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

László Erdős; Benjamin Schlein; Horng-Tzer Yau; Jun Yin

Annales de l'I.H.P. Probabilités et statistiques (2012)

  • Volume: 48, Issue: 1, page 1-46
  • ISSN: 0246-0203

Abstract

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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 {xj}j=1N are close to their classical location {γj}j=1N 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 - γ j | 2 - 1 - ε 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.

How to cite

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Erdős, László, et al. "The local relaxation flow approach to universality of the local statistics for random matrices." Annales de l'I.H.P. Probabilités et statistiques 48.1 (2012): 1-46. <http://eudml.org/doc/272033>.

@article{Erdős2012,
abstract = {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 \{xj\}j=1N are close to their classical location \{γj\}j=1N 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 $\mathbb \{E\}|x_\{j\}-\gamma _\{j\}|^\{2\}^\{-1-\varepsilon \}$ 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.},
author = {Erdős, László, Schlein, Benjamin, Yau, Horng-Tzer, Yin, Jun},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {random matrix; sample covariance matrix; Wishart matrix; Wigner–Dyson statistics; Wigner-Dyson statistics},
language = {eng},
number = {1},
pages = {1-46},
publisher = {Gauthier-Villars},
title = {The local relaxation flow approach to universality of the local statistics for random matrices},
url = {http://eudml.org/doc/272033},
volume = {48},
year = {2012},
}

TY - JOUR
AU - Erdős, László
AU - Schlein, Benjamin
AU - Yau, Horng-Tzer
AU - Yin, Jun
TI - The local relaxation flow approach to universality of the local statistics for random matrices
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2012
PB - Gauthier-Villars
VL - 48
IS - 1
SP - 1
EP - 46
AB - 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 {xj}j=1N are close to their classical location {γj}j=1N 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 $\mathbb {E}|x_{j}-\gamma _{j}|^{2}^{-1-\varepsilon }$ 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.
LA - eng
KW - random matrix; sample covariance matrix; Wishart matrix; Wigner–Dyson statistics; Wigner-Dyson statistics
UR - http://eudml.org/doc/272033
ER -

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