Symmetries of the Quaternionic Ginibre Ensemble

Abstract

We establish a few properties of eigenvalues and eigenvectors of the quaternionic Ginibre ensemble (QGE), analogous to what is known in the complex Ginibre case. We first recover a version of Kostlan's theorem that was already noticed by Rider: the set of the squared radii of the eigenvalues is distributed as a set of independent gamma variables. Our proof technique uses the De Bruijn identity and properties of Pfaffians; it also allows to prove that the high powers of these eigenvalues are independent. These results extend to any potential beyond the Gaussian case, as long as radial symmetry holds; this includes for instance truncations of quaternionic unitary matrices, products of quaternionic Ginibre matrices, and the quaternionic spherical ensemble. We then study the eigenvectors of quaternionic Ginibre matrices. The angle between eigenvectors and the matrix of overlaps both exhibit some specific features that can be compared to the complex case. In particular, we compute the distribution and the limit of the diagonal overlap associated to an eigenvalue that is conditioned to be zero.

arXiv comment19 pages

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Cited by in corpus (7)

• The distribution of overlaps between eigenvectors of Ginibre matrices
• Condition numbers for real eigenvalues in the real Elliptic Gaussian ensemble
• Wronskian structures of planar symplectic ensembles
• Determinantal structure and bulk universality of conditional overlaps in the complex Ginibre ensemble
• Condition numbers for real eigenvalues of real elliptic ensemble: weak non-normality at the edge
• Pfaffian structure of the eigenvector overlap for the symplectic Ginibre ensemble
• Truncations of random symplectic unitary matrices