{
  "abstract": "We establish the normalized covariance bound conjectured by Anari et al. (2026, Conjecture 3) for fixed-rank external-field measures. Let $d\\ge2$ and $m\\in[d-1]$. For $w\\in(0,+\\infty)^d$, let $\\mathsf S$ be an $m$-subset of $[d]$ with rank-$m$ external-field law $\\mathbb P(\\mathsf S=S)=\\frac{\\prod_{i\\in S}w_i}{e_m(w)},S\\subseteq[d],|S|=m,$ where $e_m(w):=\\sum_{T\\subseteq[d], |T|=m}\\prod_{\\ell\\in T}w_\\ell$ is the $m$th elementary symmetric polynomial in $w_1,\\dots,w_d$. Let $X:=(X_1,\\dots,X_d)^\\top$ be its indicator vector, i.e., $X_i=\\mathbb I\\{i\\in\\mathsf S\\},i\\in\\{1,\\dots,d\\}.$ Let $Σ:=\\mathrm{Cov}(X)$, put $v_i:=Σ_{ii}$ for each $i\\in[d]$, and define $v:=(v_1,\\dots,v_d)^\\top,D:=\\mathrm{diag}(v),V:=\\sum_{i=1}^dv_i.$ We prove $Σ\\succeq D-\\frac{vv^\\top}{V}.$ This improves the coefficient $1/2$ in the first version of our work (Cesari and Colomboni, 2026, Corollary 1.3) to the optimal universal value $1$. As a first corollary, we improve the coefficient in the pseudoinverse bound of Bacchiocchi et al. (2026, Lemma 2) from $2$ to the optimal value $1$. Specializing this bound to coordinate differences gives an alternative proof of our effective-resistance theorem from the first version of our work (Cesari and Colomboni, 2026, Theorem 1.1). The framework of Anari et al. (2026, Theorem 1 and Corollary 2) also yields unconditional correlated-sampling guarantees with stretch $6$ on the hypersimplex and $12$ on its at-most variant. Unconditional constant-stretch guarantees were first established in the first version of our work (Cesari and Colomboni, 2026, Corollaries 1.4 and 1.5), with constants $16$ and $32$, which we improve here to $6$ and $12$. These improved constants strengthen the positive resolution, established in the first version of our work, of the constant-stretch question posed by Naor et al. (2026, Theorem 2 and Section 5).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.13990",
  "author_total": 2,
  "authors": [
    {
      "name": "Tommaso Cesari"
    },
    {
      "name": "Roberto Colomboni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ctw2zrzv7w60e.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.13990",
  "fulltext_url": null,
  "id": "ctw2zrzv7w60e",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.13990",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7169091321",
  "pdf_url": "https://arxiv.org/pdf/2607.13990",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ctw2zrzv7w60e.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:01Z",
      "native_id": "2607.13990",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.13990"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Sharp Normalized Covariance Bounds and Constant-Stretch Correlated Sampling on the Hypersimplex",
  "title_text": "Sharp Normalized Covariance Bounds and Constant-Stretch Correlated Sampling on the Hypersimplex",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:01Z",
  "url": "https://research.draconitic.com/paper/ctw2zrzv7w60e",
  "venue": "arXiv",
  "versions": []
}