{"paper":{"title":"Sharp Log-Sobolev Inequalities for Finite Cyclic Groups with Word-Length","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.FA","math.PR"],"primary_cat":"math.CA","authors_text":"Haonan Zhang, Xinyuan Xie","submitted_at":"2026-06-01T20:11:29Z","abstract_excerpt":"Let $\\mathbb Z_n$ be the cyclic group equipped with the uniform probability measure $\\pi$, and let $-A_{\\psi_n}$ be the Laplacian with respect to the word length $\n  \\psi_n(k) = \\min(k,n-k). $ We prove the sharp log-Sobolev inequality $$\n  \\operatorname{Ent}_{\\pi}(f^2)\n  \\le 2\\pi\\bigl(f A_{\\psi_n} f\\bigr),\n  \\qquad f:\\mathbb Z_n \\to \\mathbb C, $$ for every $n \\ge 4$. The proof is inspired by the recent work of Frank and Ivanisvili~\\cite{FrankIvanisvili2026} on a sharp log-Sobolev inequality for nearest-neighbor simple random walk. We use their idea of cubic-majorant reduction, but replace thei"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.02847","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/2606.02847/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"}