Multi-Point Functional Central Limit Theorem for Wigner Matrices

TL;DR

This paper establishes a Gaussian fluctuation limit for traces of products of functions of Wigner matrices and deterministic matrices, providing explicit covariance structures and error bounds, extending previous results to arbitrary product lengths.

Contribution

It extends the functional central limit theorem for Wigner matrices to products of arbitrary length, with explicit covariance and error bounds, generalizing prior work.

Findings

Fluctuations are Gaussian with explicit covariance structure.

Derived error bounds depend on function scaling and matrix tracelessness.

Provided explicit variance formula for large-time fluctuations of trace expressions.

Abstract

Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded deterministic matrices $A_1,\dots,A_k$. We give a functional central limit theorem showing that the fluctuations around the expectation are Gaussian. Moreover, we determine the limiting covariance structure and give explicit error bounds in terms of the scaling of $f_1,\dots,f_k$ and the number of traceless matrices among $A_1,\dots,A_k$, thus extending the results of [Cipolloni, Erd\H{o}s, Schr\"oder 2023] to products of arbitrary length $k\geq2$. As an application, we consider the fluctuation of $\mathrm{Tr}(\mathrm{e}^{\mathrm{i} tW}A_1\mathrm{e}^{-\mathrm{i} tW}A_2)$ around its thermal value $\mathrm{Tr}(A_1)\mathrm{Tr}(A_2)$ when $t$ is large and give an explicit formula for the variance.