Functional Central Limit Theorems for Wigner Matrices
Abstract
We consider the fluctuations of regular functions f of a Wigner matrix W viewed as an entire matrix f(W). Going beyond the well studied tracial mode, Tr[f(W)], which is equivalent to the customary linear statistics of eigenvalues, we show that Tr[f(W)] is asymptotically normal for any non-trivial bounded deterministic matrix A. We identify three different and asymptotically independent modes of this fluctuation, corresponding to the tracial part, the traceless diagonal part and the off-diagonal part of f(W) in the entire mesoscopic regime, where we find that the off-diagonal modes fluctuate on a much smaller scale than the tracial mode. In addition, we determine the fluctuations in the Eigenstate Thermalisation Hypothesis [Deutsch 1991], i.e. prove that the eigenfunction overlaps with any deterministic matrix are asymptotically Gaussian after a small spectral averaging. In particular, in the macroscopic regime our result generalises [Lytova 2013] to complex W and to all crossover ensembles in between. The main technical inputs are the recent multi-resolvent local laws with traceless deterministic matrices from the companion paper [Cipolloni, Erdos, Schr\"oder 2020].