{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:D7OSIYT3TXHBFIKGEMIHRZPC7P","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":"2b3ea77446da645812f001db0bb683abee20c88d9486b3834720b99e28358cf8","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-25T18:50:54Z","title_canon_sha256":"85f168bae5505de2cd951fd1351ebc01aa88f3e142e97c9a449f7671fa5189a6"},"schema_version":"1.0","source":{"id":"1611.08549","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.08549","created_at":"2026-05-17T23:59:44Z"},{"alias_kind":"arxiv_version","alias_value":"1611.08549v1","created_at":"2026-05-17T23:59:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.08549","created_at":"2026-05-17T23:59:44Z"},{"alias_kind":"pith_short_12","alias_value":"D7OSIYT3TXHB","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"D7OSIYT3TXHBFIKG","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"D7OSIYT3","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:90e12da29f9ce78139723d5c2590a7a616fe1c14428273292a2c99b35b673915","target":"graph","created_at":"2026-05-17T23:59:44Z","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 percolation on finite transitive graphs, Nachmias and Peres suggested a characterization of the critical probability based on the logarithmic derivative of the susceptibility. As a first test-case, we study their suggestion for the Erd\\H{o}s-R\\'enyi random graph G_{n,p}, and confirm that the logarithmic derivative has the desired properties: (i) its maximizer lies inside the critical window p=1/n+\\Theta(n^{-4/3}), and (ii) the inverse of its maximum value coincides with the \\Theta(n^{-4/3})-width of the critical window. We also prove that the maximizer is not located at p=1/n or p=1/(n-1),","authors_text":"Lutz Warnke, Svante Janson","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-25T18:50:54Z","title":"On the critical probability in percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.08549","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:a0f191880c2aaf9280949f1319cda928cb1562bf067353db31f7ffe65a8663e1","target":"record","created_at":"2026-05-17T23:59:44Z","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":"2b3ea77446da645812f001db0bb683abee20c88d9486b3834720b99e28358cf8","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-11-25T18:50:54Z","title_canon_sha256":"85f168bae5505de2cd951fd1351ebc01aa88f3e142e97c9a449f7671fa5189a6"},"schema_version":"1.0","source":{"id":"1611.08549","kind":"arxiv","version":1}},"canonical_sha256":"1fdd24627b9dce12a146231078e5e2fbf2aa4a4ab184567fa6688188d9230313","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"1fdd24627b9dce12a146231078e5e2fbf2aa4a4ab184567fa6688188d9230313","first_computed_at":"2026-05-17T23:59:44.016592Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:59:44.016592Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mbNEViL5UKLHcI2OknrYu2Io2ArqJffIkd2oQXHYKfh8XXen1ponY2dUkPVz0wiAZDxcXWe+AkAzcSyiDtedDw==","signature_status":"signed_v1","signed_at":"2026-05-17T23:59:44.017017Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.08549","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a0f191880c2aaf9280949f1319cda928cb1562bf067353db31f7ffe65a8663e1","sha256:90e12da29f9ce78139723d5c2590a7a616fe1c14428273292a2c99b35b673915"],"state_sha256":"9a729f39c95a75ac1e38d6c2f44a9b81376e3ff9c3d1590581e31221d0e93efa"}