{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2013:5SRWNGITX47TFG3P43FVKFAIH4","short_pith_number":"pith:5SRWNGIT","schema_version":"1.0","canonical_sha256":"eca3669913bf3f329b6fe6cb5514083f21e956a0a84af792df1027917dfaa9fa","source":{"kind":"arxiv","id":"1303.3995","version":2},"attestation_state":"computed","paper":{"title":"Distances in critical long range percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Allan Sly, Jian Ding","submitted_at":"2013-03-16T16:14:05Z","abstract_excerpt":"We study the long range percolation model on $\\mathbb{Z}$ where sites $i$ and $j$ are connected with probability $\\beta |i-j|^{-s}$. Graph distances are now well understood for all exponents $s$ except in the case $s=2$ where the model exhibits non-trivial self-similar scaling. Establishing a conjecture of Benjamini and Berger \\cite{BenBer:01}, we prove that the typical distance from site 0 to $n$ grows as a power law $n^{\\theta(\\beta)}$ up to a multiplicative constant for some exponent $0<\\theta(\\beta)<1$ as does the diameter of the graph on a box of length $n$."},"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":"1303.3995","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2013-03-16T16:14:05Z","cross_cats_sorted":[],"title_canon_sha256":"adbe07b3c94a7438e70e48042edc0ca0f0bcda4cd44a079bd7c76763116728ec","abstract_canon_sha256":"e42f76dfcc1ff1152898162893da010cc9ee62896629f3b813a79d3120272dcc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:27:37.162120Z","signature_b64":"hBw1Y7F3qr2sJdrAuhnFsumvqSWK/SCwnPdGUWEVoiKR735KbGXLdrdiPWkMWtVabGCmVzUT6st60NbCYpw5CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eca3669913bf3f329b6fe6cb5514083f21e956a0a84af792df1027917dfaa9fa","last_reissued_at":"2026-05-18T01:27:37.161643Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:27:37.161643Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distances in critical long range percolation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Allan Sly, Jian Ding","submitted_at":"2013-03-16T16:14:05Z","abstract_excerpt":"We study the long range percolation model on $\\mathbb{Z}$ where sites $i$ and $j$ are connected with probability $\\beta |i-j|^{-s}$. Graph distances are now well understood for all exponents $s$ except in the case $s=2$ where the model exhibits non-trivial self-similar scaling. Establishing a conjecture of Benjamini and Berger \\cite{BenBer:01}, we prove that the typical distance from site 0 to $n$ grows as a power law $n^{\\theta(\\beta)}$ up to a multiplicative constant for some exponent $0<\\theta(\\beta)<1$ as does the diameter of the graph on a box of length $n$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.3995","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":""},"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":"1303.3995","created_at":"2026-05-18T01:27:37.161721+00:00"},{"alias_kind":"arxiv_version","alias_value":"1303.3995v2","created_at":"2026-05-18T01:27:37.161721+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.3995","created_at":"2026-05-18T01:27:37.161721+00:00"},{"alias_kind":"pith_short_12","alias_value":"5SRWNGITX47T","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_16","alias_value":"5SRWNGITX47TFG3P","created_at":"2026-05-18T12:27:34.582898+00:00"},{"alias_kind":"pith_short_8","alias_value":"5SRWNGIT","created_at":"2026-05-18T12:27:34.582898+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5SRWNGITX47TFG3P43FVKFAIH4","json":"https://pith.science/pith/5SRWNGITX47TFG3P43FVKFAIH4.json","graph_json":"https://pith.science/api/pith-number/5SRWNGITX47TFG3P43FVKFAIH4/graph.json","events_json":"https://pith.science/api/pith-number/5SRWNGITX47TFG3P43FVKFAIH4/events.json","paper":"https://pith.science/paper/5SRWNGIT"},"agent_actions":{"view_html":"https://pith.science/pith/5SRWNGITX47TFG3P43FVKFAIH4","download_json":"https://pith.science/pith/5SRWNGITX47TFG3P43FVKFAIH4.json","view_paper":"https://pith.science/paper/5SRWNGIT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1303.3995&json=true","fetch_graph":"https://pith.science/api/pith-number/5SRWNGITX47TFG3P43FVKFAIH4/graph.json","fetch_events":"https://pith.science/api/pith-number/5SRWNGITX47TFG3P43FVKFAIH4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5SRWNGITX47TFG3P43FVKFAIH4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5SRWNGITX47TFG3P43FVKFAIH4/action/storage_attestation","attest_author":"https://pith.science/pith/5SRWNGITX47TFG3P43FVKFAIH4/action/author_attestation","sign_citation":"https://pith.science/pith/5SRWNGITX47TFG3P43FVKFAIH4/action/citation_signature","submit_replication":"https://pith.science/pith/5SRWNGITX47TFG3P43FVKFAIH4/action/replication_record"}},"created_at":"2026-05-18T01:27:37.161721+00:00","updated_at":"2026-05-18T01:27:37.161721+00:00"}