{
  "abstract": "Let $A_1,\\ldots,A_n$ be independent $d \\times d$ real symmetric Gaussian random matrices, and consider the linear operator $A(x) = n^{-1/2}\\sum_{i=1}^n x_i A_i$, $x\\in \\mathbb{R}^n$. We construct an iterative algorithm in the Approximate Message Passing family which iterates over $A$ and its adjoint $A^*$, and establish a state evolution result which characterizes its behavior in the limit $d\\rightarrow \\infty, 2n/d^2 \\rightarrow α$ in terms of a correlated Gaussian-semicircular process in a free probability space, in the sense of strong convergence of operators. We then apply this iteration to the random matrix discrepancy problem which asks for a binary vector $x \\in \\{-1,+1\\}^n$ such that $A(x)$ has a small operator norm. Our algorithm achieves an operator norm $2σ(α)$, for an explicit expression of the standard deviation $σ(α)<1$ for all $0<α<α_* \\simeq 5.74$. This resolves the algorithmic question of Kunisky-Zhang (2023) and Maillard (2025) in this interval.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11836",
  "author_total": 2,
  "authors": [
    {
      "name": "August Y. Chen"
    },
    {
      "name": "Ahmed El Alaoui"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/axe9dt48n1rbr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11836",
  "fulltext_url": null,
  "id": "axe9dt48n1rbr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11836",
  "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",
      "cs.DS"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7212233199",
  "pdf_url": "https://arxiv.org/pdf/2609.11836",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/axe9dt48n1rbr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11836",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11836"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Free-Probabilistic State Evolution and Random Matrix Discrepancy",
  "title_text": "Free-Probabilistic State Evolution and Random Matrix Discrepancy",
  "type": "preprint",
  "updated_at": "2026-11-06T18:21:01Z",
  "url": "https://research.draconitic.com/paper/axe9dt48n1rbr",
  "venue": "arXiv",
  "versions": []
}