{
  "abstract": "We prove a variational formula for the Fuglede-Kadison determinant of a self-adjoint matrix semicircle $S_a = a \\otimes 1 + \\sum_j A_j \\otimes s_j$ with deterministic mean $a$. For every real $z$, $\\log Δ(S_a - z) = -1/2 + \\inf_b F_z(b)$, where the infimum is over positive definite matrices $b$ such that the map $X \\mapsto bXb - \\sum_j A_j X A_j$ is positive semidefinite, and $F_z$ is an explicit geodesically convex function. Both sides equal $-\\infty$ exactly when $z$ is an atom of the spectral measure. The formula was proposed, without proof, in a recent post of Speicher. In the centered case $a=0$ it gives a new proof of the Mai-Speicher capacity formula, and by hermitization it yields a finite-dimensional variational formula for the logarithmic potential of the Brown measure of an arbitrary matrix-valued semicircular element. The proof identifies the determinant with the value of the Dyson action at its critical point and rewrites this critical-point problem as a saddle problem in three matrix variables. For a spectral parameter $z + i\\varepsilon$ in the upper half-plane, eliminating two of the variables gives a geodesically convex minimization whose global minimizer is the positive factor $b_\\varepsilon$ in a symmetric unitary factorization of the matrix Cauchy transform. The formula on the real axis follows by letting $\\varepsilon \\to 0$ in the optimal values, with no compactness assumption on the minimizers. At points where the boundary value of the Cauchy transform is nondegenerate, the scales $b_\\varepsilon$ converge and the infimum is attained at their limit; this minimizer lies on the boundary of the feasible set when $z$ is in the support of the spectral measure and in its interior when $z$ is in the resolvent set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.15180",
  "author_total": 1,
  "authors": [
    {
      "name": "Vladislav Kargin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/640y2h6pp0gne.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.15180",
  "fulltext_url": null,
  "id": "640y2h6pp0gne",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.15180",
  "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.OA"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7221814283",
  "pdf_url": "https://arxiv.org/pdf/2610.15180",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/640y2h6pp0gne.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.15180",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.15180"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.OA",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Fuglede--Kadison Determinants of Shifted Matrix Semicircles",
  "title_text": "Fuglede--Kadison Determinants of Shifted Matrix Semicircles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/640y2h6pp0gne",
  "venue": "arXiv",
  "versions": []
}