Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem
Squared eigenvalue condition numbers and eigenvector correlations from the single ring theorem
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DOI：
10.1088/1751-8121/aa5451
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发表时间：
2016-08
期刊：
影响因子：
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通讯作者：
S. Belinschi;M. Nowak;R. Speicher;W. Tarnowski
中科院分区：
文献类型：
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作者：
S. Belinschi;M. Nowak;R. Speicher;W. Tarnowski
我们扩展了所谓的“单环定理”（Feinberg和Zee 1997 Nucl. Phys. B 504 579），也称为Haagerup-Larsen定理（Haagerup和Larsen 2000 J. Funct. Anal. 176 331）。我们这样做，表明在限制时，矩阵的大小趋于无穷大的一个特定的相关器之间的左和右特征向量的相关的非厄米特矩阵X，是频谱密度加权的平方特征值条件数，是由一个简单的公式只涉及径向谱累积分布函数的X。我们表明，这个对象允许计算的平方特征值条件数的条件期望。我们给出的例子，并提供了一个交叉检查的分析预测的大规模数值。
We extend the so-called ‘single ring theorem’ (Feinberg and Zee 1997 Nucl. Phys. B 504 579), also known as the Haagerup–Larsen theorem (Haagerup and Larsen 2000 J. Funct. Anal. 176 331). We do this by showing that in the limit when the size of the matrix goes to infinity a particular correlator between left and right eigenvectors of the relevant non-hermitian matrix X, being the spectral density weighted by the squared eigenvalue condition number, is given by a simple formula involving only the radial spectral cumulative distribution function of X. We show that this object allows the calculation of the conditional expectation of the squared eigenvalue condition number. We give examples and provide a cross-check of the analytic prediction by the large scale numerics.