The local relaxation flow approach to universality of the local statistics for random matrices

Laszlo Erdos Institute of Mathematics, University of Munich Benjamin Schlein Department of Pure Mathematics and Mathematical Statistics,University of Cambridge Horng-Tzer Yau Department of Mathematics, Harvard University Jun Yin Department of Mathematics, Harvard University

Probability mathscidoc:1608.28022

Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 48, (1), 1-46, 2012
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 ${x_j}_{j=1}^N$ are close to their classical location ${\gamma_j}_{j=1}^N$ 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 $\E |x_j - \gamma_j |^2 \le N^{-1-\e}$ 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 Wishart matrices.
Random matrix, sample covariance matrix, Wishart matrix, Wigner-Dyson statistics
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@inproceedings{laszlo2012the,
  title={The local relaxation flow approach to universality of the local statistics for random matrices},
  author={Laszlo Erdos, Benjamin Schlein, Horng-Tzer Yau, and Jun Yin},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824104853161408430},
  booktitle={Annales de l'Institut Henri Poincaré, Probabilités et Statistiques},
  volume={48},
  number={1},
  pages={1-46},
  year={2012},
}
Laszlo Erdos, Benjamin Schlein, Horng-Tzer Yau, and Jun Yin. The local relaxation flow approach to universality of the local statistics for random matrices. 2012. Vol. 48. In Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. pp.1-46. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160824104853161408430.
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