{
  "abstract": "We study large deviations of the spectral radius of $X_n=n^{-1/2}(ξ_{ij})_{i,j=1}^n$, where the entries are iid, centered, variance-one, and subgaussian, with zero pseudovariance in the complex case. We write $ρ(X_n)$ for its spectral radius, i.e. the largest modulus among its eigenvalues. For every fixed $r>1$, we prove $$ \\liminf_{n\\to\\infty}\\frac1n\\log\\mathbb{P}\\{ρ(X_n)>r\\} \\ge -\\frac\\beta2\\bigl(r^2-1-2\\log r\\bigr), $$ where $β=1$ for real entries and $β=2$ for complex entries. We prove a matching upper bound for real symmetric laws with Gaussian-dominated even moments and for complex laws satisfying the sharp Gaussian Laplace-transform bound. These classes include discrete distributions. In the complex sharp class, we also identify the exponential rate for an eigenvalue to enter a fixed disk outside the unit disk. We then turn to lower deviations and establish a quadratic-speed bound. Under the additional assumption that the entry law $μ$ has a bounded density, we prove $$ \\mathbb{P}\\{ρ(X_n)\\le r\\}\\le e^{-c_{μ,r}n^2} $$ for every fixed $0<r<1$ and all sufficiently large $n$, where $c_{μ,r}>0$. For the matching upper-tail classes with a bounded density, these bounds give a full speed-$n$ large deviation principle, with the displayed upper-tail rate for $r\\ge1$ and infinite rate for $r<1$. The upper-tail proof uses a change of measure that preserves the entry support and creates an outlying eigenvalue. The lower-tail proof develops a weighted comparison for adaptive orthonormal observations and applies it to Arnoldi residuals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.05498",
  "author_total": 1,
  "authors": [
    {
      "name": "Yi Han"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/75vwrq947ejsc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.05498",
  "fulltext_url": null,
  "id": "75vwrq947ejsc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.05498",
  "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.PR"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7220737051",
  "pdf_url": "https://arxiv.org/pdf/2610.05498",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-09",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/75vwrq947ejsc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:28:40Z",
      "native_id": "2610.05498",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.05498"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Large deviations of the spectral radius of iid subgaussian random matrices",
  "title_text": "Large deviations of the spectral radius of iid subgaussian random matrices",
  "type": "preprint",
  "updated_at": "2026-10-30T12:28:40Z",
  "url": "https://research.draconitic.com/paper/75vwrq947ejsc",
  "venue": "arXiv",
  "versions": []
}