Freeness over the diagonal and global fluctuations of complex Wigner matrices

TL;DR

This paper investigates the second order distributional limits of complex Wigner matrices using free probability concepts, providing a framework for understanding their fluctuations and extending existing notions of freeness.

Contribution

It extends the theory of second order freeness to operator-valued variables and characterizes the fluctuations of complex Wigner matrices over the diagonal.

Findings

Characterization of second order limits for complex Wigner matrices

Extension of second order freeness to operator-valued variables

Universal fluctuation rules for Gaussian Wigner matrices

Abstract

We characterize the limiting second order distributions of certain independent complex Wigner and deterministic matrices using Voiculescu's notions of freeness over the diagonal. If the Wigner matrices are Gaussian, Mingo and Speicher's notion of second order freeness gives a universal rule, in terms of marginal first and second order distribution. We adapt and reformulate this notion for operator-valued random variables in a second order probability space. The Wigner matrices are assumed to be permutation invariant with null pseudo variance and the deterministic matrices to satisfy a restrictive property