Bulk universality of random regular graphs

We consider the adjacency matrix of a uniformly random simple $d$-regular graph on $N$ vertices. For every fixed $d\ge 3$ and every fixed bulk energy, we prove that the rescaled eigenvalue point process converges to the ${\mathrm{S}\mathrm{i}\mathrm{n}\mathrm{e}}_1$ process with intensity $1\mathrm{/}\pi$. We also establish universality of consecutive gaps at deterministic bulk labels and of fixed-energy correlation measures. The eigenvector input consists of polynomial estimates for mixed fourth moments on a uniformly sampled star. These estimates prevent loss of variance in the qualitative joint Gaussian limits of Backhausz–Szegedy, yielding independent variance-one Gaussian waves for fixed tuples at nearby energies. Exact switching identities then give a stationary marked flow and the microscopic loop hierarchy, which identifies the point-process limit. The appendices prove the stronger correlation and gap statements using counting-moment bounds and conditional log-gas laws for tagged limits.