{"paper":{"title":"Critical exponents 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":"Gordon Slade","submitted_at":"2016-11-18T17:34:32Z","abstract_excerpt":"We consider the critical behaviour of long-range $O(n)$ models ($n \\ge 0$) on ${\\mathbb Z}^d$, with interaction that decays with distance $r$ as $r^{-(d+\\alpha)}$, for $\\alpha \\in (0,2)$. For $n \\ge 1$, we study the $n$-component $|\\varphi|^4$ lattice spin model. For $n =0$, we study the weakly self-avoiding walk via an exact representation as a supersymmetric spin model. These models have upper critical dimension $d_c=2\\alpha$. For dimensions $d=1,2,3$ and small $\\epsilon>0$, we choose $\\alpha = \\frac 12 (d+\\epsilon)$, so that $d=d_c-\\epsilon$ is below the upper critical dimension. For small "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06169","kind":"arxiv","version":4},"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"}