# Comparison theorem for some extremal eigenvalue statistics

@article{Landon2020ComparisonTF,
title={Comparison theorem for some extremal eigenvalue statistics},
author={Benjamin Landon and Patrick Lopatto and Jake Marcinek},
journal={The Annals of Probability},
year={2020}
}
• Published 25 December 2018
• Mathematics
• The Annals of Probability
We introduce a method for the comparison of some extremal eigenvalue statistics of random matrices. For example, it allows one to compare the maximal eigenvalue gap in the bulk of two generalized Wigner ensembles, provided that the first four moments of their matrix entries match. As an application, we extend results of Bourgade--Ben Arous and Feng--Wei that identify the limit of the maximal eigenvalue gap in the bulk of the GUE to all complex Hermitian generalized Wigner matrices.
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## References

SHOWING 1-10 OF 38 REFERENCES
Random matrices: Universality of local eigenvalue statistics
• Mathematics
• 2009
In this paper, we consider the universality of the local eigenvalue statistics of random matrices. Our main result shows that these statistics are determined by the first four moments of the
Extreme gaps between eigenvalues of random matrices
• Mathematics
• 2013
This paper studies the extreme gaps between eigenvalues of random matrices. We give the joint limiting law of the smallest gaps for Haar-distributed unitary matrices and matrices from the Gaussian
Extreme gaps between eigenvalues of Wigner matrices
• Mathematics
Journal of the European Mathematical Society
• 2021
This paper proves universality of the distribution of the smallest and largest gaps between eigenvalues of generalized Wigner matrices, under some smoothness assumption for the density of the
Universality of random matrices and local relaxation flow
• Mathematics
• 2009
Consider the Dyson Brownian motion with parameter β, where β=1,2,4 corresponds to the eigenvalue flows for the eigenvalues of symmetric, hermitian and quaternion self-dual ensembles. For any β≥1, we
Eigenvector distribution of Wigner matrices
• Mathematics
• 2011
We consider N × N Hermitian or symmetric random matrices with independent entries. The distribution of the (i, j)-th matrix element is given by a probability measure νij whose first two moments
The Eigenvector Moment Flow and Local Quantum Unique Ergodicity
• Mathematics
• 2013
We prove that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge. This includes a probabilistic version of local quantum unique ergodicity
Tail bounds for gaps between eigenvalues of sparse random matrices
• Computer Science
• 2019
The first eigenvalue repulsion bound for sparse random matrices is proved, and it is shown that these matrices have simple spectrum, improving the range of sparsity and error probability from the work of the second author and Vu.
Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge
• Mathematics
• 2010
This is a continuation of our earlier paper (Tao and Vu, http://arxiv.org/abs/0908.1982v4[math.PR], 2010) on the universality of the eigenvalues of Wigner random matrices. The main new results of
Wegner estimate and level repulsion for Wigner random matrices
• Mathematics
• 2008
We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive