General heuristics suggest that quantum chaotic systems (such as manifolds with ergodic geodesic flow) exhibit the spectral statistics of random matrices. This behavior extends to the discrete setting, where the spectral properties of operators have important applications to statistical physics, computer science, and theoretical statistics.

In this talk, I will provide an overview of recent results demonstrating this chaotic behavior, with a focus on proving that for the adjacency operator of a random regular graph, all eigenvalues exhibit optimally small fluctuations. Throughout the talk, I will introduce key techniques used in these proofs and highlight the rich connections between continuous and discrete settings.