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We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance $r$ as $r^{-(d+\\alpha)}$ with $\\alpha \\in (0,2)$. The upper critical dimension is $d_c=2\\alpha$. For $\\epsilon >0$, and $\\alpha = \\frac 12 (d+\\epsilon)$, the dimension $d=d_c-\\epsilon$ is below the upper critical dimension. For small $\\epsilon$, weak coupling, and all integers $n \\ge 0$, we prove t"},"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":"1705.08540","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math-ph","submitted_at":"2017-05-23T21:22:29Z","cross_cats_sorted":["math.MP","math.PR"],"title_canon_sha256":"f8fea18de5e9b4c0dfda0e360611b4c9d47d80a2b579aa1dd4f66fd41a30a771","abstract_canon_sha256":"3ebf67caa1c46805c0b13e838c1ca2db305c72d88667fb2a682921121080132f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:28:55.608226Z","signature_b64":"HhMIJsse3SeMut4QVM+qsaRRAs4QzcP4x6XZlftVsDgeubsNY0DsRbItf+MCtsFAP4fB9WLmC8rZv/287j0oAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ce564ce182c040c17b52c514402a2d5fc7589e3b390481654936dfd0779b428","last_reissued_at":"2026-05-18T00:28:55.607717Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:28:55.607717Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Critical two-point function for long-range $O(n)$ models below the upper critical dimension","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP","math.PR"],"primary_cat":"math-ph","authors_text":"Benjamin C. Wallace, Gordon Slade, Martin Lohmann","submitted_at":"2017-05-23T21:22:29Z","abstract_excerpt":"We consider the $n$-component $|\\varphi|^4$ lattice spin model ($n \\ge 1$) and the weakly self-avoiding walk ($n=0$) on $\\mathbb{Z}^d$, in dimensions $d=1,2,3$. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance $r$ as $r^{-(d+\\alpha)}$ with $\\alpha \\in (0,2)$. The upper critical dimension is $d_c=2\\alpha$. For $\\epsilon >0$, and $\\alpha = \\frac 12 (d+\\epsilon)$, the dimension $d=d_c-\\epsilon$ is below the upper critical dimension. 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