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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2303.00724","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.PR","submitted_at":"2023-03-01T18:29:03Z","cross_cats_sorted":[],"title_canon_sha256":"ffc572d6dae8d0ad6a3deb8a6a1018598869e9734b5847e5f9c617c44a4bccb2","abstract_canon_sha256":"f01d85e48b5706ac5ebd7ed73d807973d744e07773d2e0a8073a5dfed303f679"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:21:35.825324Z","signature_b64":"F4cY4FXZRJrHr5YGAK7p5YJOM3msuMQYrl79PhysnpbVYWh7O0k7sbbJKerPFFSlrloN+la6W+DChytPA8JTDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6649247876bf4a736624f39008630051180e675e26571fa220cc01b66c15e61b","last_reissued_at":"2026-07-05T09:21:35.824855Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:21:35.824855Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cluster-size decay in supercritical kernel-based spatial random graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Dieter Mitsche, Joost Jorritsma, J\\'ulia Komj\\'athy","submitted_at":"2023-03-01T18:29:03Z","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. 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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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2303.00724","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2303.00724/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2303.00724","created_at":"2026-07-05T09:21:35.824912+00:00"},{"alias_kind":"arxiv_version","alias_value":"2303.00724v3","created_at":"2026-07-05T09:21:35.824912+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2303.00724","created_at":"2026-07-05T09:21:35.824912+00:00"},{"alias_kind":"pith_short_12","alias_value":"MZESI6DWX5FH","created_at":"2026-07-05T09:21:35.824912+00:00"},{"alias_kind":"pith_short_16","alias_value":"MZESI6DWX5FHGZRE","created_at":"2026-07-05T09:21:35.824912+00:00"},{"alias_kind":"pith_short_8","alias_value":"MZESI6DW","created_at":"2026-07-05T09:21:35.824912+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2411.10333","citing_title":"Subcritical annulus crossing in spatial random graphs","ref_index":35,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE","json":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE.json","graph_json":"https://pith.science/api/pith-number/MZESI6DWX5FHGZRE6OIAQYYAKE/graph.json","events_json":"https://pith.science/api/pith-number/MZESI6DWX5FHGZRE6OIAQYYAKE/events.json","paper":"https://pith.science/paper/MZESI6DW"},"agent_actions":{"view_html":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE","download_json":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE.json","view_paper":"https://pith.science/paper/MZESI6DW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2303.00724&json=true","fetch_graph":"https://pith.science/api/pith-number/MZESI6DWX5FHGZRE6OIAQYYAKE/graph.json","fetch_events":"https://pith.science/api/pith-number/MZESI6DWX5FHGZRE6OIAQYYAKE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/action/storage_attestation","attest_author":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/action/author_attestation","sign_citation":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/action/citation_signature","submit_replication":"https://pith.science/pith/MZESI6DWX5FHGZRE6OIAQYYAKE/action/replication_record"}},"created_at":"2026-07-05T09:21:35.824912+00:00","updated_at":"2026-07-05T09:21:35.824912+00:00"}