{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:DP3I4BGRG2KULXRADFTZ4C2RHI","short_pith_number":"pith:DP3I4BGR","canonical_record":{"source":{"id":"1107.2276","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-12T13:14:07Z","cross_cats_sorted":[],"title_canon_sha256":"c4e302ee2047e3fa3787e65d2a4e7da7d83c79350bc96406d69a210882d04280","abstract_canon_sha256":"1b142d51ac316188bb57788a1bb3a28531d0230899a926a1ec0d8aca8fb96179"},"schema_version":"1.0"},"canonical_sha256":"1bf68e04d1369545de2019679e0b513a120cc69b891b014fbac7e005c1e32d0f","source":{"kind":"arxiv","id":"1107.2276","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2276","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2276v2","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2276","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"pith_short_12","alias_value":"DP3I4BGRG2KU","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"DP3I4BGRG2KULXRA","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"DP3I4BGR","created_at":"2026-05-18T12:26:26Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:DP3I4BGRG2KULXRADFTZ4C2RHI","target":"record","payload":{"canonical_record":{"source":{"id":"1107.2276","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-12T13:14:07Z","cross_cats_sorted":[],"title_canon_sha256":"c4e302ee2047e3fa3787e65d2a4e7da7d83c79350bc96406d69a210882d04280","abstract_canon_sha256":"1b142d51ac316188bb57788a1bb3a28531d0230899a926a1ec0d8aca8fb96179"},"schema_version":"1.0"},"canonical_sha256":"1bf68e04d1369545de2019679e0b513a120cc69b891b014fbac7e005c1e32d0f","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:52.633381Z","signature_b64":"m23x6wXXFN2RD5i5nmeIvBWzg0bgIte/hRiINBTlVPi1JnA503FPGNWARgCnCfUYe/jplf6nUzVow7zyJ5BYAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"1bf68e04d1369545de2019679e0b513a120cc69b891b014fbac7e005c1e32d0f","last_reissued_at":"2026-05-18T02:17:52.632641Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:52.632641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1107.2276","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":"2026-05-18T02:17:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"65K47TqWa5ZyTtNV0WjzfCnMPOfVcNebY9lFagSQJUrxaRORNwzdlpxOUhchvTiGY4v3m1tw0+5CG0ElCewZAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:07:13.382456Z"},"content_sha256":"14ffc9f95e64569f1d9ec089c97b255cd8f29a086c3613f950abc789753f2f3d","schema_version":"1.0","event_id":"sha256:14ffc9f95e64569f1d9ec089c97b255cd8f29a086c3613f950abc789753f2f3d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:DP3I4BGRG2KULXRADFTZ4C2RHI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Asymptotics of first-passage percolation on 1-dimensional graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Daniel Ahlberg","submitted_at":"2011-07-12T13:14:07Z","abstract_excerpt":"In this paper we consider first-passage percolation on certain 1-dimensional periodic graphs, such as the $\\Z\\times\\{0,1,\\ldots,K-1\\}^{d-1}$ nearest neighbour graph for $d,K\\geq1$. We find that both length and weight of minimal-weight paths present a typical 1-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of the minimizing path between two points are monotone in the distance betwee"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2276","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":"2026-05-18T02:17:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NzR4suKuSzVQxQY9C+QKvyFX1X3Ez8T2ENMKLWvYE/Pf6sPqdRLR1DEQIvVMXuztRjrZc+7P9jMvLRpNzgepDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T15:07:13.382790Z"},"content_sha256":"142b90a635fbfe48d0931c24ec31129031a606e4c78383bd86f219465522a8f3","schema_version":"1.0","event_id":"sha256:142b90a635fbfe48d0931c24ec31129031a606e4c78383bd86f219465522a8f3"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/DP3I4BGRG2KULXRADFTZ4C2RHI/bundle.json","state_url":"https://pith.science/pith/DP3I4BGRG2KULXRADFTZ4C2RHI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/DP3I4BGRG2KULXRADFTZ4C2RHI/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":"2026-06-02T15:07:13Z","links":{"resolver":"https://pith.science/pith/DP3I4BGRG2KULXRADFTZ4C2RHI","bundle":"https://pith.science/pith/DP3I4BGRG2KULXRADFTZ4C2RHI/bundle.json","state":"https://pith.science/pith/DP3I4BGRG2KULXRADFTZ4C2RHI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/DP3I4BGRG2KULXRADFTZ4C2RHI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:DP3I4BGRG2KULXRADFTZ4C2RHI","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":"1b142d51ac316188bb57788a1bb3a28531d0230899a926a1ec0d8aca8fb96179","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-12T13:14:07Z","title_canon_sha256":"c4e302ee2047e3fa3787e65d2a4e7da7d83c79350bc96406d69a210882d04280"},"schema_version":"1.0","source":{"id":"1107.2276","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1107.2276","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"arxiv_version","alias_value":"1107.2276v2","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1107.2276","created_at":"2026-05-18T02:17:52Z"},{"alias_kind":"pith_short_12","alias_value":"DP3I4BGRG2KU","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_16","alias_value":"DP3I4BGRG2KULXRA","created_at":"2026-05-18T12:26:26Z"},{"alias_kind":"pith_short_8","alias_value":"DP3I4BGR","created_at":"2026-05-18T12:26:26Z"}],"graph_snapshots":[{"event_id":"sha256:142b90a635fbfe48d0931c24ec31129031a606e4c78383bd86f219465522a8f3","target":"graph","created_at":"2026-05-18T02:17:52Z","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":"In this paper we consider first-passage percolation on certain 1-dimensional periodic graphs, such as the $\\Z\\times\\{0,1,\\ldots,K-1\\}^{d-1}$ nearest neighbour graph for $d,K\\geq1$. We find that both length and weight of minimal-weight paths present a typical 1-dimensional asymptotic behaviour. Apart from a strong law of large numbers, we derive a central limit theorem, a law of the iterated logarithm, and a Donsker theorem for these quantities. In addition, we prove that the mean and variance of the length and weight of the minimizing path between two points are monotone in the distance betwee","authors_text":"Daniel Ahlberg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-12T13:14:07Z","title":"Asymptotics of first-passage percolation on 1-dimensional graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1107.2276","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":"sha256:14ffc9f95e64569f1d9ec089c97b255cd8f29a086c3613f950abc789753f2f3d","target":"record","created_at":"2026-05-18T02:17:52Z","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":"1b142d51ac316188bb57788a1bb3a28531d0230899a926a1ec0d8aca8fb96179","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2011-07-12T13:14:07Z","title_canon_sha256":"c4e302ee2047e3fa3787e65d2a4e7da7d83c79350bc96406d69a210882d04280"},"schema_version":"1.0","source":{"id":"1107.2276","kind":"arxiv","version":2}},"canonical_sha256":"1bf68e04d1369545de2019679e0b513a120cc69b891b014fbac7e005c1e32d0f","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1bf68e04d1369545de2019679e0b513a120cc69b891b014fbac7e005c1e32d0f","first_computed_at":"2026-05-18T02:17:52.632641Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:17:52.632641Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"m23x6wXXFN2RD5i5nmeIvBWzg0bgIte/hRiINBTlVPi1JnA503FPGNWARgCnCfUYe/jplf6nUzVow7zyJ5BYAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:17:52.633381Z","signed_message":"canonical_sha256_bytes"},"source_id":"1107.2276","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:14ffc9f95e64569f1d9ec089c97b255cd8f29a086c3613f950abc789753f2f3d","sha256:142b90a635fbfe48d0931c24ec31129031a606e4c78383bd86f219465522a8f3"],"state_sha256":"c7c2415936f3ac9558a808c77521a8ad2c45a0b7a7b350bf886176394a62ca83"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"AN7uUejW0N/K7v3RQ/Mu1Xn6KXzzHws+kYo7uUhL4Dw+C0l6gt884MjDqbVmLD/2LgGUTOsWSgIhqy4oFJCHAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T15:07:13.384571Z","bundle_sha256":"62907f07aa92ae64540d71a10b175787ff8b1cb487892e7426f3476092ea74ce"}}