The distribution of overlaps between eigenvectors of Ginibre matrices
The distribution of overlaps between eigenvectors of Ginibre matrices
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DOI：
10.1007/s00440-019-00953-x
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发表时间：
2020-06-01
影响因子：
2
通讯作者：
Dubach, G.
中科院分区：
文献类型：
--
作者：
Bourgade, P.;Dubach, G.
研究了非正规矩阵特征向量的重叠问题。他们量化的稳定性的频谱，并表征联合特征值增量戴森型动力学。众所周知的工作Chalker和Mehlig计算的期望，这些重叠复杂的Ginibre矩阵。对于同一模型，我们通过推导对角重叠（条件数）的分布及其相关性来扩展他们的结果。我们证明：（i）批量特征值的条件数收敛到逆伽马分布;更一般地，我们分解淬火重叠（ii）非对角重叠的渐近期望，无论是微观或介观分离的相应特征值;（iii）以维度上的多项式速度，在介观距离处与本征值相关联的条件数的去相关;（iv）二阶矩渐近性，当相关特征值被任何介观尺度分离时，用于识别非对角重叠的涨落阶数;（v）一个新的本征值重叠之间的相关性公式，在微观距离，对角线和非对角线。这些结果意味着估计的极端条件数，体积的伪谱和扩散演化的特征值戴森型动力学下，在平衡。
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.