{
  "abstract": "Consider a symmetric function $\\mathcal{C}(x,y)$ on $[0,1]\\times[0,1]$ which is twice continuously differentiable up to the boundary, and which satisfies $ \\mathcal{C}(x,y)=\\mathcal{C}(1-x,1-y)$. Let $A^{(n)} = \\big(a^{(n)}_{i,j}\\, :\\, i,j \\in [n]\\big)$ be the matrix with entries $a^{(n)}_{i,j}\\, =\\, \\exp(-\\mathcal{C}(i/n,j/n))$. Soumik Pal conjectured the asymptotics $$\\operatorname{perm}\\big(A^{(n)}\\big)/n!\\sim \\exp\\big(n Λ[\\mathcal{C}]\\big)/ \\sqrt{\\mathcal{D}[\\mathcal{C}]}$$ as $n \\to \\infty$ for known functionals that arise naturally in the context of entropy regularized optimal transport. The functional $Λ[\\mathcal{C}]$ is the known large deviation rate function, already proved rigorously by Sumit Mukherjee. It is $\\int_{0}^1 \\int_0^{1} (α(x)+β(y))\\, dx\\, dy$ where $α(x)+β(y)$ is chosen such that $ρ(x,y) := \\exp(-\\mathcal{C}(x,y)-α(x)-β(y))$ has uniform marginals. The algebraic term $\\mathcal{D}[c]$ is given by Peter McCullagh's formula for doubly stochastic matrices: $\\operatorname{det}_F(I+J-T^*T)$, the Fredholm determinant, where $I$ is the identity on $L^2([0,1])$, $Jf(x) \\equiv \\int_{0}^1 f(z)\\, dz$ (for all $x$) and $Tf(x) = \\int_0^1 ρ(x,y) f(y)\\, dy$. We prove the conjecture for functions $\\mathcal C$ that are constant on blocks, exploiting a well-known Ross Pinsky's combinatorial decomposition of permutations in blocks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2605.25274",
  "author_total": 2,
  "authors": [
    {
      "name": "Andrea Ottolini"
    },
    {
      "name": "Shannon Starr"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/e19mxhh8yszdj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2605.25274",
  "fulltext_url": null,
  "id": "e19mxhh8yszdj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2605.25274",
  "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-ph"
    ],
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7162457902",
  "pdf_url": "https://arxiv.org/pdf/2605.25274",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/e19mxhh8yszdj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:19Z",
      "native_id": "2605.25274",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2605.25274"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Pal's permanent conjecture: proof for block uniform matrices",
  "title_text": "Pal's permanent conjecture: proof for block uniform matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/e19mxhh8yszdj",
  "venue": "arXiv",
  "versions": []
}