{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:Y44CEBTMJY4R77DA5LYSUE7IP3","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":"c3b6a33d9a9bf63402a64c0314d0ae3bfd38dae2e9c4fa231eb6b1833bbde708","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-08T07:12:07Z","title_canon_sha256":"86639be5f79d7f3e785d6ec63bd0e26b61e46f2a22e6459a927acdc1f064f829"},"schema_version":"1.0","source":{"id":"1506.02372","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1506.02372","created_at":"2026-05-18T01:23:29Z"},{"alias_kind":"arxiv_version","alias_value":"1506.02372v3","created_at":"2026-05-18T01:23:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1506.02372","created_at":"2026-05-18T01:23:29Z"},{"alias_kind":"pith_short_12","alias_value":"Y44CEBTMJY4R","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"Y44CEBTMJY4R77DA","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"Y44CEBTM","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:2935f52e452c3501efb00fc2d51891854e346631c1a4189eca11e18c40722b17","target":"graph","created_at":"2026-05-18T01:23:29Z","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":"Recently, by introducing  the notion of cumulatively merged partition, M\\'enard and Singh provide   a sufficient condition on graphs ensuring that the critical value of   the contact process is positive.  In this note, we show  that    the one-dimensional long range percolation with high exponent satisfies their condition and thus the contact process exhibits a non-trivial phase transition.","authors_text":"Van Hao Can (I2M)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-08T07:12:07Z","title":"Contact process on one-dimensional long-range percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.02372","kind":"arxiv","version":3},"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":"sha256:2ce476f55250307d7333d72d35a7bcfd46371bdabb9515c27efb3f1d9cae6161","target":"record","created_at":"2026-05-18T01:23:29Z","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":"c3b6a33d9a9bf63402a64c0314d0ae3bfd38dae2e9c4fa231eb6b1833bbde708","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-06-08T07:12:07Z","title_canon_sha256":"86639be5f79d7f3e785d6ec63bd0e26b61e46f2a22e6459a927acdc1f064f829"},"schema_version":"1.0","source":{"id":"1506.02372","kind":"arxiv","version":3}},"canonical_sha256":"c73822066c4e391ffc60eaf12a13e87edf2957a1bc89c026cb239ffc4b9a06e6","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c73822066c4e391ffc60eaf12a13e87edf2957a1bc89c026cb239ffc4b9a06e6","first_computed_at":"2026-05-18T01:23:29.273877Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:23:29.273877Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"28JwORJDmq7IGpmQpCDBkAaAjL7/zQfgGOAp60OlHye2+Hu5/MmGNpeuUeg8bbaTwxL6+tqAygaRaCWHvjaSDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:23:29.274516Z","signed_message":"canonical_sha256_bytes"},"source_id":"1506.02372","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2ce476f55250307d7333d72d35a7bcfd46371bdabb9515c27efb3f1d9cae6161","sha256:2935f52e452c3501efb00fc2d51891854e346631c1a4189eca11e18c40722b17"],"state_sha256":"58d14247dd80efa253f66675c711d4986990daab6fbf0a40c72fb24a4a437643"}