The local relaxation flow approach to universality of the local statistics for random matrices
The local relaxation flow approach to universality of the local statistics for random matrices
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DOI：
10.1214/10-aihp388
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发表时间：
2009-11
影响因子：
1.5
通讯作者：
L. Erdős;B. Schlein;H. Yau;J. Yin
中科院分区：
文献类型：
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作者：
L. Erdős;B. Schlein;H. Yau;J. Yin
为了证明一类大随机矩阵的局部谱统计量的普适性，我们对局部松弛流方法进行了推广。我们证明了如果单个矩阵元素的分布是光滑的，并且特征值fxjg N=1接近由特征值的极限密度决定的经典位置fjgn =1，那么特征值的局部分布与相应的高斯系综的局部统计量是一致的。在相邻特征值之间的典型距离为1阶=N的尺度下，对特征值位置的必要先验估计只需要知道平均Ejxj−jj 2°N 1”。这些信息可以通过各种矩阵系综的成熟方法得到。我们通过证明样本协方差矩阵的局部谱通用性来证明该方法。
We present a generalization of the method of the local relaxation flow to establish the universality of local spectral statistics of a broad class of large random matrices. We show that the local distribution of the eigenvalues coincides with the local statistics of the corresponding Gaussian ensemble provided the distribution of the individual matrix element is smooth and the eigenvalues fxjg N=1 are close to their classical location fjg N=1 determined by the limiting density of eigenvalues. Under the scaling where the typical distance between neighboring eigenvalues is of order 1=N, the necessary apriori estimate on the location of eigenvalues requires only to know that Ejxj − jj 2 � N 1 " on average. This information can be obtained by well established methods for various matrix ensembles. We demonstrate the method by proving local spectral universality for sample covariance matrices.