Random-matrix bulk statistics for random regular graphs of fixed degree
In plain words
A random network in which every node has exactly d links is a discrete model of a chaotic system, and its eigenvalues should be spaced like those of random matrices. This is proven at the spectrum's edge and for slowly growing d, but not in the bulk at fixed d.
Precise statement
Adjacency matrix of a uniform random d-regular graph on $N$ vertices with $d\ge 3$ fixed. Prove that bulk local eigenvalue statistics (e.g. the gap distribution at a fixed energy inside $(-2\sqrt{d-1}, 2\sqrt{d-1})$) converge to GOE as $N\to\infty$. Answer: a proof.
What would settle it
A proof of GOE bulk gap or correlation universality for one fixed d.
Status in the literature
Unverified note: it names recent work and was partly written without a live literature search; it awaits a check by an editor.
Bulk universality is proven for $N^{\alpha} \le d \le N^{2/3-\alpha}$ (Bauerschmidt, Huang, Knowles and Yau, Annals of Probability 2017, arXiv 1505.06700); at fixed d, optimal eigenvalue rigidity and Tracy-Widom edge universality were proven by Huang, McKenzie and Yau (arXiv 2412.20263, 2024).
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