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

By L. Erdős, B. Schlein, Horng-Tzer Yau, Jun Yin. 2009

177 citations recorded by the source corpus.

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.

Subfields: Radiology, Nuclear Medicine and Imaging, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/78e5c7a715ee36f73df59a5fa4295abb97d9d02c

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In Radiology, Nuclear Medicine and Imaging

In Statistics and Probability

Citations in the corpus

Universality for the largest eigenvalue of sample covariance matrices with general population (2013, unknown): “The delocalization property for the eigenvectors of random matrices was also raised in [12] and improved in the series work [13, 14, 19] for Wigner matrices and extended to sample covariance matrices in [35, 39, 16, 31], which asserts that the coefficients of the unit eigenvectors are in the order o”
A necessary and sufficient condition for edge universality of Wigner matrices (2012, methodology): “ee also the work by Ben Arous and P´ech´e [6].) For general Wigner matrices, a new approach was introduced to prove the bulk universality in a series of papers by Er˝odos, Schlein, Yau, and others in [17, 18, 19, 20, 21, 24, 25, 26, 15, 16]. The bulk university for Wigner matrices was also obtained ”
Edge universality of correlation matrices (2011, unknown): “5) of [10]), it can be shown that TrG (1) Q (z)−TrGQ(z) =O(η), η =Iz.”
Fixed Energy Universality for Generalized Wigner Matrices (2014, methodology): “A completely new method based on relaxation of the Dyson Brownian Motion (DBM) to local equilibrium was developed in a series of papers [13,14,19].”
Edge universality for deformed Wigner matrices (2014, background): “Another advantage of the method of the present paper is that it does not require that the eigenvalues evolve autonomously under DBM, i.e., that the stochastic differential equations for the eigenvalue and the eigenvectors decouple under DBM.”
The local semicircle law for a general class of random matrices (2012, methodology): “The techniques used here have been developed in a series of papers [3, 10,12,16].”
Universality of random matrices and local relaxation flow (2009, methodology): “The scale parameter R has a meaning in [ 15 ], but it is purely a choice of convention here.”
Dyson Brownian motion for general $$\beta $$ and potential at the edge (2018, background): “For a good review of the general framework regarding this type of analysis, we refer to the book [21] by Erdős and Yau.”