{
  "abstract": "Covariance-sensitive analyses of Gaussian annealing for sampling from a convex body require controlling how much covariance can grow under a radial Gaussian tilt. For an isotropic convex body $K\\subseteq\\mathbb{R}^n$, let $μ_{K,t} (\\mathrm{d} x) \\propto e^{-t\\| x \\| ^2} \\mathbb{1}_K(x)\\,\\mathrm{d} x$, and let $Q_n$ be the supremum of $\\|\\operatorname{Cov}(μ_{K,t})\\|_{\\mathrm{op}}$ over all such $K$ and all $t>0$. We prove the sharp bound $Q_n=Θ(n^{2/5})$, closing the gap between the known $Ω(n^{1/3})$ lower bound and the $O(\\sqrt{n\\log(en)})$ upper bound. The upper bound applies not only to uniform measures on convex bodies but to every compactly supported isotropic logconcave probability measure. It combines a dimension-free variance bound for quadratic forms with a Rényi comparison at a nearby time, projected moment estimates, and relative-entropy control along the Gaussian-tilt path. For the matching lower bound, we construct an explicit unconditional convex body whose axial coordinate is coupled to the transverse quadratic energy. Moderate-deviation estimates show that an appropriate tilt creates directional variance $Ω(n^{2/5})$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08930",
  "author_total": 3,
  "authors": [
    {
      "name": "Minbo Gao"
    },
    {
      "name": "Zhengfeng Ji"
    },
    {
      "name": "Chenghua Liu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2pv1h8kgw078p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08930",
  "fulltext_url": null,
  "id": "2pv1h8kgw078p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08930",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MG"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7212087137",
  "pdf_url": "https://arxiv.org/pdf/2609.08930",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2pv1h8kgw078p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2609.08930",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08930"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.FA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.MG",
      "subfield": "geometry-topology"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Optimal Covariance Inflation under Gaussian Tilts",
  "title_text": "Optimal Covariance Inflation under Gaussian Tilts",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/2pv1h8kgw078p",
  "venue": "arXiv",
  "versions": []
}