{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2023:MZESI6DWX5FHGZRE6OIAQYYAKE","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":"f01d85e48b5706ac5ebd7ed73d807973d744e07773d2e0a8073a5dfed303f679","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2023-03-01T18:29:03Z","title_canon_sha256":"ffc572d6dae8d0ad6a3deb8a6a1018598869e9734b5847e5f9c617c44a4bccb2"},"schema_version":"1.0","source":{"id":"2303.00724","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2303.00724","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"arxiv_version","alias_value":"2303.00724v3","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.00724","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"pith_short_12","alias_value":"MZESI6DWX5FH","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"pith_short_16","alias_value":"MZESI6DWX5FHGZRE","created_at":"2026-07-05T09:21:35Z"},{"alias_kind":"pith_short_8","alias_value":"MZESI6DW","created_at":"2026-07-05T09:21:35Z"}],"graph_snapshots":[{"event_id":"sha256:fb648da02bc732f3b1036d929beecc8ded7b039f296474f493c03534d82516a1","target":"graph","created_at":"2026-07-05T09:21:35Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2303.00724/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We consider a large class of spatially-embedded random graphs that includes among others long-range percolation, continuum scale-free percolation and the age-dependent random connection model. We assume that the model is supercritical: there is an infinite component. We identify the stretch-exponent $\\zeta\\in(0,1)$ of the decay of the cluster-size distribution. That is, with $|\\mathcal{C}(0)|$ denoting the number of vertices in the component of the vertex at $0\\in \\mathbb{R}^d$, we prove \\[\n  \\mathbb{P}(k< |\\mathcal{C}(0)|<\\infty)=\\exp\\big(-\\Theta(k^{\\zeta})\\big), \\qquad \\text{as }k\\to\\infty.\n","authors_text":"Dieter Mitsche, Joost Jorritsma, J\\'ulia Komj\\'athy","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2023-03-01T18:29:03Z","title":"Cluster-size decay in supercritical kernel-based spatial random graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.00724","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":"sha256:2b26dc6958cd979163124d703942a2250733089c687d3263cb3ead6083f70439","target":"record","created_at":"2026-07-05T09:21:35Z","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":"f01d85e48b5706ac5ebd7ed73d807973d744e07773d2e0a8073a5dfed303f679","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2023-03-01T18:29:03Z","title_canon_sha256":"ffc572d6dae8d0ad6a3deb8a6a1018598869e9734b5847e5f9c617c44a4bccb2"},"schema_version":"1.0","source":{"id":"2303.00724","kind":"arxiv","version":3}},"canonical_sha256":"6649247876bf4a736624f39008630051180e675e26571fa220cc01b66c15e61b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6649247876bf4a736624f39008630051180e675e26571fa220cc01b66c15e61b","first_computed_at":"2026-07-05T09:21:35.824855Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:21:35.824855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"F4cY4FXZRJrHr5YGAK7p5YJOM3msuMQYrl79PhysnpbVYWh7O0k7sbbJKerPFFSlrloN+la6W+DChytPA8JTDA==","signature_status":"signed_v1","signed_at":"2026-07-05T09:21:35.825324Z","signed_message":"canonical_sha256_bytes"},"source_id":"2303.00724","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2b26dc6958cd979163124d703942a2250733089c687d3263cb3ead6083f70439","sha256:fb648da02bc732f3b1036d929beecc8ded7b039f296474f493c03534d82516a1"],"state_sha256":"b10855e3dcf0e64048e29c0482e059ec328beeee7206b567a54da9f1ec1770bb"}