{
  "abstract": "Let $K\\subset\\mathbb{R}^n$ be an isotropic convex body. We prove that the hit-and-run walk, started from any $M$-warm distribution, reaches total-variation distance $\\varepsilon$ from the uniform distribution on $K$ in $O\\!\\left(n^2ψ_n^{-2}\\log^3(M/\\varepsilon)\\right)$ steps, where $ψ_n^{-1}$ is the Kannan-Lovász-Simonovits (KLS) constant. Up to logarithmic factors, this matches the best-known warm-start mixing time for the ball walk. Chen and Eldan [Discrete Comput. Geom. 2026] obtained the same $n^2ψ_n^{-2}$ dependence for hit-and-run, but with polynomial dependence on $M/\\varepsilon$. Our result improves that polynomial dependence to a polylogarithmic one, fully resolving their open question about warm-start mixing of hit-and-run in isotropic convex bodies.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.13487",
  "author_total": 1,
  "authors": [
    {
      "name": "Ruizhe Zhang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5skjmfvx0m2xf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.13487",
  "fulltext_url": null,
  "id": "5skjmfvx0m2xf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.13487",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7203469506",
  "pdf_url": "https://arxiv.org/pdf/2608.13487",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5skjmfvx0m2xf.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2608.13487",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.13487"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Hit-and-Run Mixes as Fast as the Ball Walk",
  "title_text": "Hit-and-Run Mixes as Fast as the Ball Walk",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/5skjmfvx0m2xf",
  "venue": "arXiv",
  "versions": []
}