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GOE bulk universality for fixed-degree random regular graphs
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Abstract
We prove the fixed-degree bulk-universality conjecture for random regular graphs. For every fixed integer d ≥ 3 and every fixed energy in the open Kesten–McKay bulk, the unfolded eigenvalue point process of the adjacency matrix of a uniform simple labelled d-regular graph converges to the GOE bulk process. The convergence holds along all admissible graph sizes, without additional conditioning.
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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.
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 same clean model for every fixed degree at least three, stating fixed-energy GOE (Sine₁) point-process convergence, the statement of Theorem 1.1 here, together with deterministic-label gap universality and fixed-energy correlation-measure universality; its LLM disclosure states "GPT-6 assisted the proof development at the level of a coauthor." [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)
- Jiaoyang Huang, Theo McKenzie, Horng-Tzer Yau, Optimal Eigenvalue Rigidity of Random Regular Graphs (arXiv:2405.12161, May 2024). The dedicated paper on optimal rigidity for fixed d ≥ 3, relevant to the introduction's account of rigidity inputs, which cites only arXiv:2412.20263 for this.
- Jiaoyang Huang, Horng-Tzer Yau, Lecture Notes on Edge Universality for Random Regular Graphs (arXiv:2602.00975, Feb 2026). Expository notes on the local law and loop-equation approach for fixed-degree random regular graphs, the spectral input on which this paper relies.
- Leander Geisinger, Convergence of the density of states and delocalization of eigenvectors on random regular graphs (arXiv:1305.1039, 2013; J. Spectral Theory 2015). Earlier fixed-degree Green-function estimates giving convergence to the Kesten–McKay/tree density of states and eigenvector delocalization.
- Jiaoyang Huang, Spectral Statistics of Sparse Random Graphs (Proceedings of the International Congress of Mathematicians 2026, volume 6, pp. 43–63, 13 July 2026). This survey supplies context for the paper's bulk-universality claim by reviewing sparse-graph bulk, edge and eigenvector universality and the role of loop equations.
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)
- VibeMathed: Problem record
- Emergent Mind: family 219
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026,
author = {{OpenAI}},
title = {{GOE bulk universality for fixed-degree random regular graphs}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026/paper.pdf}{OAI:GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026}},
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: