{
  "abstract": "We study the vector Dyson equation $$-\\frac{1}{m(z)}=z\\mathbf{1}-\\mathbf{a}+Sm(z),$$ with parameter $z$ in the complex upper half-plane $\\mathbb{C}_+$, where $\\mathbf{a}\\in\\mathbb R^d$ and $S$ is a nonnegative matrix, not necessarily symmetric. This equation has a unique vector solution $m(z)\\in\\mathbb{C}_+^d$, for which we establish a complete measure decomposition and prove regularity. We then develop a graph-theoretic approach to the singularity and stability problem for non-symmetric matrices $S$. The graph structure of $S$ identifies the possible degeneracies of the stability operator as $z$ approaches the real axis. In particular, for non-backtracking matrices, we prove square-root growth at regular edges, cubic-root growth at regular cusps, and complete stability estimates. We also obtain the corresponding estimates for symmetric matrices in the periodic setting.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.16333",
  "author_total": 3,
  "authors": [
    {
      "name": "Jiaoyang Huang"
    },
    {
      "name": "Zhonggen Su"
    },
    {
      "name": "Ruizhe Xu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/ehpy26c5m5ss0.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.16333",
  "fulltext_url": null,
  "id": "ehpy26c5m5ss0",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.16333",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.CV"
    ],
    [
      "arxiv",
      "math.FA"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "arxiv",
      "math.SP"
    ],
    [
      "openalex",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7169885682",
  "pdf_url": "https://arxiv.org/pdf/2607.16333",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/ehpy26c5m5ss0.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:18Z",
      "native_id": "2607.16333",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.16333"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CV",
      "subfield": "analysis-pdes"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Non-symmetric vector dyson equations",
  "title_text": "Non-symmetric vector dyson equations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:22Z",
  "url": "https://research.draconitic.com/paper/ehpy26c5m5ss0",
  "venue": "arXiv",
  "versions": []
}