On the Global Fluctuations of Block Gaussian Matrices

TL;DR

This paper investigates the global fluctuations of block Gaussian matrices using second-order free probability, introducing new tools like matricial second-order expectations to compute their second-order Cauchy transforms.

Contribution

It introduces a matricial second-order conditional expectation and applies linearization to analyze fluctuations of non-commutative rational functions on Gaussian matrices.

Findings

Derived the second-order Cauchy transform for block Gaussian matrices

Developed a matricial second-order conditional expectation framework

Applied linearization to non-commutative rational functions

Abstract

In this paper we study the global fluctuations of block Gaussian matrices within the framework of second-order free probability theory. In order to compute the second-order Cauchy transform of these matrices, we introduce a matricial second-order conditional expectation and compute the matricial second-order Cauchy transform of a certain type of non-commutative random variables. As a by-product, using the linearization technique, we obtain the second-order Cauchy transform of non-commutative rational functions evaluated on selfadjoint Gaussian matrices.