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A Real Quaternion Spherical Ensemble of Random Matrices

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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.

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Acknowledgements

AM was partly supported by the Australian Mathematical Society (AustMS) LiftOff Fellowship during the work leading to this paper. Special thanks to Peter Forrester for reading an early draft of this work and providing many useful comments. The author would also like to thank Gernot Akemann, Tilo Wettig and Anita Ponsaing for several interesting discussions, and the following for their kind hospitality: Department of Mathematics, Vanderbilt University; Department of Mathematics, University of Geneva; Faculty of Physics, University of Bielefeld; and Faculty of Physics, University of Regensburg. Thanks to Joshua Feinberg for pointing me to an earlier work of his which is relevant to this paper.

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  1. Chaire ASAPGONE, Alcatel Lucent Chair on Flexible Radio, Supélec, Gif-sur-Yvette, France

    Anthony Mays

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Mays, A. A Real Quaternion Spherical Ensemble of Random Matrices. J Stat Phys 153, 48–69 (2013). https://doi.org/10.1007/s10955-013-0808-7

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