{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:276GJNW5SMGM6EX72JNTPQVL6K","short_pith_number":"pith:276GJNW5","canonical_record":{"source":{"id":"1307.3084","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-07-11T12:24:52Z","cross_cats_sorted":[],"title_canon_sha256":"b157e38746440993156779902ee66ca333fdc688ccea6397f7d724694995729f","abstract_canon_sha256":"d3e3fbd1e5c57750e70c84f63450740d57318b9b386370ad72e92c7a1ce1a1a8"},"schema_version":"1.0"},"canonical_sha256":"d7fc64b6dd930ccf12ffd25b37c2abf2b43c9a6d04ed8d2547b19f20ddcbe270","source":{"kind":"arxiv","id":"1307.3084","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.3084","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"arxiv_version","alias_value":"1307.3084v1","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.3084","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"pith_short_12","alias_value":"276GJNW5SMGM","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"276GJNW5SMGM6EX7","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"276GJNW5","created_at":"2026-05-18T12:27:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:276GJNW5SMGM6EX72JNTPQVL6K","target":"record","payload":{"canonical_record":{"source":{"id":"1307.3084","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-07-11T12:24:52Z","cross_cats_sorted":[],"title_canon_sha256":"b157e38746440993156779902ee66ca333fdc688ccea6397f7d724694995729f","abstract_canon_sha256":"d3e3fbd1e5c57750e70c84f63450740d57318b9b386370ad72e92c7a1ce1a1a8"},"schema_version":"1.0"},"canonical_sha256":"d7fc64b6dd930ccf12ffd25b37c2abf2b43c9a6d04ed8d2547b19f20ddcbe270","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:18:41.270480Z","signature_b64":"h5Z+Rv3xY2mzZsfxQuC+ZFkdsUACEfa/JGbr145aEqKS3oEScsg5dkyX6WV4YDG/DUh2d5iLwPe084iBnc/KBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7fc64b6dd930ccf12ffd25b37c2abf2b43c9a6d04ed8d2547b19f20ddcbe270","last_reissued_at":"2026-05-18T03:18:41.269814Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:18:41.269814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1307.3084","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":"2026-05-18T03:18:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aWvZfWFteER7EWdAvI+GZ/4lVZgAnmLd7CZyZhtSebiLI28oJSIAt9Q6aEtkRgybiPEIEaBQ/eIoR1z9jN7UAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T11:07:45.323755Z"},"content_sha256":"e533a2e65c39916c92e129a633db03c96295b1a15471abb1acb0800cf5ad9bc9","schema_version":"1.0","event_id":"sha256:e533a2e65c39916c92e129a633db03c96295b1a15471abb1acb0800cf5ad9bc9"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:276GJNW5SMGM6EX72JNTPQVL6K","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The travel time in a finite box in supercritical Bernoulli percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Rapha\\\"el Cerf","submitted_at":"2013-07-11T12:24:52Z","abstract_excerpt":"We consider the standard site percolation model on the three dimensional cubic lattice. Starting solely with the hypothesis that $\\theta(p)>0$, we prove that, for any $\\alpha>0$, there exists $\\kappa>0$ such that, with probability larger than $1-1/n^\\alpha$, every pair of vertices inside the box $\\Lambda(n)$ are joined by a path having at most $\\kappa(\\ln n)^2$ closed sites."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3084","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":"2026-05-18T03:18:41Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vNZX+6gaUAinWZ/d+jgsOqdVz1O/s7NnAV11lYjn0+gmD4gWHrJxptx73WWQ7HAYebjLzwhDLkXU+Jw+QB5hCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T11:07:45.324105Z"},"content_sha256":"438c7ec78bc560feedc3afc1901e9c81928d2da4f6e3cd49bb6b806fb1d37cf8","schema_version":"1.0","event_id":"sha256:438c7ec78bc560feedc3afc1901e9c81928d2da4f6e3cd49bb6b806fb1d37cf8"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/276GJNW5SMGM6EX72JNTPQVL6K/bundle.json","state_url":"https://pith.science/pith/276GJNW5SMGM6EX72JNTPQVL6K/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/276GJNW5SMGM6EX72JNTPQVL6K/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":"2026-05-28T11:07:45Z","links":{"resolver":"https://pith.science/pith/276GJNW5SMGM6EX72JNTPQVL6K","bundle":"https://pith.science/pith/276GJNW5SMGM6EX72JNTPQVL6K/bundle.json","state":"https://pith.science/pith/276GJNW5SMGM6EX72JNTPQVL6K/state.json","well_known_bundle":"https://pith.science/.well-known/pith/276GJNW5SMGM6EX72JNTPQVL6K/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:276GJNW5SMGM6EX72JNTPQVL6K","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":"d3e3fbd1e5c57750e70c84f63450740d57318b9b386370ad72e92c7a1ce1a1a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-07-11T12:24:52Z","title_canon_sha256":"b157e38746440993156779902ee66ca333fdc688ccea6397f7d724694995729f"},"schema_version":"1.0","source":{"id":"1307.3084","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1307.3084","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"arxiv_version","alias_value":"1307.3084v1","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1307.3084","created_at":"2026-05-18T03:18:41Z"},{"alias_kind":"pith_short_12","alias_value":"276GJNW5SMGM","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"276GJNW5SMGM6EX7","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"276GJNW5","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:438c7ec78bc560feedc3afc1901e9c81928d2da4f6e3cd49bb6b806fb1d37cf8","target":"graph","created_at":"2026-05-18T03:18:41Z","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 the standard site percolation model on the three dimensional cubic lattice. Starting solely with the hypothesis that $\\theta(p)>0$, we prove that, for any $\\alpha>0$, there exists $\\kappa>0$ such that, with probability larger than $1-1/n^\\alpha$, every pair of vertices inside the box $\\Lambda(n)$ are joined by a path having at most $\\kappa(\\ln n)^2$ closed sites.","authors_text":"Rapha\\\"el Cerf","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-07-11T12:24:52Z","title":"The travel time in a finite box in supercritical Bernoulli percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1307.3084","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:e533a2e65c39916c92e129a633db03c96295b1a15471abb1acb0800cf5ad9bc9","target":"record","created_at":"2026-05-18T03:18:41Z","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":"d3e3fbd1e5c57750e70c84f63450740d57318b9b386370ad72e92c7a1ce1a1a8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-07-11T12:24:52Z","title_canon_sha256":"b157e38746440993156779902ee66ca333fdc688ccea6397f7d724694995729f"},"schema_version":"1.0","source":{"id":"1307.3084","kind":"arxiv","version":1}},"canonical_sha256":"d7fc64b6dd930ccf12ffd25b37c2abf2b43c9a6d04ed8d2547b19f20ddcbe270","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d7fc64b6dd930ccf12ffd25b37c2abf2b43c9a6d04ed8d2547b19f20ddcbe270","first_computed_at":"2026-05-18T03:18:41.269814Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:18:41.269814Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"h5Z+Rv3xY2mzZsfxQuC+ZFkdsUACEfa/JGbr145aEqKS3oEScsg5dkyX6WV4YDG/DUh2d5iLwPe084iBnc/KBA==","signature_status":"signed_v1","signed_at":"2026-05-18T03:18:41.270480Z","signed_message":"canonical_sha256_bytes"},"source_id":"1307.3084","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e533a2e65c39916c92e129a633db03c96295b1a15471abb1acb0800cf5ad9bc9","sha256:438c7ec78bc560feedc3afc1901e9c81928d2da4f6e3cd49bb6b806fb1d37cf8"],"state_sha256":"2808220b87a5438ac841262db8ac19edcd1ae1aade1d2a5fe259adf27836986e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YBSUl4H1pI6ZkyT7PaafbA5DrL288aAu6McQPB/CLVHBB3buxT241/kbM/rKbN+H426Ky7kDIf/PeKqcuWGhAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T11:07:45.326059Z","bundle_sha256":"6672592a29b51c203058ffb7741782197b77525fbfdac319bc559e0abef054c9"}}