{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:SJGEZP2YKCOAZFGQDM5IC7F7KL","short_pith_number":"pith:SJGEZP2Y","schema_version":"1.0","canonical_sha256":"924c4cbf58509c0c94d01b3a817cbf52e7516edb69603ab4469ffd102ee11293","source":{"kind":"arxiv","id":"1808.06789","version":3},"attestation_state":"computed","paper":{"title":"Critical two-point function for long-range models with power-law couplings: The marginal case for $d\\ge d_c$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Akira Sakai, Lung-Chi Chen","submitted_at":"2018-08-21T07:13:31Z","abstract_excerpt":"Consider the long-range models on $\\mathbb{Z}^d$ of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as $|x|^{-d-\\alpha}$ for some $\\alpha>0$. In the previous work (Ann. Probab., 43, 639--681, 2015), we have shown in a unified fashion for all $\\alpha\\ne2$ that, assuming a bound on the \"derivative\" of the $n$-step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function $G_{p_c}(x)$ decays as $|x|^{\\alpha\\wedge2-d}$ above the upper-critical dimension $d"},"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":"1808.06789","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2018-08-21T07:13:31Z","cross_cats_sorted":["math.MP","math.PR"],"title_canon_sha256":"955ff88c76405a7105f436ee9b398cb94235ebf0878ec890c912e0ffa0519c58","abstract_canon_sha256":"9e54a61c530b24ada827e953af24447f52fdcfd32c3cefb30f80687ed17857c4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:50:24.346954Z","signature_b64":"TtElNOyjI42SQQEdnbPTcR2HgmSp9AVNjUDFOShl69dnHmM6KUJawu5RiS45cotSNv1HnuNtzqUoNM1WB1OdAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"924c4cbf58509c0c94d01b3a817cbf52e7516edb69603ab4469ffd102ee11293","last_reissued_at":"2026-05-17T23:50:24.346208Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:50:24.346208Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical two-point function for long-range models with power-law couplings: The marginal case for $d\\ge d_c$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Akira Sakai, Lung-Chi Chen","submitted_at":"2018-08-21T07:13:31Z","abstract_excerpt":"Consider the long-range models on $\\mathbb{Z}^d$ of random walk, self-avoiding walk, percolation and the Ising model, whose translation-invariant 1-step distribution/coupling coefficient decays as $|x|^{-d-\\alpha}$ for some $\\alpha>0$. In the previous work (Ann. Probab., 43, 639--681, 2015), we have shown in a unified fashion for all $\\alpha\\ne2$ that, assuming a bound on the \"derivative\" of the $n$-step distribution (the compound-zeta distribution satisfies this assumed bound), the critical two-point function $G_{p_c}(x)$ decays as $|x|^{\\alpha\\wedge2-d}$ above the upper-critical dimension $d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1808.06789","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":""},"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":"1808.06789","created_at":"2026-05-17T23:50:24.346335+00:00"},{"alias_kind":"arxiv_version","alias_value":"1808.06789v3","created_at":"2026-05-17T23:50:24.346335+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1808.06789","created_at":"2026-05-17T23:50:24.346335+00:00"},{"alias_kind":"pith_short_12","alias_value":"SJGEZP2YKCOA","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"SJGEZP2YKCOAZFGQ","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"SJGEZP2Y","created_at":"2026-05-18T12:32:53.628368+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/SJGEZP2YKCOAZFGQDM5IC7F7KL","json":"https://pith.science/pith/SJGEZP2YKCOAZFGQDM5IC7F7KL.json","graph_json":"https://pith.science/api/pith-number/SJGEZP2YKCOAZFGQDM5IC7F7KL/graph.json","events_json":"https://pith.science/api/pith-number/SJGEZP2YKCOAZFGQDM5IC7F7KL/events.json","paper":"https://pith.science/paper/SJGEZP2Y"},"agent_actions":{"view_html":"https://pith.science/pith/SJGEZP2YKCOAZFGQDM5IC7F7KL","download_json":"https://pith.science/pith/SJGEZP2YKCOAZFGQDM5IC7F7KL.json","view_paper":"https://pith.science/paper/SJGEZP2Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1808.06789&json=true","fetch_graph":"https://pith.science/api/pith-number/SJGEZP2YKCOAZFGQDM5IC7F7KL/graph.json","fetch_events":"https://pith.science/api/pith-number/SJGEZP2YKCOAZFGQDM5IC7F7KL/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SJGEZP2YKCOAZFGQDM5IC7F7KL/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SJGEZP2YKCOAZFGQDM5IC7F7KL/action/storage_attestation","attest_author":"https://pith.science/pith/SJGEZP2YKCOAZFGQDM5IC7F7KL/action/author_attestation","sign_citation":"https://pith.science/pith/SJGEZP2YKCOAZFGQDM5IC7F7KL/action/citation_signature","submit_replication":"https://pith.science/pith/SJGEZP2YKCOAZFGQDM5IC7F7KL/action/replication_record"}},"created_at":"2026-05-17T23:50:24.346335+00:00","updated_at":"2026-05-17T23:50:24.346335+00:00"}