Emergent random matrix universality in quantum operator dynamics

The high complexity of many-body quantum dynamics means that essentially all approaches either exploit special structure or are approximate in nature. One such approach--the memory function formalism--involves a carefully chosen split into fast and slow modes. An approximate model for the fast modes can then be used to solve for Green's functions G (z) of the slow modes. Using a formulation in operator Krylov space known as the recursion method, we prove the emergence of a universal random matrix description of the fast mode dynamics. This is captured by the level-n Green's function Gₙ (z), which we show approaches universal scaling forms in the fast limit n. Notably, this emergent universality can occur in both chaotic and non-chaotic systems, provided their spectral functions are smooth. This universality of Gₙ (z) is precisely analogous to the universality of eigenvalue correlations in random matrix theory (RMT), even though there is no explicit randomness present in the Hamiltonian. At finite z we show that Gₙ (z) approaches the Wigner semicircle law, while if G (z) is the Green's function of certain hydrodynamical variables, we show that at low frequencies Gₙ (z) is instead governed by the Bessel universality class from RMT. As an application of this universality, we give a numerical method--the spectral bootstrap--for approximating spectral functions from Lanczos coefficients. Our proof involves a map to a Riemann-Hilbert problem which we solve using a steepest-descent-type method, rigorously controlled in the n limit. We are led via steepest-descent to a Coulomb gas optimization problem, and we discuss how a recent conjecture--the `Operator Growth Hypothesis--is linked to a confinement transition in this Coulomb gas. These results elevate the recursion method to a theoretically principled framework with universal content.