Functional Central Limit Theorems for Wigner Matrices

TL;DR

This paper establishes central limit theorems for functions of Wigner matrices, revealing independent fluctuation modes and extending previous results to complex matrices and broader ensembles, with applications to eigenstate thermalization.

Contribution

It introduces a comprehensive analysis of fluctuations of functions of Wigner matrices beyond the trace, identifying independent fluctuation modes and generalizing prior results to complex and crossover ensembles.

Findings

Asymptotic normality of trace of functions of Wigner matrices.

Identification of three independent fluctuation modes.

Extension of eigenstate thermalization results to complex 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, $\mathrm{Tr}[f(W)]$, 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, Erd\H{o}s, Schr\"oder 2020].