{
  "abstract": "Let $d>1$ be fixed and let $A_n$ be an $n\\times n$ matrix with independent $\\Ber(d/n)$ entries. For every $0<r<\\sqrt d$, we prove that, with high probability, a positive proportion of the eigenvalues of $A_n$ have modulus larger than $r$. Together with the known upper bound, this implies that the modulus of the second largest eigenvalue converges in probability to $\\sqrt d$. Our proof works with the Brown measure $μ_d$ of the adjacency operator of the directed Poisson--Galton--Watson tree. The convergence theorem of Sah, Sahasrabudhe, and Sawhney, together with the Brown-measure identification following it, gives weak convergence of the empirical spectral measure of $A_n$ to $μ_d$ in probability. We prove that the outer radius of its support is $\\sqrt d$, using a resolvent recursion on Poisson--Galton--Watson trees.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.18369",
  "author_total": 2,
  "authors": [
    {
      "name": "Simon Coste"
    },
    {
      "name": "Yizhe Zhu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/661hzm2gg5y2f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.18369",
  "fulltext_url": null,
  "id": "661hzm2gg5y2f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.18369",
  "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"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2613|Random Matrices and Applications"
    ]
  ],
  "openalex": "W7213511120",
  "pdf_url": "https://arxiv.org/pdf/2609.18369",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-16",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/661hzm2gg5y2f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:04Z",
      "native_id": "2609.18369",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.18369"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.SP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "The spectral edge of sparse directed Erdös-Rényi graphs",
  "title_text": "The spectral edge of sparse directed Erdös-Rényi graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:04Z",
  "url": "https://research.draconitic.com/paper/661hzm2gg5y2f",
  "venue": "arXiv",
  "versions": []
}