Global Fluctuations of Gaussian Elliptic Matrices

Abstract

We introduce a spoke-arc decomposition of non-crossing annular pair partitions $N{C}_2(p,q)$ that records spoke type and orientation, isolates spoke-level contributions, and factorizes the dependence on the ellipticity parameter ${\hat{I}}^3$ 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.

v2: refined the statement of the main theorem and added new references