{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:UGIBHIATFOXXAXAASRRJ5PMYQN","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":"5c09f9a9a011b01161101bd63e01c5bf5b6cb3a4847adcb7b4465658ba122349","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-10-24T12:50:28Z","title_canon_sha256":"c9892105dbbfc3274931edbf9e16e1714a655188995930454b287ed406030af2"},"schema_version":"1.0","source":{"id":"1610.07391","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1610.07391","created_at":"2026-05-18T00:43:01Z"},{"alias_kind":"arxiv_version","alias_value":"1610.07391v2","created_at":"2026-05-18T00:43:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1610.07391","created_at":"2026-05-18T00:43:01Z"},{"alias_kind":"pith_short_12","alias_value":"UGIBHIATFOXX","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"UGIBHIATFOXXAXAA","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"UGIBHIAT","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:46619e47be4510e3457564a5a6d33575a1db8017309f6848de83791c18b989e8","target":"graph","created_at":"2026-05-18T00:43:01Z","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":"The continuum random cluster model is a Gibbs modification of the standard boolean model of intensity $z > 0$ and law of radii $Q$. The formal unormalized density is given by $q^{N_{cc}}$ where $q$ is a fixed parameter and $N_{cc}$ is the number of connected components in the random structure. We prove for a large class of parameters that percolation occurs for $z$ large enough and does not occur for $z$ small enough. An application to the phase transition of the Widom-Rowlinson model with random radii is given. Our main tools are stochastic domination properties, a fine study of the interacti","authors_text":"Pierre Houdebert","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-10-24T12:50:28Z","title":"Percolation results for the Continuum Random Cluster Model"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.07391","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:a1adc942837e6aba92736450a0fa07136fb7cae2332597b68ab55c00facf898c","target":"record","created_at":"2026-05-18T00:43:01Z","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":"5c09f9a9a011b01161101bd63e01c5bf5b6cb3a4847adcb7b4465658ba122349","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2016-10-24T12:50:28Z","title_canon_sha256":"c9892105dbbfc3274931edbf9e16e1714a655188995930454b287ed406030af2"},"schema_version":"1.0","source":{"id":"1610.07391","kind":"arxiv","version":2}},"canonical_sha256":"a19013a0132baf705c0094629ebd9883615883f4d4815aa2284a7fd461e2bb6b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a19013a0132baf705c0094629ebd9883615883f4d4815aa2284a7fd461e2bb6b","first_computed_at":"2026-05-18T00:43:01.972986Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:43:01.972986Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1b0ADC88620hBCBzeYg+s65mOUO8FlhudN73XAaqVc8mvkFGj/vC7HgDOhkdYZqGRu0lhDMbbBFxNeUT/p7CAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:43:01.973542Z","signed_message":"canonical_sha256_bytes"},"source_id":"1610.07391","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a1adc942837e6aba92736450a0fa07136fb7cae2332597b68ab55c00facf898c","sha256:46619e47be4510e3457564a5a6d33575a1db8017309f6848de83791c18b989e8"],"state_sha256":"1d5a4d726a1b79decbf6e6a06786c7fd7d157b3481478714c09d032103076502"}