{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:PLQRT3ZXKC3THRZLTL73F3HHOH","short_pith_number":"pith:PLQRT3ZX","canonical_record":{"source":{"id":"2507.19288","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-07-25T14:02:03Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"2f8a3b7c8cdf1ba4d734efbca69c90147f12b8ad4ebcbba89e5eaa7e086fdc2a","abstract_canon_sha256":"0a5441b35ebc319e88859440634f228b75210504af988d23948e286aa6c205b5"},"schema_version":"1.0"},"canonical_sha256":"7ae119ef3750b733c72b9affb2ece771ce6ec76de8d07680d0d759f6db57a1ef","source":{"kind":"arxiv","id":"2507.19288","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.19288","created_at":"2026-05-28T02:04:42Z"},{"alias_kind":"arxiv_version","alias_value":"2507.19288v2","created_at":"2026-05-28T02:04:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.19288","created_at":"2026-05-28T02:04:42Z"},{"alias_kind":"pith_short_12","alias_value":"PLQRT3ZXKC3T","created_at":"2026-05-28T02:04:42Z"},{"alias_kind":"pith_short_16","alias_value":"PLQRT3ZXKC3THRZL","created_at":"2026-05-28T02:04:42Z"},{"alias_kind":"pith_short_8","alias_value":"PLQRT3ZX","created_at":"2026-05-28T02:04:42Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:PLQRT3ZXKC3THRZLTL73F3HHOH","target":"record","payload":{"canonical_record":{"source":{"id":"2507.19288","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-07-25T14:02:03Z","cross_cats_sorted":["math-ph","math.MP"],"title_canon_sha256":"2f8a3b7c8cdf1ba4d734efbca69c90147f12b8ad4ebcbba89e5eaa7e086fdc2a","abstract_canon_sha256":"0a5441b35ebc319e88859440634f228b75210504af988d23948e286aa6c205b5"},"schema_version":"1.0"},"canonical_sha256":"7ae119ef3750b733c72b9affb2ece771ce6ec76de8d07680d0d759f6db57a1ef","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-28T02:04:42.103812Z","signature_b64":"NhMuHVtpNKf5lpIeVfG2zuUPXlVh+jH25YKe7ITIytWy1HnFLmN/XV8K1Ss3L3jIqVu0h61vsnJQu9E03ARCBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7ae119ef3750b733c72b9affb2ece771ce6ec76de8d07680d0d759f6db57a1ef","last_reissued_at":"2026-05-28T02:04:42.103292Z","signature_status":"signed_v1","first_computed_at":"2026-05-28T02:04:42.103292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2507.19288","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-28T02:04:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Rk7MDzabiRFce5qSHEtUOlP4S0VIqrUXMe4CFc0mEoB1TKLgEAsGv3kR+21bPpkPUHfM/XIsUPGR+nZKteQFDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T18:37:58.759084Z"},"content_sha256":"599a7064996bbda7ee1b9034c74bc8302bdcbcc4c7b8c186ea73c6cf0d69866b","schema_version":"1.0","event_id":"sha256:599a7064996bbda7ee1b9034c74bc8302bdcbcc4c7b8c186ea73c6cf0d69866b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:PLQRT3ZXKC3THRZLTL73F3HHOH","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Decay of connection probability in high-dimensional continuum percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Matthew Dickson, Yucheng Liu","submitted_at":"2025-07-25T14:02:03Z","abstract_excerpt":"We study a percolation model on $\\mathbb R^d$ called the random connection model. For $d$ large, we use the lace expansion to prove that the critical two-point connection probability decays like $|x|^{-(d-2)}$ as $|x| \\to \\infty$, with possible anisotropic decay. Our proof also applies to nearest-neighbour Bernoulli percolation on $\\mathbb Z^d$ in $d \\ge 11$ and simplifies considerably the proof given by Hara in 2008. The method is based on the recent deconvolution strategy of Liu and Slade and uses an $L^p$ version of Hara's induction argument."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.19288","kind":"arxiv","version":2},"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/2507.19288/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-28T02:04:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"vO/J2OPOLU7J2nkhMvqtIvZcTmwzWcVZ3OgVi+zdke/KVzgv04zas74m22w6qAMgqSmogXChqxQCfhgVjEkTCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T18:37:58.759480Z"},"content_sha256":"f83fcbd4fe50c839d0332e6d77d5237d7bbe32a082a2a2c4527b3c7e922fc3ae","schema_version":"1.0","event_id":"sha256:f83fcbd4fe50c839d0332e6d77d5237d7bbe32a082a2a2c4527b3c7e922fc3ae"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/PLQRT3ZXKC3THRZLTL73F3HHOH/bundle.json","state_url":"https://pith.science/pith/PLQRT3ZXKC3THRZLTL73F3HHOH/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/PLQRT3ZXKC3THRZLTL73F3HHOH/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-02T18:37:58Z","links":{"resolver":"https://pith.science/pith/PLQRT3ZXKC3THRZLTL73F3HHOH","bundle":"https://pith.science/pith/PLQRT3ZXKC3THRZLTL73F3HHOH/bundle.json","state":"https://pith.science/pith/PLQRT3ZXKC3THRZLTL73F3HHOH/state.json","well_known_bundle":"https://pith.science/.well-known/pith/PLQRT3ZXKC3THRZLTL73F3HHOH/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:PLQRT3ZXKC3THRZLTL73F3HHOH","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":"0a5441b35ebc319e88859440634f228b75210504af988d23948e286aa6c205b5","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-07-25T14:02:03Z","title_canon_sha256":"2f8a3b7c8cdf1ba4d734efbca69c90147f12b8ad4ebcbba89e5eaa7e086fdc2a"},"schema_version":"1.0","source":{"id":"2507.19288","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2507.19288","created_at":"2026-05-28T02:04:42Z"},{"alias_kind":"arxiv_version","alias_value":"2507.19288v2","created_at":"2026-05-28T02:04:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2507.19288","created_at":"2026-05-28T02:04:42Z"},{"alias_kind":"pith_short_12","alias_value":"PLQRT3ZXKC3T","created_at":"2026-05-28T02:04:42Z"},{"alias_kind":"pith_short_16","alias_value":"PLQRT3ZXKC3THRZL","created_at":"2026-05-28T02:04:42Z"},{"alias_kind":"pith_short_8","alias_value":"PLQRT3ZX","created_at":"2026-05-28T02:04:42Z"}],"graph_snapshots":[{"event_id":"sha256:f83fcbd4fe50c839d0332e6d77d5237d7bbe32a082a2a2c4527b3c7e922fc3ae","target":"graph","created_at":"2026-05-28T02:04:42Z","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/2507.19288/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We study a percolation model on $\\mathbb R^d$ called the random connection model. For $d$ large, we use the lace expansion to prove that the critical two-point connection probability decays like $|x|^{-(d-2)}$ as $|x| \\to \\infty$, with possible anisotropic decay. Our proof also applies to nearest-neighbour Bernoulli percolation on $\\mathbb Z^d$ in $d \\ge 11$ and simplifies considerably the proof given by Hara in 2008. The method is based on the recent deconvolution strategy of Liu and Slade and uses an $L^p$ version of Hara's induction argument.","authors_text":"Matthew Dickson, Yucheng Liu","cross_cats":["math-ph","math.MP"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-07-25T14:02:03Z","title":"Decay of connection probability in high-dimensional continuum percolation"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.19288","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":"sha256:599a7064996bbda7ee1b9034c74bc8302bdcbcc4c7b8c186ea73c6cf0d69866b","target":"record","created_at":"2026-05-28T02:04:42Z","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":"0a5441b35ebc319e88859440634f228b75210504af988d23948e286aa6c205b5","cross_cats_sorted":["math-ph","math.MP"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2025-07-25T14:02:03Z","title_canon_sha256":"2f8a3b7c8cdf1ba4d734efbca69c90147f12b8ad4ebcbba89e5eaa7e086fdc2a"},"schema_version":"1.0","source":{"id":"2507.19288","kind":"arxiv","version":2}},"canonical_sha256":"7ae119ef3750b733c72b9affb2ece771ce6ec76de8d07680d0d759f6db57a1ef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7ae119ef3750b733c72b9affb2ece771ce6ec76de8d07680d0d759f6db57a1ef","first_computed_at":"2026-05-28T02:04:42.103292Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-28T02:04:42.103292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NhMuHVtpNKf5lpIeVfG2zuUPXlVh+jH25YKe7ITIytWy1HnFLmN/XV8K1Ss3L3jIqVu0h61vsnJQu9E03ARCBw==","signature_status":"signed_v1","signed_at":"2026-05-28T02:04:42.103812Z","signed_message":"canonical_sha256_bytes"},"source_id":"2507.19288","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:599a7064996bbda7ee1b9034c74bc8302bdcbcc4c7b8c186ea73c6cf0d69866b","sha256:f83fcbd4fe50c839d0332e6d77d5237d7bbe32a082a2a2c4527b3c7e922fc3ae"],"state_sha256":"8e1cae866a6b6038f4ca68d22ed4b50ea992b96de4faaeadd258a7d51e1a6b77"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"e0/BYEimTHFJGD9Pay9ko/LSv+C9NLn7nqhGe62xchCk1zvPL6K8U7kAdS6OQt2gYPMMuW1J0xwy2Uw+AUO4CA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T18:37:58.761857Z","bundle_sha256":"4c9bd87bc5b3af8a0864ff41a37a4309a771dcaff6c17cd71668e9d6602209b6"}}