Global Fluctuations of Gaussian Elliptic Matrices

TL;DR

This paper introduces a spoke-arc decomposition for non-crossing annular pair partitions to analyze the covariance structure of Gaussian elliptic matrices, revealing their asymptotic second-order freeness.

Contribution

It presents a novel spoke-arc decomposition method that isolates contributions and factorizes dependence on ellipticity, providing explicit formulas for covariance and second-order freeness.

Findings

Closed-form descriptions of limiting covariance for Gaussian elliptic matrices

Demonstration of asymptotic second-order freeness of independent Gaussian elliptic matrices

Introduction of spoke-arc decomposition for non-crossing annular pair partitions

Abstract

We introduce a spoke-arc decomposition of non-crossing annular pair partitions $NC_2(p,q)$ that records spoke type and orientation, isolates spoke-level contributions, and factorizes the dependence on the ellipticity parameter $\gamma$ into a spoke factor and arc weights. This yields closed-form descriptions of the limiting covariance of Gaussian elliptic matrices. As a corollary, we show that an independent family of Gaussian elliptic random matrices is asymptotically second-order free.