Joint Global Fluctuations of complex Wigner and deterministic Matrices

TL;DR

This paper studies the joint fluctuations of multiple Wigner and deterministic matrices, revealing that their trace fluctuations follow a CLT but are not asymptotically free of second order, with covariance depending on detailed matrix information.

Contribution

It provides a detailed characterization of the limiting fluctuations of traces of independent Wigner and deterministic matrices, highlighting non-freeness at second order.

Findings

CLT holds for trace fluctuations

Families are not asymptotically free of second order

Covariance depends on detailed deterministic matrix information

Abstract

We characterize the limiting fluctuations of traces of several independent Wigner matrices and deterministic matrices under mild conditions. A CLT holds but in general the families are not asymptotically free of second order and the limiting covariance depends on more information on the deterministic matrices than their limiting *-distribution.