{
  "abstract": "Let $K=\\{x\\in\\R^d:Ax\\ge b\\}$ be a bounded full-dimensional polytope with $m$ inequalities. For targets on $K$ with density proportional to $\\exp(-\\vartheta^\\transpose x)$, we prove a $\\widetilde O(d^2)$ bound, measured in Markov chain iterations, for the chi-square mixing of the $1/2$-lazy Metropolis Dikin walk based on a polylogarithmically rescaled Lee--Sidford metric. The estimate holds for every initial law with finite chi-square divergence. The proof controls the reverse Gaussian proposal through a quantitative average self-concordance estimate at the required $O(1/d)$ scale. We establish this estimate from uniform analytic derivative bounds for the Lewis-weight subspace and an all-order projection-weighted contraction inequality for Gaussian chaoses. For order $j$, the latter bounds each base-point polynomial at scale $d^{j/2}$, up to $j$-dependent and logarithmic factors. Truncating at logarithmic order then absorbs the remaining pathwise loss. This resolves, in the affirmative, the open problem of whether the Gaussian Lee--Sidford Dikin walk mixes in $\\widetilde O(d^2)$ iterations: the previously stated quadratic bounds \\cite[Theorem~1.4]{LLV20}, \\cite[Theorem~1.2]{GKM24} rest on reverse-proposal steps identified as unresolved in \\cite[Section~1]{Kook26}, whose $\\widetilde O(d^{9/4})$ bound was the best rigorously established rate prior to this work. The proof is found by ChatGPT 5.6 Pro.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22374",
  "author_total": 1,
  "authors": [
    {
      "name": "Yiping Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0xwcg64fxx013.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22374",
  "fulltext_url": null,
  "id": "0xwcg64fxx013",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22374",
  "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",
      "math.PR"
    ],
    [
      "arxiv",
      "stat.CO"
    ],
    [
      "openalex",
      "2613|Markov Chains and Monte Carlo Methods"
    ]
  ],
  "openalex": "W7222915310",
  "pdf_url": "https://arxiv.org/pdf/2610.22374",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0xwcg64fxx013.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22374",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22374"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:stat.CO",
      "subfield": "statistics"
    }
  ],
  "title": "Quadratic mixing for the Lee--Sidford Dikin walk",
  "title_text": "Quadratic mixing for the Lee--Sidford Dikin walk",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:38Z",
  "url": "https://research.draconitic.com/paper/0xwcg64fxx013",
  "venue": "arXiv",
  "versions": []
}