{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:BF73DRDHOA4POCVRFLQOOOSARI","short_pith_number":"pith:BF73DRDH","canonical_record":{"source":{"id":"1208.4014","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-08-20T14:07:39Z","cross_cats_sorted":[],"title_canon_sha256":"56f6c79a98edabe63165d2c25c2c28bcc321b91a081cad165690a866288698f5","abstract_canon_sha256":"90edc74e75f10f79fe0501f661b8407352e774fdf94035884d8f16a9f47bd2a4"},"schema_version":"1.0"},"canonical_sha256":"097fb1c4677038f70ab12ae0e73a408a3b5cb634cd5116cbc3d1c258d1fee635","source":{"kind":"arxiv","id":"1208.4014","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4014","created_at":"2026-05-18T03:48:14Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4014v2","created_at":"2026-05-18T03:48:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4014","created_at":"2026-05-18T03:48:14Z"},{"alias_kind":"pith_short_12","alias_value":"BF73DRDHOA4P","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"BF73DRDHOA4POCVR","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"BF73DRDH","created_at":"2026-05-18T12:26:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:BF73DRDHOA4POCVRFLQOOOSARI","target":"record","payload":{"canonical_record":{"source":{"id":"1208.4014","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-08-20T14:07:39Z","cross_cats_sorted":[],"title_canon_sha256":"56f6c79a98edabe63165d2c25c2c28bcc321b91a081cad165690a866288698f5","abstract_canon_sha256":"90edc74e75f10f79fe0501f661b8407352e774fdf94035884d8f16a9f47bd2a4"},"schema_version":"1.0"},"canonical_sha256":"097fb1c4677038f70ab12ae0e73a408a3b5cb634cd5116cbc3d1c258d1fee635","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:48:14.465282Z","signature_b64":"1QzB4VK77bp3D0SHdQBdpIOkiCDosbue01dc0mCEe4AZW6js0xYr8QpD2tViAKOb9N1gPHC0VhSmrAAx/NCbDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"097fb1c4677038f70ab12ae0e73a408a3b5cb634cd5116cbc3d1c258d1fee635","last_reissued_at":"2026-05-18T03:48:14.464507Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:48:14.464507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1208.4014","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":"2026-05-18T03:48:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"py3rGAnR85pZ1uM8zkm/Ckm+veFiCC4lj3njTY/Y7ld4omaObZVDifgA30MWjxFqSFQv53OC+AOATkfQhecxBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T02:35:37.350373Z"},"content_sha256":"9d4f60e619ca2c4d63f5f213a436bfe7db4299c10ac7e0393c5828d2d0c6cf65","schema_version":"1.0","event_id":"sha256:9d4f60e619ca2c4d63f5f213a436bfe7db4299c10ac7e0393c5828d2d0c6cf65"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:BF73DRDHOA4POCVRFLQOOOSARI","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the size of the largest cluster in 2D critical percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Jacob van den Berg, Rene Conijn","submitted_at":"2012-08-20T14:07:39Z","abstract_excerpt":"We consider (near-)critical percolation on the square lattice. Let M_n be the size of the largest open cluster contained in the box [-n,n]^2, and let pi(n) be the probability that there is an open path from O to the boundary of the box. It is well-known that for all 0< a < b the probability that M_n is smaller than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are bounded away from 0 as n tends to infinity. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that M_n is bet"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4014","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":"2026-05-18T03:48:14Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"73CccM18PjRIdr5EKA7ERu47us/a8BB9qg5SZ5py8LYl2rM/8+2pWTV9MbvMHX2cSwQcpRNJSxVYum1HbYK6BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T02:35:37.350983Z"},"content_sha256":"62b2109048d327d0f264625c829ac852c04530c6c12af1dd58744b25176db18b","schema_version":"1.0","event_id":"sha256:62b2109048d327d0f264625c829ac852c04530c6c12af1dd58744b25176db18b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BF73DRDHOA4POCVRFLQOOOSARI/bundle.json","state_url":"https://pith.science/pith/BF73DRDHOA4POCVRFLQOOOSARI/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BF73DRDHOA4POCVRFLQOOOSARI/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":"2026-05-30T02:35:37Z","links":{"resolver":"https://pith.science/pith/BF73DRDHOA4POCVRFLQOOOSARI","bundle":"https://pith.science/pith/BF73DRDHOA4POCVRFLQOOOSARI/bundle.json","state":"https://pith.science/pith/BF73DRDHOA4POCVRFLQOOOSARI/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BF73DRDHOA4POCVRFLQOOOSARI/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:BF73DRDHOA4POCVRFLQOOOSARI","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":"90edc74e75f10f79fe0501f661b8407352e774fdf94035884d8f16a9f47bd2a4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-08-20T14:07:39Z","title_canon_sha256":"56f6c79a98edabe63165d2c25c2c28bcc321b91a081cad165690a866288698f5"},"schema_version":"1.0","source":{"id":"1208.4014","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1208.4014","created_at":"2026-05-18T03:48:14Z"},{"alias_kind":"arxiv_version","alias_value":"1208.4014v2","created_at":"2026-05-18T03:48:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.4014","created_at":"2026-05-18T03:48:14Z"},{"alias_kind":"pith_short_12","alias_value":"BF73DRDHOA4P","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_16","alias_value":"BF73DRDHOA4POCVR","created_at":"2026-05-18T12:26:58Z"},{"alias_kind":"pith_short_8","alias_value":"BF73DRDH","created_at":"2026-05-18T12:26:58Z"}],"graph_snapshots":[{"event_id":"sha256:62b2109048d327d0f264625c829ac852c04530c6c12af1dd58744b25176db18b","target":"graph","created_at":"2026-05-18T03:48:14Z","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 (near-)critical percolation on the square lattice. Let M_n be the size of the largest open cluster contained in the box [-n,n]^2, and let pi(n) be the probability that there is an open path from O to the boundary of the box. It is well-known that for all 0< a < b the probability that M_n is smaller than an^2 pi(n) and the probability that M_n is larger than bn^2 pi(n) are bounded away from 0 as n tends to infinity. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that M_n is bet","authors_text":"Jacob van den Berg, Rene Conijn","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-08-20T14:07:39Z","title":"On the size of the largest cluster in 2D critical percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.4014","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":"sha256:9d4f60e619ca2c4d63f5f213a436bfe7db4299c10ac7e0393c5828d2d0c6cf65","target":"record","created_at":"2026-05-18T03:48:14Z","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":"90edc74e75f10f79fe0501f661b8407352e774fdf94035884d8f16a9f47bd2a4","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2012-08-20T14:07:39Z","title_canon_sha256":"56f6c79a98edabe63165d2c25c2c28bcc321b91a081cad165690a866288698f5"},"schema_version":"1.0","source":{"id":"1208.4014","kind":"arxiv","version":2}},"canonical_sha256":"097fb1c4677038f70ab12ae0e73a408a3b5cb634cd5116cbc3d1c258d1fee635","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"097fb1c4677038f70ab12ae0e73a408a3b5cb634cd5116cbc3d1c258d1fee635","first_computed_at":"2026-05-18T03:48:14.464507Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:48:14.464507Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1QzB4VK77bp3D0SHdQBdpIOkiCDosbue01dc0mCEe4AZW6js0xYr8QpD2tViAKOb9N1gPHC0VhSmrAAx/NCbDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:48:14.465282Z","signed_message":"canonical_sha256_bytes"},"source_id":"1208.4014","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9d4f60e619ca2c4d63f5f213a436bfe7db4299c10ac7e0393c5828d2d0c6cf65","sha256:62b2109048d327d0f264625c829ac852c04530c6c12af1dd58744b25176db18b"],"state_sha256":"9dff5f72f0b4b251dd8d469e3388793646ff9bcb7ef0e685359898fcb64ea89a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J1RfOX3+2SmTMXi+ZwrJfJlAjhcSZZpDxpH4uIdTwtF2UydfSPEDvx+4EJTuTzMGtpcMda5Kqwn2/TsZ7GQbAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T02:35:37.354473Z","bundle_sha256":"32ea24166f9a264a011bc899bcc004f39e90a5cac19421e011e51e308f27c5e2"}}