{
  "abstract": "We study $(\\mathbf{O}^\\top \\mathbf{A} \\mathbf{O} + \\mathbf{B})^{-1}$, where $\\mathbf{A}, \\mathbf{B} \\in \\mathbb{R}^{N \\times N}$ are symmetric positive definite, with limiting eigenvalue distributions ${\\rm F}_{\\mathbf{A}}, {\\rm F}_{\\mathbf{B}}$ and spectra bounded and bounded away from zero, and $\\mathbf{O}$ is Haar orthogonal. We analyze the deviation of deterministic-equivalent-type approximation $\\boldsymbolΔ\\doteq (\\mathbf{O}^\\top \\mathbf{A} \\mathbf{O} + \\mathbf{B})^{-1}- \\mathbf{S}^{-1}$ where $\\mathbf{S} = \\mathbf{B} - z^\\star \\mathbf{I}$. First, let $\\mathbf{O}$ be independent of $(\\mathbf{A}, \\mathbf{B}, \\mathbf{u})$, $\\|\\mathbf{u}\\| = 1$. There is sequence of a standard Gaussian vectors $\\mathbf{g}_N$ independent of $(\\mathbf{A}, \\mathbf{B}, \\mathbf{u})$ with $\\| \\mathbf{S} \\boldsymbolΔ \\mathbf{S} \\mathbf{u} - \\sqrt{\\frac τN}\\, \\mathbf{g}_N \\| \\to 0$ a.s. Here $z^\\star = -{\\rm R}(-χ)$ and $τ= {\\rm R}'(-χ)(1 + η{\\rm R}'(-χ))$, where ${\\rm R}$ is the R-transform of ${\\rm F}_{\\mathbf{A}}$ and $(χ, η)$ are the first two inverse moments of the free additive convolution of ${\\rm F}_{\\mathbf{A}}, {\\rm F}_{\\mathbf{B}}$. Let $\\mathbf{K} = \\sum_{i \\le \\lfloor N^q \\rfloor} λ_i \\mathbf{u}_i \\mathbf{v}_i^\\top$, $q \\in [0, C]$, with unit-norm, not necessarily orthogonal $\\mathbf{u}_i, \\mathbf{v}_i$ and $\\limsup_N \\max_i |λ_i| < \\infty$ a.s. If $\\mathbf{O}$ is independent of $(\\mathbf{A}, \\mathbf{B}, \\{\\mathbf{u}_i, \\mathbf{v}_i\\})$, then $N^{-q} {\\rm tr}(\\mathbf{K} \\boldsymbolΔ) \\to 0$ a.s. The proofs use a dynamical mean-field approach: the generating functions of the free Appell polynomials give a linear dynamics approximating the quantity of interest in the iterated large-$N$, large-time limit, whose fluctuations we analyze.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21140",
  "author_total": 2,
  "authors": [
    {
      "name": "Burak Çakmak"
    },
    {
      "name": "Manfred Opper"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/edvcpma64hkkm.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21140",
  "fulltext_url": null,
  "id": "edvcpma64hkkm",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21140",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7213943837",
  "pdf_url": "https://arxiv.org/pdf/2609.21140",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/edvcpma64hkkm.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21140",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21140"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Inverse of a Sum of Random Matrices: A Dynamical Mean-Field Approach",
  "title_text": "Inverse of a Sum of Random Matrices: A Dynamical Mean-Field Approach",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/edvcpma64hkkm",
  "venue": "arXiv",
  "versions": []
}