# Functional central limit theorems for Wigner matrices

Giorgio Cipolloni, László Erdős, Dominik Schröder

Ann. Appl. Probab.Vol. 33 (2023)

## Summary

We extend the classical CLTs for linear statistics functionally, in the sense that we prove the CLT for traces auf functions of Wigner matrices multiplied with arbitrary deterministic observables.## 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, $\mathrm{Tr}[f(W)]$, which is equivalent to the customary linear statistics of eigenvalues, we show that $\mathrm{Tr}[f(W)]$ 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, Erdős, Schröder 2020].