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 (see [7, 11, 14]). We first recover a version of Kostlan’s theorem that was already at the heart of an argument by Rider [1], namely, that 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. Angles 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 at the origin. This complements a recent study of overlaps in quaternionic ensembles by Akemann, Förster and Kieburg [1, 2].

中文翻译：

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四元数 Ginibre 系综的对称性

我们建立了四元数 Ginibre 系综 (QGE) 的特征值和特征向量的一些性质，类似于在复 Ginibre 情况下已知的性质（参见 [7, 11, 14]）。我们首先恢复了一个版本的 Kostlan 定理，它已经成为 Rider [1] 论证的核心，即特征值的平方半径的集合作为一组独立的 gamma 变量分布。我们的证明技术使用了 Pfaffians 的 De Bruijn 恒等式和属性；它还可以证明这些特征值的高次幂是独立的。只要径向对称成立，这些结果就可以扩展到高斯情况之外的任何潜力；这包括例如四元数酉矩阵的截断、四元数 Ginibre 矩阵的乘积和四元数球面系综。然后我们研究四元数 Ginibre 矩阵的特征向量。特征向量和重叠矩阵之间的角度都表现出一些可以与复杂情况进行比较的特定特征。特别是，我们计算与以原点为条件的特征值相关的对角线重叠的分布和限制。这补充了 Akemann、Förster 和 Kieburg 最近对四元数系综重叠的研究 [1, 2]。