Out-of-time-ordered correlators for Wigner matrices

Abstract: We consider the time evolution of the out-of-time-ordered correlator (OTOC) of two general observables $A$ and $B$ in a mean field chaotic quantum system described by a random Wigner matrix as its Hamiltonian. We rigorously identify three time regimes separated by the physically relevant scrambling and relaxation times. The main feature of our analysis is that we express the error terms in the optimal Schatten (tracial) norms of the observables, allowing us to track the exact dependence of the errors on their rank. In particular, for significantly overlapping observables with low rank the OTOC is shown to exhibit a significant local maximum at the scrambling time, a feature that may not have been noticed in the physics literature before. Our main tool is a novel multi-resolvent local law with Schatten norms that unifies and improves previous local laws involving either the much cruder operator norm (cf. [G. Cipolloni, L. Erd\H{o}s, D. Schr\"oder. Elect. J. Prob. 27, 1-38, 2022]) or the Hilbert-Schmidt norm (cf. [G. Cipolloni, L. Erd\H{o}s, D. Schr\"oder. Forum Math., Sigma 10, E96, 2022]).