Skip to Main content Skip to Navigation
Journal articles

A real quaternion spherical ensemble of random 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 Y=A −1 B, where A and B are independent N×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 joint probability density function and correlation functions are calculated. This completes the exploration of spherical matrices using the traditional Dyson indices β=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 Y=A −1 B, where A and B are independent, iid matrices the (stereographically projected) eigenvalue density tends to uniformity on the sphere.
Document type :
Journal articles
Complete list of metadata

Cited literature [31 references]  Display  Hide  Download
Contributor : Azary Abboud <>
Submitted on : Tuesday, January 21, 2014 - 10:30:15 AM
Last modification on : Monday, December 14, 2020 - 2:36:02 PM
Long-term archiving on: : Tuesday, April 22, 2014 - 9:49:28 AM


Files produced by the author(s)



Anthony Mays. A real quaternion spherical ensemble of random matrices. Journal of Statistical Physics, Springer Verlag, 2013, 153 (1), pp.48-69. ⟨10.1007/s10955-013-0808-7⟩. ⟨hal-00933791⟩



Record views


Files downloads