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Accepted · first revision · record

Title: Random-matrix bulk statistics for random regular graphs of fixed degree
Plain: 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: 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.
Settled by: A proof of GOE bulk gap or correlation universality for one fixed d.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: 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).
Posed since: (empty)