{
  "abstract": "The eigenvalues of a large normal random matrix form a two-dimensional Coulomb gas. At fixed weight, the spectral droplet grows with the matrix rank, and its semiclassical density increment is harmonic measure. We freeze an $N$-eigenvalue configuration and sample $M$ further eigenvalues using the projection onto polynomials of degree less than $N+M$ that vanish at the frozen points. Their conditional law is an exact polynomial ensemble in the potential of the frozen charges. For a regular analytic droplet, we prove that the spatial distribution of the new eigenvalues converges to harmonic measure on the old boundary when $M/\\log N\\to\\infty$ and $M/N\\to0$. This convergence holds conditionally on the frozen spectrum, outside a set of frozen configurations whose probability tends to zero. The equilibrium process formed from orthogonal polynomials of degrees $N,\\ldots,N+M-1$ has the same limit. An exact finite-rank identity relates polynomial moments of the conditional mean to fluctuations of the frozen spectrum, with a remainder from polynomial modes of degree $N+M$ and above.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03912",
  "author_total": 1,
  "authors": [
    {
      "name": "Oleg Alekseev"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3fwvcrzt98gzd.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03912",
  "fulltext_url": null,
  "id": "3fwvcrzt98gzd",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03912",
  "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.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7220742659",
  "pdf_url": "https://arxiv.org/pdf/2610.03912",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3fwvcrzt98gzd.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:27:01Z",
      "native_id": "2610.03912",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03912"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Quenched Growth of Normal Matrices and Harmonic Measure",
  "title_text": "Quenched Growth of Normal Matrices and Harmonic Measure",
  "type": "preprint",
  "updated_at": "2026-10-30T12:27:10Z",
  "url": "https://research.draconitic.com/paper/3fwvcrzt98gzd",
  "venue": "arXiv",
  "versions": []
}