Pickrell has fully characterized the unitarily invariant probability measures on infinite Hermitian matrices, and an alternative proof of this classification has been found by Olshanski and Vershik. Borodin and Olshanski deduced from this proof that under any of these invariant measures, the extreme eigenvalues of the minors, divided by the dimension, converge almost surely. In this paper, we prove that one also has a weak convergence for the eigenvectors, in a sense which is made precise. After mapping Hermitian to unitary matrices via the Cayley transform, our result extends a convergence proven in our paper with Maples and Nikeghbali, for which a coupling of the Circular Unitary Ensemble of all dimensions is considered.

中文翻译：

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酉不变无限随机矩阵的次要特征向量收敛

Pickrell 已经充分表征了无限 Hermitian 矩阵上的酉不变概率测度，并且 Olshanski 和 Vershik 找到了这种分类的替代证明。Borodin 和 Olshanski 从这个证明中推断出，在这些不变测度中的任何一个下，次要的极值特征值除以维度，几乎肯定会收敛。在本文中，我们证明了特征向量也具有弱收敛性，在某种意义上是精确的。在通过 Cayley 变换将 Hermitian 映射到酉矩阵之后，我们的结果扩展了在我们的论文中与 Maples 和 Nikeghbali 证明的收敛性，为此考虑了所有维度的圆形酉系综的耦合。