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Fixed-energy universality for weak Anderson disorder on random regular graphs
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Abstract
For every fixed degree d ≥ 3, we prove fixed-energy GOE universality for the adjacency matrix of a uniformly random simple labelled d-regular graph with independent uniform diagonal disorder. The disorder strength is positive, sufficiently small, and fixed as the graph grows. At each fixed energy in a compact subinterval of the clean spectral band, the full microscopic eigenvalue point process converges to the GOE bulk process after rescaling by the positive density of states of the corresponding infinite-tree operator. The permitted disorder strength depends on the degree and the distance from the clean spectral edges.
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Follow-up works
- Yukun He, Jiaoyang Huang and Xiaoyu Wang, Bulk universality of random regular graphs (arXiv:2610.08127, 6 October 2026). This contemporaneous work treats the family's clean model for every fixed degree at least three, stating fixed-energy GOE point-process convergence together with deterministic-label gap universality and fixed-energy correlation-measure universality; this is the w = 0 counterpart that this paper takes from its companion. [family-level]
- Hexagon, contributed by Patrick Lopatto, Mobility Edge for Sparse Complex Gaussian Matrices (hexagon:2610.00168v1, October 2026). Section 1.1 cites both family manuscripts and describes He–Huang–Wang's clean-graph result as independent and simultaneous, within a study of a different model with complex Gaussian edge weights on sparse Erdős–Rényi graphs. [AI-generated per source] [family-level]
Possibly relevant literature (found by the agent, not checked)
- Michael Aizenman, Simone Warzel, Resonant delocalization for random Schrödinger operators on tree graphs (arXiv:1104.0969, 2011; JEMS 2013). Studies absolutely continuous spectrum of the adjacency operator plus a random potential on regular trees at weak disorder, relevant to the tree-side spectral input (Klein's theorem and its refinements) discussed in the introduction.
- Leander Geisinger, Convergence of the density of states and delocalization of eigenvectors on random regular graphs (arXiv:1305.1039, 2013; J. Spectral Theory 2015). Proves convergence of the integrated density of states of random Schrödinger operators on fixed-degree random regular graphs to that of the tree, the density ν_{d,w} used here for unfolding.
- Leander Geisinger, Poisson eigenvalue statistics for random Schrödinger operators on regular graphs (arXiv:1406.1608, 2014; Ann. Henri Poincaré 2015). Proves Poisson microscopic statistics in localization regimes of the same model, the counterpart to the GOE statistics claimed here at weak disorder.
- Suhan Liu and Patrick Lopatto, Mobility Edge for the Anderson Model on Random Regular Graphs (arXiv:2603.14230v2, 19 March 2026). Studies localization and delocalization for Gaussian disorder at sufficiently large fixed degree, a comparison point for the manuscript's finite-graph resolvent and delocalization arguments under weak uniform disorder at every fixed degree at least three.
Digests and expositions
- Latent Seminar, Microscopic spectra of random regular graphs (6 October 2026). This family-specific exposition explains unfolding, the claimed Gaussian eigenvector marks, switching and Gaussian insertion, while explicitly limiting its review to statements and selected proof mechanisms rather than complete proof verification. [family-level]
- Gradient Damage, Six-Vertex Model, Lipschitz Heights, The O(3) Model Continuum Limit and 6 More (15 October 2026). The chapter beginning at 2:27 introduces the regular-graph eigenvalue problem, and its accompanying text states both family claims as unreviewed preprint results. [family-level]
Elsewhere
- Hexagon: Mobility Edge for Sparse Complex Gaussian Matrices (AI-generated; human understanding: some)
- Emergent Mind: family 219
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026,
author = {{OpenAI}},
title = {{Fixed-energy universality for weak Anderson disorder on random regular graphs}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026/fixed-energy-universality-weak-anderson-disorder-random-regular-graphs.pdf}{OAI:Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026}},
year = {2026}
}Family 219: GOE bulk universality for regular graphs with weak Anderson disorder
For every fixed degree d ≥ 3, the bulk adjacency-eigenvalue point process of a uniform simple random d-regular graph converges to the Gaussian orthogonal ensemble law, including for cubic graphs. The same fixed-energy universality persists under sufficiently weak fixed iid uniform diagonal disorder, throughout compact bands strictly inside the clean spectral edges.
Other papers in this family:
- GOE bulk universality for fixed-degree random regular graphs · September 23, 2026