","alias_value":"6V5ZLRBFVDTS","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"6V5ZLRBFVDTS36P5","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"6V5ZLRBF","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:efec2a253df4158395d4695f00a583fff15f7c87a82ce3340a54659eea6bbd74","target":"graph","created_at":"2026-05-18T03:43:53Z","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":"We consider, in the continuous time version, $\\gamma$ independent random walks on $\\mathbb{Z_+}$ in random environment in the Sinai's regime. Let $T_\\gam$ be the first meeting time of one pair of the $\\gamma$ random walks starting at different positions. We first show that the tail of the quenched distribution of $T_\\gamma$, after a suitable rescaling, converges in probability, to some functional of the Brownian motion. Then we compute the law of this functional. Eventually, we obtain results about the moments of this meeting time. Being $\\Eo$ the quenched expectation, we show that, for almost","authors_text":"Christophe Gallesco","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-27T17:23:31Z","title":"On the moments of the meeting time of independent random walks in random environment"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0903.4697","kind":"arxiv","version":5},"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":"sha256:8fcee62d3a5513046086336b7b9b7f527827b184e20fea656bd3cfccc2d78dde","target":"record","created_at":"2026-05-18T03:43:53Z","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":"0dd67f0a045cdf8e47a4a1bc810fa44ef4c1a293cc337724ff914b2e40ba5a74","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-03-27T17:23:31Z","title_canon_sha256":"bf279ed44c8729855ebc03ea3354324658b4019f133784d77763938ed8a37d9c"},"schema_version":"1.0","source":{"id":"0903.4697","kind":"arxiv","version":5}},"canonical_sha256":"f57b95c425a8e72df9fdcdaa0510145e3e5a46e5f3520fc4489bd9f699b539f8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f57b95c425a8e72df9fdcdaa0510145e3e5a46e5f3520fc4489bd9f699b539f8","first_computed_at":"2026-05-18T03:43:53.162820Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:43:53.162820Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"/ohCJOYQ/hxhrWDQakL7IHjV/k+bNg2/zEhif7tkq1v3ZfJSqZNZLxXqedJR9xSCqzvuQP8nwOg/DN1cnoxKCg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:43:53.163242Z","signed_message":"canonical_sha256_bytes"},"source_id":"0903.4697","source_kind":"arxiv","source_version":5}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8fcee62d3a5513046086336b7b9b7f527827b184e20fea656bd3cfccc2d78dde","sha256:efec2a253df4158395d4695f00a583fff15f7c87a82ce3340a54659eea6bbd74"],"state_sha256":"a1a8bda9dd37980ee51b104a8b5dcd654eea376cc060a13937a336f4c8a5fbf7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"gVdhL2w+bAwj4GWeZBcUIEllr9vJPZRFXYNCgjmEWN14Z8OWw7x59WikrOjfvW1LuH2aE57IRiESsMLMSBc8AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T14:09:38.919410Z","bundle_sha256":"6991a8af1e907f4cf4874b7c52464f8fe4ab7acd538be78647f49e9b87b5dad3"}}