{"paper":{"title":"Critical two-point functions for long-range statistical-mechanical models in high dimensions","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":"2012-04-05T11:08:24Z","abstract_excerpt":"We consider long-range self-avoiding walk, percolation and the Ising model on $\\mathbb{Z}^d$ that are defined by power-law decaying pair potentials of the form $D(x)\\asymp|x|^{-d-\\alpha}$ with $\\alpha>0$. The upper-critical dimension $d_{\\mathrm{c}}$ is $2(\\alpha\\wedge2)$ for self-avoiding walk and the Ising model, and $3(\\alpha\\wedge2)$ for percolation. Let $\\alpha\\ne2$ and assume certain heat-kernel bounds on the $n$-step distribution of the underlying random walk. We prove that, for $d>d_{\\mathrm{c}}$ (and the spread-out parameter sufficiently large), the critical two-point function $G_{p_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1204.1180","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"}