{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2009:7UW5WPELE62TND5FNPTDRIDCOE","short_pith_number":"pith:7UW5WPEL","canonical_record":{"source":{"id":"0907.2843","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-16T13:53:46Z","cross_cats_sorted":[],"title_canon_sha256":"4c2acb310c365bdb2b05fab45d7af29f54ecfae74898ae477fd09fb2c99263e9","abstract_canon_sha256":"dcfe1ba7966e88924df0a2127b8feef2ee91832646578321f4903ec05a3dc4c8"},"schema_version":"1.0"},"canonical_sha256":"fd2ddb3c8b27b5368fa56be638a062711e5d119267de789718d6327ebc61b0b6","source":{"kind":"arxiv","id":"0907.2843","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.2843","created_at":"2026-05-18T04:32:00Z"},{"alias_kind":"arxiv_version","alias_value":"0907.2843v2","created_at":"2026-05-18T04:32:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.2843","created_at":"2026-05-18T04:32:00Z"},{"alias_kind":"pith_short_12","alias_value":"7UW5WPELE62T","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"7UW5WPELE62TND5F","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"7UW5WPEL","created_at":"2026-05-18T12:25:58Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2009:7UW5WPELE62TND5FNPTDRIDCOE","target":"record","payload":{"canonical_record":{"source":{"id":"0907.2843","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-16T13:53:46Z","cross_cats_sorted":[],"title_canon_sha256":"4c2acb310c365bdb2b05fab45d7af29f54ecfae74898ae477fd09fb2c99263e9","abstract_canon_sha256":"dcfe1ba7966e88924df0a2127b8feef2ee91832646578321f4903ec05a3dc4c8"},"schema_version":"1.0"},"canonical_sha256":"fd2ddb3c8b27b5368fa56be638a062711e5d119267de789718d6327ebc61b0b6","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:32:00.102533Z","signature_b64":"ikrIPwRIS2WIVyMSPV9IQbmYXgBq3akdf059k7TtFkcmKYZk/VlH9VyZnIDmGnEmg3nMDCrsxyDZy0ycoB3fAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"fd2ddb3c8b27b5368fa56be638a062711e5d119267de789718d6327ebc61b0b6","last_reissued_at":"2026-05-18T04:32:00.101956Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:32:00.101956Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"0907.2843","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":"2026-05-18T04:32:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BXED1vTJBydI/2BRRxYxlr6xwCLh9iY6SXzfQgTVbo9ttMIk6m/H2UYjierBZRrap6Qs3DsxxIdiusV7WZ5YDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T17:59:14.002251Z"},"content_sha256":"427fd7be89499de41fb3802c0b91a0c1ff8f02374ed458a2f9d26252c7f2fd9d","schema_version":"1.0","event_id":"sha256:427fd7be89499de41fb3802c0b91a0c1ff8f02374ed458a2f9d26252c7f2fd9d"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2009:7UW5WPELE62TND5FNPTDRIDCOE","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sharpness of the percolation transition in the two-dimensional contact process","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"J. van den Berg","submitted_at":"2009-07-16T13:53:46Z","abstract_excerpt":"For ordinary (independent) percolation on a large class of lattices it is well known that below the critical percolation parameter $p_c$ the cluster size distribution has exponential decay and that power-law behavior of this distribution can only occur at $p_c$. This behavior is often called ``sharpness of the percolation transition.'' For theoretical reasons, as well as motivated by applied research, there is an increasing interest in percolation models with (weak) dependencies. For instance, biologists and agricultural researchers have used (stationary distributions of) certain two-dimension"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2843","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":"2026-05-18T04:32:00Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"s4SOw368KOKBcn3m/qfUKSxwJkCdxWZr8WlkgyMOQpqrnftVIUCWBOtoR7At4mziU/psD086VTQcehA1I3LQBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T17:59:14.002608Z"},"content_sha256":"155681fa1f17a057dbf23d07541a25625aca8314fb518c2d9c551375c6363da1","schema_version":"1.0","event_id":"sha256:155681fa1f17a057dbf23d07541a25625aca8314fb518c2d9c551375c6363da1"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/7UW5WPELE62TND5FNPTDRIDCOE/bundle.json","state_url":"https://pith.science/pith/7UW5WPELE62TND5FNPTDRIDCOE/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/7UW5WPELE62TND5FNPTDRIDCOE/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":"2026-06-30T17:59:14Z","links":{"resolver":"https://pith.science/pith/7UW5WPELE62TND5FNPTDRIDCOE","bundle":"https://pith.science/pith/7UW5WPELE62TND5FNPTDRIDCOE/bundle.json","state":"https://pith.science/pith/7UW5WPELE62TND5FNPTDRIDCOE/state.json","well_known_bundle":"https://pith.science/.well-known/pith/7UW5WPELE62TND5FNPTDRIDCOE/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2009:7UW5WPELE62TND5FNPTDRIDCOE","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":"dcfe1ba7966e88924df0a2127b8feef2ee91832646578321f4903ec05a3dc4c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-16T13:53:46Z","title_canon_sha256":"4c2acb310c365bdb2b05fab45d7af29f54ecfae74898ae477fd09fb2c99263e9"},"schema_version":"1.0","source":{"id":"0907.2843","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"0907.2843","created_at":"2026-05-18T04:32:00Z"},{"alias_kind":"arxiv_version","alias_value":"0907.2843v2","created_at":"2026-05-18T04:32:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0907.2843","created_at":"2026-05-18T04:32:00Z"},{"alias_kind":"pith_short_12","alias_value":"7UW5WPELE62T","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_16","alias_value":"7UW5WPELE62TND5F","created_at":"2026-05-18T12:25:58Z"},{"alias_kind":"pith_short_8","alias_value":"7UW5WPEL","created_at":"2026-05-18T12:25:58Z"}],"graph_snapshots":[{"event_id":"sha256:155681fa1f17a057dbf23d07541a25625aca8314fb518c2d9c551375c6363da1","target":"graph","created_at":"2026-05-18T04:32:00Z","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":"For ordinary (independent) percolation on a large class of lattices it is well known that below the critical percolation parameter $p_c$ the cluster size distribution has exponential decay and that power-law behavior of this distribution can only occur at $p_c$. This behavior is often called ``sharpness of the percolation transition.'' For theoretical reasons, as well as motivated by applied research, there is an increasing interest in percolation models with (weak) dependencies. For instance, biologists and agricultural researchers have used (stationary distributions of) certain two-dimension","authors_text":"J. van den Berg","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-16T13:53:46Z","title":"Sharpness of the percolation transition in the two-dimensional contact process"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0907.2843","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":"sha256:427fd7be89499de41fb3802c0b91a0c1ff8f02374ed458a2f9d26252c7f2fd9d","target":"record","created_at":"2026-05-18T04:32:00Z","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":"dcfe1ba7966e88924df0a2127b8feef2ee91832646578321f4903ec05a3dc4c8","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2009-07-16T13:53:46Z","title_canon_sha256":"4c2acb310c365bdb2b05fab45d7af29f54ecfae74898ae477fd09fb2c99263e9"},"schema_version":"1.0","source":{"id":"0907.2843","kind":"arxiv","version":2}},"canonical_sha256":"fd2ddb3c8b27b5368fa56be638a062711e5d119267de789718d6327ebc61b0b6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fd2ddb3c8b27b5368fa56be638a062711e5d119267de789718d6327ebc61b0b6","first_computed_at":"2026-05-18T04:32:00.101956Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:32:00.101956Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ikrIPwRIS2WIVyMSPV9IQbmYXgBq3akdf059k7TtFkcmKYZk/VlH9VyZnIDmGnEmg3nMDCrsxyDZy0ycoB3fAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:32:00.102533Z","signed_message":"canonical_sha256_bytes"},"source_id":"0907.2843","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:427fd7be89499de41fb3802c0b91a0c1ff8f02374ed458a2f9d26252c7f2fd9d","sha256:155681fa1f17a057dbf23d07541a25625aca8314fb518c2d9c551375c6363da1"],"state_sha256":"382b9f5f6803936033c297db56177d261477dbb9c5f2e8ee769749c01461454a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wEZGawzVH7E7Hik33SDP20aH9KFaqyc2hVX46uCL+EeCKkm9HVYpL8YERI2HGnDdKjIk8Tc2vhCF3BjUpXH2CQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T17:59:14.004518Z","bundle_sha256":"a96a8ed13b8ed25d359fa994fb2a973fd4310b049f94ca63f9785700846c7462"}}