{
  "abstract": "We study the eigenvalues and eigenvectors of the adjacency matrix $A$ of the Erdős-Rényi graph $\\mathbb G(N,p)$ in the regime $Np \\gg (\\log N)^2$. We establish optimal isotropic delocalization for the bulk eigenvectors $\\boldsymbol u$, meaning that $\\langle \\boldsymbol v, \\boldsymbol u\\rangle^2 \\leq \\frac{C \\log N}{N}$ with very high probability for any deterministic normalized $\\boldsymbol v$. In addition, we prove local spectral universality in the bulk by showing that the local spectral statistics coincide with those of the GOE. The main tool of our proof is a local law for the resolvent of $A$, down to optimal spectral scales and with optimal error bounds. Its proof relies on a new approach to local laws that we call the Bernoulli flow. It is a characteristic flow method in which the usual Brownian process is replaced by a Bernoulli process, where each edge of the graph forms an independent Markov process that jumps at unit rate from closed to open. The spectral parameter flows according to a suitably constructed matrix-valued Bernoulli characteristic flow.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.15566",
  "author_total": 2,
  "authors": [
    {
      "name": "Joscha Henheik"
    },
    {
      "name": "Antti Knowles"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/azbsejkqp084t.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.15566",
  "fulltext_url": null,
  "id": "azbsejkqp084t",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.15566",
  "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": "W7213345187",
  "pdf_url": "https://arxiv.org/pdf/2609.15566",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/azbsejkqp084t.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:25Z",
      "native_id": "2609.15566",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.15566"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Bernoulli flow for Erdős-Rényi graphs",
  "title_text": "Bernoulli flow for Erdős-Rényi graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:29Z",
  "url": "https://research.draconitic.com/paper/azbsejkqp084t",
  "venue": "arXiv",
  "versions": []
}