Dirichlet-Constrained Random Operators and Quenched Reflection Quantum Walks: Tree Spectra and Poisson Criticality at High Degree
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 √d, a weak limit exists if and only if dαd2 converges in [0,∞], where αd 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 dαd2→0, 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 dαd. 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.
Related
- Generalized Stević–Sharma operators on weighted Dirichlet spaces
- Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators
- Numerical range and invariant subspaces of weighted composition operators on the Hardy space of Dirichlet series
- Volterra operators on the BMOA space of Dirichlet series
- Weighted Composition-differentiation operators on weighted Dirichlet spaces
BibTeX
@misc{nagata2026dirichletconstrained,
title = {{Dirichlet-Constrained Random Operators and Quenched Reflection Quantum Walks: Tree Spectra and Poisson Criticality at High Degree}},
author = {Nino Nagata and Ryoichi Suzuki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36326},
eprint = {2610.36326},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36326},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Dirichlet-Constrained Random Operators and Quenched Reflection Quantum Walks: Tree Spectra and Poisson Criticality at High Degree
AU - Nino Nagata
AU - Ryoichi Suzuki
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36326
UR - https://arxiv.org/abs/2610.36326
AB - 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 √d, a weak limit exists if and only if dα_d^2 converges in [0,∞], 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→0, 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.
ER -