{
  "abstract": "Let $\\mathbb Z_n$ be the cyclic group equipped with the uniform probability measure $π$, and let $A_{ψ_n}$ be the Laplacian with word length $$ ψ_n(k) = \\min(k,n-k). $$ For every $n\\ge4$, we prove the sharp log-Sobolev inequality $$ \\text{Ent}_π(|f|^2) \\le 2π(\\bar{f}A_{ψ_n} f), \\qquad f:\\mathbb Z_n \\to \\mathbb{C}, $$ where $\\text{Ent}_π$ is the relative entropy with respect to $π$. Equivalently, the Poisson semigroup $P_t=e^{-tA_{ψ_n}}$ satisfies the optimal hypercontractivity. The proof is inspired by the recent work of Frank and Ivanisvili [FI26] on a sharp log-Sobolev inequality for the nearest-neighbor simple random walk. Similar arguments yield a simple proof of Weissler's sharp log-Sobolev inequality for the Poisson semigroup on the circle \\cite{Weissler1980}. The same inequalities were independently obtained by Yao~\\cite{Yao2026} using a different method. We also prove quantitative stability estimates. For $n\\ge 4$ and $f:\\mathbb Z_n\\to[0,\\infty)$ with $π(f^2)=1$, $$ 2π(fA_{ψ_n}f)-\\operatorname{Ent}_π(f^2) \\ge \\frac1{12}\\|f-1\\|_{L^2(π)}^4, $$ with coefficient $1/(12d)$ on products $(\\mathbb Z_n)^d$. The quartic order and the $d^{-1}$ dependence are optimal.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.02847",
  "author_total": 2,
  "authors": [
    {
      "name": "Xinyuan Xie"
    },
    {
      "name": "Haonan Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3vd1dhgw81057.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.02847",
  "fulltext_url": null,
  "id": "3vd1dhgw81057",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.02847",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CA"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2604|Geometric Analysis and Curvature Flows"
    ]
  ],
  "openalex": "W7163422559",
  "pdf_url": "https://arxiv.org/pdf/2606.02847",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-31",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3vd1dhgw81057.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:48Z",
      "native_id": "2606.02847",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.02847"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp log-Sobolev inequalities and quartic stability on finite cyclic groups",
  "title_text": "Sharp log-Sobolev inequalities and quartic stability on finite cyclic groups",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:48Z",
  "url": "https://research.draconitic.com/paper/3vd1dhgw81057",
  "venue": "arXiv",
  "versions": []
}