GOE bulk universality for regular graphs with weak Anderson disorder

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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.

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What the papers state

The paper states

Theorem 1.1 (Fixed-energy universality)

In Fixed-energy universality for weak Anderson disorder on random regular graphs, 5 October 2026

Possible match to OpenAI’s description, selected from shared wording. No review confirming the selection is recorded.

For every d\ge3 and every 0<\kappa<2\sqrt{d-1}, there is w_0=w_0(d,\kappa)>0 with the following property. For each fixed 0<w<w_0, the restriction of \nu_{d,w} to an open neighborhood of

I_{d,\kappa}=[-2\sqrt{d-1}+\kappa,\,2\sqrt{d-1}-\kappa]

has a continuous density \rho_{d,w}, strictly positive on I_{d,\kappa}. For every fixed E\in I_{d,\kappa}, set

\Xi_{n,E}=\sum_{i=1}^n \delta_{n\rho_{d,w}(E)(\lambda_i-E)}.

Then, for every nonnegative f\in C_c(\RR),

\left| \EE\exp\left(-\int f\,d\Xi_{n,E}\right) -\EE\exp\left(-\int f\,d\Xi_n^{\GOE}\right) \right|\longrightarrow0

as n\to\infty through all sizes n>d with nd even.

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One more possible match

Theorem 1.1 in GOE bulk universality for fixed-degree random regular graphs, 23 September 2026.

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Papers

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

5 October 2026, current edition.

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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.

GOE bulk universality for fixed-degree random regular graphs

23 September 2026, current edition.

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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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Fixed-energy universality for weak Anderson disorder on random regular graphs cites GOE bulk universality for fixed-degree random regular graphs.

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