标题： 实数和复数椭圆Ginibre系综在强非厄米和弱非厄米情况下的平均特征向量自重叠 显示英文标题

标题： Mean eigenvector self-overlap in the real and complex elliptic Ginibre ensembles at strong and weak non-Hermiticity

摘要： 我们研究了与复数特征值相关的左和右特征向量的平均对角重叠，在$N\times N$非厄米随机高斯矩阵中。在 Chalker 和 Mehlig 著名的工作中，该（自）重叠的期望值在复数 Ginibre 系综中被计算为$N\to \infty$。在本工作中，我们考虑了由非对角线元素之间的相关性由参数$\tau\in[0,1]$控制的真实和复数椭圆 Ginibre 系综中的相同量，其中$\tau=1$对应于厄米极限。我们在任何有限的$N$下，对于实轴以外的任何特征值，推导了这两个系综中平均对角重叠的精确表达式。 我们进一步研究了多个尺度范围，即$N\rightarrow \infty$，在强非厄米特性极限下保持固定$\tau\in[0,1)$，以及在弱非厄米特性极限下，$\tau$以使得$N(1-\tau)$保持有限的方式接近单位值。 显示英文摘要

摘要： We study the mean diagonal overlap of left and right eigenvectors associated with complex eigenvalues in $N\times N$ non-Hermitian random Gaussian matrices. In well known works by Chalker and Mehlig the expectation of this (self-)overlap was computed for the complex Ginibre ensemble as $N\to \infty$. In the present work, we consider the same quantity in the real and complex elliptic Ginibre ensembles characterized by correlations between off-diagonal entries controlled by a parameter $\tau\in[0,1]$, with $\tau=1$ corresponding to the Hermitian limit. We derive exact expressions for the mean diagonal overlap in both ensembles at any finite $N$, for any eigenvalue off the real axis. We further investigate several scaling regimes as $N\rightarrow \infty$, both in the limit of strong non-Hermiticity keeping a fixed $\tau\in[0,1)$ and in the weak non-Hermiticity limit, with $\tau$ approaching unity in such a way that $N(1-\tau)$ remains finite.