{"paper":{"title":"Fluctuations of the Sherrington--Kirkpatrick free energy at critical temperature","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cond-mat.dis-nn","math-ph","math.MP"],"primary_cat":"math.PR","authors_text":"Brice Huang, Hang Du","submitted_at":"2026-07-02T13:44:13Z","abstract_excerpt":"We consider the Sherrington--Kirkpatrick spin glass model at the critical inverse temperature $\\beta = 1$ with zero external field. We prove that the free energy $F_N = F_{N,\\beta=1}$ of this model has variance\n  \\[\n  \\mathrm{Var}(F_N) = \\frac16 \\log N + O(1)\\,,\n  \\] confirming a physics prediction of Aspelmeier \\cite{aspelmeier2008free}, and that the centered and scaled $F_N$ satisfies a Gaussian CLT. We also identify the critical two-replica overlap scale, proving\n  \\[\n  \\mathbb{E} \\langle R_{1,2}^2\\rangle \\asymp N^{-2/3}\\,,\n  \\] as conjectured by Talagrand \\cite{talagrand2011mean2}, togethe"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02172","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.02172/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}