{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:Y22NHSYBL6JXF3ETNNJQBZQDVD","short_pith_number":"pith:Y22NHSYB","canonical_record":{"source":{"id":"1201.0658","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-03T15:10:28Z","cross_cats_sorted":[],"title_canon_sha256":"0c0e6a437780abd9f894dc3d2ba0a3cd4d8a2d0ec0c5d841c1c10fc180a9a2de","abstract_canon_sha256":"763d0485bd7756f59cab6cd874b1b8eb80b15ef8211cffe4767b4b0cb3efa219"},"schema_version":"1.0"},"canonical_sha256":"c6b4d3cb015f9372ec936b5300e603a8caa1b236c5c5001a76e74e0feea9cbc2","source":{"kind":"arxiv","id":"1201.0658","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.0658","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"arxiv_version","alias_value":"1201.0658v2","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.0658","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"pith_short_12","alias_value":"Y22NHSYBL6JX","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"Y22NHSYBL6JXF3ET","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"Y22NHSYB","created_at":"2026-05-18T12:27:27Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:Y22NHSYBL6JXF3ETNNJQBZQDVD","target":"record","payload":{"canonical_record":{"source":{"id":"1201.0658","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-03T15:10:28Z","cross_cats_sorted":[],"title_canon_sha256":"0c0e6a437780abd9f894dc3d2ba0a3cd4d8a2d0ec0c5d841c1c10fc180a9a2de","abstract_canon_sha256":"763d0485bd7756f59cab6cd874b1b8eb80b15ef8211cffe4767b4b0cb3efa219"},"schema_version":"1.0"},"canonical_sha256":"c6b4d3cb015f9372ec936b5300e603a8caa1b236c5c5001a76e74e0feea9cbc2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:46:03.544653Z","signature_b64":"uhMii6OV/ADmdKudcm9XO5MGG+XG4IAYQkTTRNJzHo+T5LUXg3pFRJm7zYR+efoXeVGC+BcNHtdqyQSWmz/wBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c6b4d3cb015f9372ec936b5300e603a8caa1b236c5c5001a76e74e0feea9cbc2","last_reissued_at":"2026-05-18T03:46:03.544058Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:46:03.544058Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1201.0658","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:46:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MdCQ14pVsocjiuAHA42nXyYWDlDaL4mJhNjskPVfQswwayS4luCTd4jb+v36ID1FFA60ssDmZYZEUBFnIvD0Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T06:32:18.709752Z"},"content_sha256":"312624341c05ca9fabbe43ab12f4ce943908e0ef08257df9e9b7e754d4ebe75d","schema_version":"1.0","event_id":"sha256:312624341c05ca9fabbe43ab12f4ce943908e0ef08257df9e9b7e754d4ebe75d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:Y22NHSYBL6JXF3ETNNJQBZQDVD","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Localization on 4 sites for Vertex-reinforced random walks on $\\mathbb Z$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Anne-Laure Basdevant (MODAL'X), Arvind Singh (LM-Orsay), Bruno Schapira (LM-Orsay)","submitted_at":"2012-01-03T15:10:28Z","abstract_excerpt":"We characterize non-decreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on 4 sites. A phase transition appears for weights of order $n\\log \\log n$: for weights growing faster than this rate, the VRRW localizes almost surely on at most 4 sites whereas for weights growing slower, the VRRW cannot localize on less than 5 sites. When $w$ is of order $n\\log \\log n$, the VRRW localizes almost surely on either 4 or 5 sites, both events happening with positive probability."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0658","kind":"arxiv","version":2},"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":"2026-05-18T03:46:03Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"JgoYC9rHCKFTrj4Z46NWP/Px/TvQSJ97YAI0WtF0Xy8kucJ3JbdW3zVIT7b1CLNYWxxa5AFOVoOXeW3zL3yOAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T06:32:18.710092Z"},"content_sha256":"1f10ef1467848b409669483cc109c52b4ff5caab5980fbcc6567c69423f2b9da","schema_version":"1.0","event_id":"sha256:1f10ef1467848b409669483cc109c52b4ff5caab5980fbcc6567c69423f2b9da"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/Y22NHSYBL6JXF3ETNNJQBZQDVD/bundle.json","state_url":"https://pith.science/pith/Y22NHSYBL6JXF3ETNNJQBZQDVD/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/Y22NHSYBL6JXF3ETNNJQBZQDVD/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":"2026-06-02T06:32:18Z","links":{"resolver":"https://pith.science/pith/Y22NHSYBL6JXF3ETNNJQBZQDVD","bundle":"https://pith.science/pith/Y22NHSYBL6JXF3ETNNJQBZQDVD/bundle.json","state":"https://pith.science/pith/Y22NHSYBL6JXF3ETNNJQBZQDVD/state.json","well_known_bundle":"https://pith.science/.well-known/pith/Y22NHSYBL6JXF3ETNNJQBZQDVD/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:Y22NHSYBL6JXF3ETNNJQBZQDVD","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":"763d0485bd7756f59cab6cd874b1b8eb80b15ef8211cffe4767b4b0cb3efa219","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-03T15:10:28Z","title_canon_sha256":"0c0e6a437780abd9f894dc3d2ba0a3cd4d8a2d0ec0c5d841c1c10fc180a9a2de"},"schema_version":"1.0","source":{"id":"1201.0658","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1201.0658","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"arxiv_version","alias_value":"1201.0658v2","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1201.0658","created_at":"2026-05-18T03:46:03Z"},{"alias_kind":"pith_short_12","alias_value":"Y22NHSYBL6JX","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_16","alias_value":"Y22NHSYBL6JXF3ET","created_at":"2026-05-18T12:27:27Z"},{"alias_kind":"pith_short_8","alias_value":"Y22NHSYB","created_at":"2026-05-18T12:27:27Z"}],"graph_snapshots":[{"event_id":"sha256:1f10ef1467848b409669483cc109c52b4ff5caab5980fbcc6567c69423f2b9da","target":"graph","created_at":"2026-05-18T03:46:03Z","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 characterize non-decreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on 4 sites. A phase transition appears for weights of order $n\\log \\log n$: for weights growing faster than this rate, the VRRW localizes almost surely on at most 4 sites whereas for weights growing slower, the VRRW cannot localize on less than 5 sites. When $w$ is of order $n\\log \\log n$, the VRRW localizes almost surely on either 4 or 5 sites, both events happening with positive probability.","authors_text":"Anne-Laure Basdevant (MODAL'X), Arvind Singh (LM-Orsay), Bruno Schapira (LM-Orsay)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-03T15:10:28Z","title":"Localization on 4 sites for Vertex-reinforced random walks on $\\mathbb Z$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1201.0658","kind":"arxiv","version":2},"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":"sha256:312624341c05ca9fabbe43ab12f4ce943908e0ef08257df9e9b7e754d4ebe75d","target":"record","created_at":"2026-05-18T03:46:03Z","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":"763d0485bd7756f59cab6cd874b1b8eb80b15ef8211cffe4767b4b0cb3efa219","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-01-03T15:10:28Z","title_canon_sha256":"0c0e6a437780abd9f894dc3d2ba0a3cd4d8a2d0ec0c5d841c1c10fc180a9a2de"},"schema_version":"1.0","source":{"id":"1201.0658","kind":"arxiv","version":2}},"canonical_sha256":"c6b4d3cb015f9372ec936b5300e603a8caa1b236c5c5001a76e74e0feea9cbc2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c6b4d3cb015f9372ec936b5300e603a8caa1b236c5c5001a76e74e0feea9cbc2","first_computed_at":"2026-05-18T03:46:03.544058Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:46:03.544058Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uhMii6OV/ADmdKudcm9XO5MGG+XG4IAYQkTTRNJzHo+T5LUXg3pFRJm7zYR+efoXeVGC+BcNHtdqyQSWmz/wBg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:46:03.544653Z","signed_message":"canonical_sha256_bytes"},"source_id":"1201.0658","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:312624341c05ca9fabbe43ab12f4ce943908e0ef08257df9e9b7e754d4ebe75d","sha256:1f10ef1467848b409669483cc109c52b4ff5caab5980fbcc6567c69423f2b9da"],"state_sha256":"25ac045bc99be3e8585568f6f672f8dd91383cef85712809b9e446958f16b91b"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"cYDV5Tb3M0x6wcscRufJMX63Y+gEBrEpQUUKIex6+Mjk2KvBSCvYAd0GSe38yw0gPAZdVaC+Ne894OGFzjVoDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T06:32:18.712359Z","bundle_sha256":"f1707d75f6c2f703c676102eb611ceae4dba1d35295c1df3dc7c4ba028fd57bc"}}