Result 219, Probability and statistical mechanics

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.

Proof

The bigger picture

Why it matters

A network's adjacency matrix records which vertices are linked. The manuscripts report that sparse random regular graphs share fine-scale eigenvalue statistics with Gaussian real symmetric matrices, even under weak random on-site perturbations, connecting two structurally different models.

What changes?

For fixed degree d at least 3, take a uniform simple labelled graph with d neighbors per vertex. The reported Gaussian orthogonal ensemble (GOE) law governs the full pattern of nearby adjacency eigenvalues at each fixed energy inside the clean spectral band. It persists with independent, identically distributed uniform diagonal disorder, sufficiently weak but positive and fixed as the graph grows. The allowed strength depends on d and distance from the clean spectral edges; the disordered claim covers compact interior bands.

What does that help mathematicians do?

For the clean model, this includes cubic graphs, where every vertex has three neighbors, and all admissible graph sizes without extra conditioning. For the disordered model, the comparison rescales eigenvalue spacings using the positive density of states of the corresponding infinite-tree operator. This lets researchers deduce limiting local spacing statistics from GOE, rather than knowing only how eigenvalues are distributed on average.

Are there practical applications?

The immediate value is foundational for probability and statistical mechanics. Diagonal disorder represents random on-site potentials in an Anderson model. The reported result identifies an interior-energy, weak-disorder regime with GOE statistics. It does not by itself establish transport properties or extend the conclusion to spectral edges or stronger disorder.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

2 manuscripts

Fixed-energy universality for weak Anderson disorder on random regular graphs

October 5, 2026 60 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

GOE bulk universality for fixed-degree random regular graphs

September 23, 2026 44 pages

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.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

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An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.