A real quaternion spherical ensemble of random matrices

TL;DR

This paper introduces a new ensemble of random matrices based on real quaternions, analyzes their eigenvalue distribution on the sphere, and shows that in large dimensions, the eigenvalues become uniformly distributed, extending spherical universality classes.

Contribution

It develops the real quaternion spherical ensemble, computes eigenvalue distributions, and demonstrates the spherical law for this new class, completing the exploration of spherical matrix ensembles.

Findings

Eigenvalue density exhibits a ring depletion along the real axis.

Eigenvalue distribution approaches uniformity on the sphere as matrix size increases.

The results extend spherical universality to real quaternion matrices.

Abstract

One can identify a tripartite classification of random matrix ensembles into geometrical universality classes corresponding to the plane, the sphere and the anti-sphere. The plane is identified with Ginibre-type (iid) matrices and the anti-sphere with truncations of unitary matrices. This paper focusses on an ensemble corresponding to the sphere: matrices of the form $\bY= \bA^{-1} \bB$, where $\bA$ and $\bB$ are independent $N\times N$ matrices with iid standard Gaussian real quaternion entries. By applying techniques similar to those used for the analogous complex and real spherical ensembles, the eigenvalue jpdf and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices $\beta=1,2,4$. We find that the eigenvalue density (after stereographic projection onto the sphere) has a depletion of eigenvalues along a ring corresponding to the real axis, with reflective symmetry about this ring. However, in the limit of large matrix dimension, this eigenvalue density approaches that of the corresponding complex ensemble, a density which is uniform on the sphere. This result is in keeping with the spherical law (analogous to the circular law for iid matrices), which states that for matrices having the spherical structure $\bY= \bA^{-1} \bB$, where $\bA$ and $\bB$ are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.