{"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. For small $\\epsilon$, weak coupling, and all integers $n \\ge 0$, we prove t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.08540","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"}