We study the overlaps between eigenvectors of nonnormal matrices. They quantify the stability of the spectrum, and characterize the joint eigenvalues increments under Dyson-type dynamics. Well known work by Chalker and Mehlig calculated the expectation of these overlaps for complex Ginibre matrices. For the same model, we extend their results by deriving the distribution of diagonal overlaps (the condition numbers), and their correlations. We prove: (i) convergence of condition numbers for bulk eigenvalues to an inverse Gamma distribution; more generally, we decompose the quenched overlap (i.e. conditioned on eigenvalues) as a product of independent random variables; (ii) asymptotic expectation of off-diagonal overlaps, both for microscopic or mesoscopic separation of the corresponding eigenvalues; (iii) decorrelation of condition numbers associated to eigenvalues at mesoscopic distance, at polynomial speed in the dimension; (iv) second moment asymptotics to identify the fluctuations order for off-diagonal overlaps, when the related eigenvalues are separated by any mesoscopic scale; (v) a new formula for the correlation between overlaps for eigenvalues at microscopic distance, both diagonal and off-diagonal. These results imply estimates on the extreme condition numbers, the volume of the pseudospectrum and the diffusive evolution of eigenvalues under Dyson-type dynamics, at equilibrium.

中文翻译：

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Ginibre矩阵的特征向量之间的重叠分布

我们研究非正规矩阵的特征向量之间的重叠。他们量化了光谱的稳定性，并表征了戴森型动力学下的联合特征值增量。Chalker 和 Mehlig 的著名工作计算了复杂 Ginibre 矩阵的这些重叠的期望。对于同一个模型，我们通过推导对角线重叠（条件数）的分布及其相关性来扩展他们的结果。我们证明： (i) 大量特征值的条件数收敛到逆 Gamma 分布；更一般地，我们将淬灭重叠（即以特征值为条件）分解为独立随机变量的乘积；(ii) 非对角线重叠的渐近期望，用于相应特征值的微观或细观分离；(iii) 与特征值相关联的条件数在维度上以多项式速度在介观距离上去相关；(iv) 当相关的特征值被任何细观尺度分开时，二阶矩渐近法用于识别非对角线重叠的波动顺序；(v) 微观距离特征值重叠之间相关性的新公式，包括对角线和非对角线。这些结果意味着在平衡状态下对极端条件数、伪谱的体积和特征值在戴森型动力学下的扩散演化的估计。(v) 微观距离特征值重叠之间相关性的新公式，包括对角线和非对角线。这些结果意味着在平衡状态下对极端条件数、伪谱的体积和特征值在戴森型动力学下的扩散演化的估计。(v) 微观距离特征值重叠之间相关性的新公式，包括对角线和非对角线。这些结果意味着在平衡状态下对极端条件数、伪谱的体积和特征值在戴森型动力学下的扩散演化的估计。