View source for GOE bulk universality for random regular graphs

'''GOE bulk universality for random regular graphs''' is the prediction that the eigenvalues of the adjacency matrix of a uniformly random ''d''-regular graph of fixed degree have, near a fixed energy inside the spectrum, the same local statistics as the Gaussian orthogonal ensemble (GOE). Two manuscripts in OpenAI's public mathematics collection ([https://github.com/openai/math github.com/openai/math]) claim this for every fixed degree ''d'' ≥ 3 and every fixed energy in the bulk. They are dated 23 September and 5 October 2026. The second manuscript extends the claim to graphs with weak independent diagonal (Anderson) disorder. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.

== Background ==
* As recounted in the introductions of the manuscripts:
** The limiting eigenvalue density of random ''d''-regular graphs is the Kesten–McKay density, which comes from the spectral measure of the infinite regular tree. Jakobson, Miller, Rivin and Rudnick formulated the GOE prediction for fixed-degree regular graphs and gave numerical evidence for it.
** For growing degrees, Bauerschmidt, Knowles and Yau proved a local semicircle law, and Bauerschmidt, Huang, Knowles and Yau proved bulk gap universality and energy-averaged correlation universality in a polynomial degree range. Bourgade and Huang obtained fixed-energy point-process universality for (log ''n'')<sup>24</sup> ≪ ''d'' ≤ ''n''<sup>1/2</sup> by loop equations, and stated the fixed-degree problem as their Conjecture 2.11.
** For fixed degree, Bauerschmidt, Huang and Yau proved the local law and delocalization for sufficiently large degree. Huang and Yau proved rigidity, complete delocalization and local resolvent estimates for every ''d'' ≥ 3. Huang, McKenzie and Yau proved optimal eigenvalue rigidity and GOE edge universality for every ''d'' ≥ 3.
** Backhausz and Szegedy proved Gaussian-wave limits for almost eigenvectors of random regular graphs by entropy inequalities, with a variance factor between zero and one in the weak limit.
** For diagonal disorder, Klein proved purely absolutely continuous spectrum for the Anderson model on the regular tree at weak disorder, with later approaches by Aizenman, Sims and Warzel and by Froese, Hasler and Spitzer. Anantharaman and Sabri proved quantum ergodicity for the Anderson model on expanding regular graphs at weak disorder.

== The original results ==
* '''Claim (clean graphs).''' Theorem 1.1 of ''GOE bulk universality for fixed-degree random regular graphs'' considers a uniform simple labelled ''d''-regular graph on ''n'' vertices, with no conditioning on connectedness or other properties. It states that for every fixed ''d'' ≥ 3 and every fixed energy ''E'' with |''E''| < 2√(''d'' − 1), the eigenvalue point process near ''E'', rescaled by ''n'' times the Kesten–McKay density at ''E'', converges in the sense of Laplace functionals to the GOE bulk point process, as ''n'' → ∞ through all admissible sizes. The manuscript states that this proves the fixed-degree bulk-universality conjecture of Bourgade and Huang (their Conjecture 2.11) in a fixed-energy formulation.
* ''Method, as described:''
** An entropy argument in the framework of Backhausz and Szegedy shows that the values of finitely many bulk eigenvectors at the two ends of a uniformly sampled edge have joint moments polynomially close to Gaussian moments with variance one. The argument uses colored-star counting, Gaussian smoothing, the logarithmic potential of the tree and a strict tensor inequality.
** Two-edge switches combined with Gaussian endpoint marks give exact continuous switching identities for the microscopic limiting law. Many weak switches then insert GOE noise.
** The fixed-energy Dyson Brownian motion universality theorem of Landon, Sosoe and Yau identifies the limit, and a decimation identity of Forrester and Rains upgrades correlation statements to the full point process.
* '''Claim (weak Anderson disorder).''' Theorem 1.1 of ''Fixed-energy universality for weak Anderson disorder on random regular graphs'' adds independent uniform potentials ''w''ω<sub>''v''</sub>, with ω<sub>''v''</sub> in [−1, 1], to the diagonal of the adjacency matrix. For every ''d'' ≥ 3 and every κ in (0, 2√(''d'' − 1)), the theorem states that there is ''w''<sub>0</sub>(''d'', κ) > 0 such that for each fixed disorder strength 0 < ''w'' < ''w''<sub>0</sub>:
** the density of states of the disordered tree has a continuous density, positive on the band [−2√(''d'' − 1) + κ, 2√(''d'' − 1) − κ];
** at every fixed energy in that band, the rescaled eigenvalue point process converges to the GOE bulk point process.
: The manuscript states that it adapts the entropy and Gaussian-insertion arguments of the clean companion, and proves within the paper the quantitative stability of the tree Green-function recursion that the finite-graph argument needs.
* '''Sources.''' The manuscripts:
** [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/GOE-bulk-universality-for-fixed-degree-random-regular-graphs-September-23-2026 GOE bulk universality for fixed-degree random regular graphs] (23 September 2026);
** [https://github.com/openai/math/tree/1581d4cbdd69a0faf34b2089614fe2a814316a6a/preprints/Fixed-energy-universality-for-weak-Anderson-disorder-on-random-regular-graphs-October-5-2026 Fixed-energy universality for weak Anderson disorder on random regular graphs] (5 October 2026).
* '''Formalization.''' The repository's Lean documentation at commit <code>1581d4c</code> has no scope note for these manuscripts.
* See also [[Localization and delocalization in the Anderson model]] for the collection's results on the Anderson model on the integer lattice.

== Public follow-up work ==
{| class="wikitable"
! Date !! Work !! Relation !! Public claim !! Public status (checked 19 October 2026)
|-
| 2026-10-06
| Y. He, J. Huang, X. Wang, ''Bulk universality of random regular graphs'' ([https://arxiv.org/abs/2610.08127v1 arXiv:2610.08127], v1)
| Parallel (independent proof)
| The paper proves the following for every fixed ''d'' ≥ 3 and every fixed bulk energy:
* the rescaled eigenvalue point process of a uniformly random simple ''d''-regular graph converges to the Sine<sub>1</sub> process;
* consecutive gaps at deterministic bulk labels are universal;
* fixed-energy correlation measures are universal.
The eigenvector input is a set of polynomial estimates for mixed fourth moments on a uniformly sampled star, followed by exact switching identities and a microscopic loop hierarchy. The paper's LLM disclosure states that GPT-6 assisted the proof development at the level of a coauthor.
| arXiv preprint (version 1 only).
|-
| 2026-10-12 || P. Lopatto, ''Mobility Edge for Sparse Complex Gaussian Matrices'' ([https://hexagonmath.org/2610.00168v1 Hexagon 2610.00168v1]) || Remark (citation) || In its discussion of recent advances, the paper cites both manuscripts. It describes them as preprints produced by an internal OpenAI large language model that announce fixed-energy GOE bulk universality for the adjacency matrix and for weak diagonal disorder. It states that the result for the adjacency matrix was obtained independently and simultaneously by He, Huang and Wang. The paper's own results concern a mobility edge for sparse Hermitian random matrices on Erdős–Rényi graphs. || Hexagon (version 1).
|-
| 2026-10-16
| J. Jardine, Lattice24, referee reviews [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_025.json rank_025] (clean graphs) and [https://github.com/JJardine919/openai-math-audit/blob/a06a8c6e77974d4c6021338a55a8e91b634f67f7/reviews/rank_046.json rank_046] (weak disorder) in [https://github.com/JJardine919/openai-math-audit JJardine919/openai-math-audit] (archived at [https://doi.org/10.5281/zenodo.23422995 Zenodo 10.5281/zenodo.23422995])
| Review (AI-assisted audit)
| Both reviews report no gap, and the reviewer states that it read every section of both manuscripts.
* For the clean case, it lists numerical checks of the tree logarithmic-potential identity and of the tensor inequality for several degrees and energies. It records two minor remarks on compressed passages, which it says do not affect the result.
* For both manuscripts, it states that the exact hypotheses of the cited external theorems (including Landon–Sosoe–Yau) were taken as quoted, not re-derived.
| Part of an audit of the collection's manuscripts without Lean that the audit describes as complete. The reviews are published under CC BY 4.0.
|}

== Notes ==
* The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.

[[Category:OpenAI results]]