Universality for generalized Wigner matrices with Bernoulli distribution

By L. Erdős, Horng-Tzer Yau, Jun Yin. 2010

142 citations recorded by the source corpus.

The universality for the eigenvalue spacing statistics of generalized Wigner matrices was established in our previous work \cite{EYY} under certain conditions on the probability distributions of the matrix elements. A major class of probability measures excluded in \cite{EYY} are the Bernoulli measures. In this paper, we extend the universality result of \cite{EYY} to include the Bernoulli measures so that the only restrictions on the probability distributions of the matrix elements are the subexponential decay and the normalization condition that the variances in each row sum up to one. The new ingredient is a strong local semicircle law which improves the error estimate on the Stieltjes transform of the empirical measure of the eigenvalues from the order $(N \eta)^{-1/2}$ to $(N \eta)^{-1}$. Here $\eta$ is the imaginary part of the spectral parameter in the definition of the Stieltjes transform and $N$ is the size of the matrix.

Subfields: Discrete Mathematics and Combinatorics, Mathematical Physics, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/88a10d127dbd58a6cb8771f51fe53f8cd461c293

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Citations in the corpus

A necessary and sufficient condition for edge universality of Wigner matrices (2012, methodology): “) For general Wigner matrices, a new approach was introduced to prove the bulk universality in a series of papers by Erdős, Schlein, Yau, and others in [17, 18, 19, 20, 21, 24, 25, 26, 15, 16].”
The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, methodology): “Another key ingredient in the proof of universality of random matrices is the Green function comparison method introduced in [26].”
Lectures on the local semicircle law for Wigner matrices (2016, unknown): “04 05 5v 3 [ m at h. PR ] 1 8 O ct 2 01 6 Contents”
The Eigenvector Moment Flow and Local Quantum Unique Ergodicity (2013, background): “We remark that although this proof is very simple, the fact that the Dyson Brownian motion preserves the detailed behaviour of eigenvalues and eigenvectors is surprising and even contradictory.”
Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, methodology): “Optimal local laws in the bulk and at the edges of the spectrum were obtained for Wigner matrices in [18, 17, 15].”
Universality of local spectral statistics of random matrices (2011, methodology): “The real Bernoulli random matrices required a more refined argument [37].”
Universality of Wigner random matrices: a survey of recent results (2010, methodology): “sitive ξbut the method has become general. The result was further generalized in [46] to quaternion self-dual Wigner matrices and sample covariance matrices and even to generalized Wigner matrices in [48, 49] (see Deﬁnition 1.1). Finally, in [50] we showed that the local equilibrium is already reach”
Universality for the largest eigenvalue of sample covariance matrices with general population (2013, methodology): “This strategy was raised in [17, 18, 19] for deriving both bulk and edge universality of a class of generalized Wigner matrices, and used in [11] for the local spectral statistics of Erdös-Rényi graphs.”
Universality of random matrices and local relaxation flow (2009, background): “ove all assumptions from Theorem 2.3 except for a decay condition and a technical condition that νis supported in at least three points. This latter technical condition was removed in our later paper [16], where we generalized our approach to a broader class of random matrix ensembles. Corollary 2.4”
Eigenvector distribution of Wigner matrices (2011, background): “We prove that the joint probability distribution of the components of eigenvectors associated with eigenvalues close to the spectral edge agrees with that of the corresponding Gaussian ensemble.”
Rigidity of eigenvalues of generalized Wigner matrices (2010, methodology): “3 was proved in [23], a similar result, with no error estimate, was obtained in [20] for the hermitian case by using an explicit formula related to Johansson’s formula [26].”
A Dynamical Approach to Random Matrix Theory (2017, methodology): “These optimal N -dependences were first achieved in [68] by introducing the fluctuation averaging mechanism.”
Isotropic local laws for sample covariance and generalized Wigner matrices (2013, background): “We remark that the fluctuation averaging mechanism was first exploited in [14].”
Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues (2011, methodology): “Neglecting the mean zero condition, the Erdős-Rényi matrix becomes a Wigner random matrix with a Bernoulli distribution when 0 p 1 is a constant independent of N .”