Fluctuation Moments for Regular Functions of Wigner Matrices

TL;DR

This paper derives formulas for mixed fluctuation moments of functions of Wigner matrices, extending previous results and confirming the validity of combinatorial structures beyond second-order free probability.

Contribution

It provides a deterministic approximation for mixed moments involving Sobolev functions of Wigner matrices, generalizing recent polynomial results and linking to variance characterizations.

Findings

Formulas reproduce known polynomial results

Combinatorial structures remain valid beyond second-order free probability

Characterize variance in a recent functional CLT

Abstract

We compute the deterministic approximation for mixed fluctuation moments of products of deterministic matrices and general Sobolev functions of Wigner matrices. Restricting to polynomials, our formulas reproduce recent results of [Male, Mingo, Pech\'e, Speicher 2022], showing that the underlying combinatorics of non-crossing partitions and annular non-crossing permutations continue to stay valid beyond the setting of second-order free probability theory. The formulas obtained further characterize the variance in the functional central limit theorem obtained recently in the companion paper [Reker 2023].