GOE bulk universality for fixed-degree random regular graphs
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Do bulk adjacency eigenvalues of a uniformly random simple labelled d-regular graph have Gaussian orthogonal ensemble statistics for each fixed d≥3? The target is convergence at a fixed energy strictly inside the Kesten–McKay bulk, along every admissible graph-size sequence and without extra conditioning.
Why it matters and what counts as progress
Why it matters: Fixed-degree, fixed-energy universality includes cubic random regular graphs beyond results that let degree grow or average the energy.
What counts as progress: Prove fixed-energy universality for additional fixed degrees, particularly d=3, or obtain quantitative convergence rates.
How the problem developed
- 30 Sept 2003 · Numerical evidence [1]
D. Jakobson, S. D. Miller, I. Rivin, and Z. Rudnick Numerical experiments suggested GOE level-spacing statistics for generic regular graphs as the number of vertices grows; they did not prove fixed-degree bulk universality.
- 23 Sept 2026 · Reported claim [2]
OpenAI The manuscript claims GOE bulk universality for each fixed degree d≥3 and each fixed interior energy, for the random regular graph model stated there.
AI contribution
GOE bulk universality for fixed-degree random regular graphs
For every fixed degree d≥3 and energy strictly inside the Kesten–McKay bulk, the unfolded adjacency-eigenvalue point process of a uniform simple labelled d-regular graph converges at that fixed energy to the GOE bulk process along all admissible graph sizes, without extra conditioning.
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