{
  "abstract": "Consider the bipartite Erdős--Rényi graph $\\mathbb{G}(n,m,d/\\sqrt{mn})$ at the critical sparsity $d = b\\log(N)$ for some $b > 0$ with $N = n + m$, and constant aspect ratio $γ= n/m \\geq 1$. We show that the spectrum of its centered and rescaled bi-adjacency matrix, whose limiting law is the Marčenko--Pastur (MP) distribution, is split into two phases: in the bulk, singular vectors are completely delocalized, while near the edges, singular vectors are semilocalized, with their mass concentrating on small disjoint balls, centered at vertices with extreme degrees. As $b$ decreases, we identify sharp thresholds for the emergence of outlier singular values. For explicit $b_{\\rm R}^{\\star}$ and $b_{\\rm L}^{\\star}$ as functions of $γ$, we prove that when $b > b_{\\rm R}^{\\star}$, there is no outlier outside the bulk; when $b_{\\rm L}^{\\star}< b < b_{\\rm R}^{\\star}$, only right outliers emerge; and when $b < b_{\\rm L}^{\\star}$, outliers are present on both sides, all with high probability. This removes an extra technical condition from our previous work arxiv:2508.01456, providing a complete characterization of the parameter space. In sparser regimes, in terms of left outliers, we provide some nontrivial generating mechanisms, demonstrating that local graph structure plays a key role in understanding extreme singular values. Our proofs rely on two technical inputs: (1) an entrywise local law in the MP bulk, where the deterministic equivalent of the resolvent depends on both variance profiles and normalized vertex degrees, different from the Gaussian setting; (2) spectral radius upper bounds for nonbacktracking matrices constructed from regularized random graphs, recently developed in arXiv: 2609.26615. Our results reveal fruitful phenomena for sparse random rectangular matrices at criticality, which are beyond the universality class of their Gaussian counterparts.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.22658",
  "author_total": 2,
  "authors": [
    {
      "name": "Ioana Dumitriu"
    },
    {
      "name": "Hai-Xiao Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/dywy7ksqg8vf6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.22658",
  "fulltext_url": null,
  "id": "dywy7ksqg8vf6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.22658",
  "language": null,
  "licence": "CC-BY-NC-ND-4.0",
  "licence_label": "CC BY-NC-ND 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-nd/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.NA"
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      "arxiv",
      "math.CO"
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      "arxiv",
      "math.NA"
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    [
      "arxiv",
      "math.PR"
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      "arxiv",
      "math.ST"
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    [
      "openalex",
      "2613|Random Matrices and Applications"
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  ],
  "openalex": "W7222957322",
  "pdf_url": "https://arxiv.org/pdf/2610.22658",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/dywy7ksqg8vf6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:30:37Z",
      "native_id": "2610.22658",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.22658"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.NA",
      "subfield": "numerical-analysis"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.ST",
      "subfield": "statistics"
    }
  ],
  "title": "Singular vectors of sparse random rectangular matrices: Transition between delocalized and semilocalized phases",
  "title_text": "Singular vectors of sparse random rectangular matrices: Transition between delocalized and semilocalized phases",
  "type": "preprint",
  "updated_at": "2026-10-30T12:33:38Z",
  "url": "https://research.draconitic.com/paper/dywy7ksqg8vf6",
  "venue": "arXiv",
  "versions": []
}