{
  "abstract": "We study spectral limits of random operators whose incident edge weights are coupled by local normalization, and the reflection quantum walks associated with them. At each vertex of a $d$-regular graph, an independent Dirichlet probability vector distributes unit mass among outgoing directions. At fixed degree, the empirical spectral measures of the associated self-adjoint operators converge almost surely on locally tree-like graph sequences. We give a closed-walk formula for every moment of the limiting tree law. Our main result identifies two distinct high-degree scales. After rescaling the deterministic expected root spectral law by $\\sqrt d$, a weak limit exists if and only if $dα_d^2$ converges in $[0,\\infty]$, where $α_d$ is the Dirichlet parameter. The limits range from a point mass at zero through non-semicircular critical laws to the semicircle law. The critical laws admit a Poisson-tree representation with an explicit Bessel intensity. The extended family, including both endpoint laws, provides a first-Wasserstein approximation uniform over positive Dirichlet parameters. In the regime $dα_d^2\\to0$, the rescaled law converges weakly to a point mass although its second moment remains one. Weighting spectral mass by its contribution to this moment and rescaling again reveals a different family of limits, classified by $dα_d$. These limits have explicit Beta-product descriptions. A quantitative comparison transfers the variance-scale classification to the centered and rescaled inherited eigenphases of the quantum walk.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36326",
  "author_total": 2,
  "authors": [
    {
      "name": "Nino Nagata"
    },
    {
      "name": "Ryoichi Suzuki"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9qnew2t37wnr6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36326",
  "fulltext_url": null,
  "id": "9qnew2t37wnr6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36326",
  "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",
      "2610|Spectral Theory in Mathematical Physics"
    ]
  ],
  "openalex": "W7224106626",
  "pdf_url": "https://arxiv.org/pdf/2610.36326",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-26",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9qnew2t37wnr6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:50Z",
      "native_id": "2610.36326",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36326"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Dirichlet-Constrained Random Operators and Quenched Reflection Quantum Walks: Tree Spectra and Poisson Criticality at High Degree",
  "title_text": "Dirichlet-Constrained Random Operators and Quenched Reflection Quantum Walks: Tree Spectra and Poisson Criticality at High Degree",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/9qnew2t37wnr6",
  "venue": "arXiv",
  "versions": []
}