Bulk universality of random regular graphs
Abstract
We consider the adjacency matrix of a uniformly random simple d-regular graph on N vertices. For every fixed d≥3 and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the Sine1 process with intensity 1/π. We also establish universality of consecutive gaps at deterministic bulk labels and of fixed-energy correlation measures. The eigenvector input consists of polynomial estimates for mixed fourth moments on a uniformly sampled star. These estimates prevent loss of variance in the qualitative joint Gaussian limits of Backhausz--Szegedy, yielding independent variance-one Gaussian waves for fixed tuples at nearby energies. Exact switching identities then give a stationary marked flow and the microscopic loop hierarchy, which identifies the point-process limit. The appendices prove the stronger correlation and gap statements using counting-moment bounds and conditional log-gas laws for tagged limits.
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BibTeX
@misc{he2026bulk,
title = {{Bulk universality of random regular graphs}},
author = {Yukun He and Jiaoyang Huang and Xiaoyu Wang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08127},
eprint = {2610.08127},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08127},
}
RIS
TY - UNPB
TI - Bulk universality of random regular graphs
AU - Yukun He
AU - Jiaoyang Huang
AU - Xiaoyu Wang
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08127
UR - https://arxiv.org/abs/2610.08127
AB - We consider the adjacency matrix of a uniformly random simple d-regular graph on N vertices. For every fixed d≥3 and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the Sine_1 process with intensity 1/π. We also establish universality of consecutive gaps at deterministic bulk labels and of fixed-energy correlation measures. The eigenvector input consists of polynomial estimates for mixed fourth moments on a uniformly sampled star. These estimates prevent loss of variance in the qualitative joint Gaussian limits of Backhausz--Szegedy, yielding independent variance-one Gaussian waves for fixed tuples at nearby energies. Exact switching identities then give a stationary marked flow and the microscopic loop hierarchy, which identifies the point-process limit. The appendices prove the stronger correlation and gap statements using counting-moment bounds and conditional log-gas laws for tagged limits.
ER -