Eigenvector distribution of Wigner matrices

By Antti Knowles, Jun Yin. 2011

138 citations recorded by the source corpus.

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 coincide with those of the corresponding Gaussian ensemble. 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. For eigenvectors associated with bulk eigenvalues, the same conclusion holds provided the first four moments of the distribution νij coincide with those of the corresponding Gaussian ensemble. More generally, we prove that the joint eigenvector–eigenvalue distributions near the spectral edge of two generalized Wigner ensembles agree, provided that the first two moments of the entries match and that one of the ensembles satisfies a level repulsion estimate. If in addition the first four moments match then this result holds also in the bulk.

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

View Paper: https://www.semanticscholar.org/paper/5da6ecfc0f6ffd354d5bfab712560bcd6e8f08b4

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

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Citations in the corpus

Edge universality of correlation matrices (2011, methodology): “= 13 will suﬃce), as proved in [4] for Wigner matrices. The central ideas in this paper are based on the general machinery for proving universality established in a series of recent papers, [3]-[13], [18]-[19], where the authors Yau, Erd˝os, et al. study the distribution of eigenvalues and eigenvect”
A Limit Theorem for Scaled Eigenvectors of Random Dot Product Graphs (2013, background): “Specifically, [27] and [16] provide central limit theorems for the eigenvectors of a Wigner matrix.”
A necessary and sufficient condition for edge universality of Wigner matrices (2012, methodology): “Wigner matrices, covariance matrices, correlation matrices, and adjacency matrices of random graphs [24, 25, 26, 34, 41, 42, 15, 16]. It was also used in the study of the distribution of eigenvectors [33] and the determinant [52] of Wigner matrices. We believe that our new method in the present pape”
The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, methodology): “the extremal bulk eigenvalues of Hecoincide with those of the GOE/GUE ensemble. Indeed, choosing ψ= ϕin Theorem 6.3 and invoking the edge universality for the Wigner matrix H proved in Theorem 1.1 of [28] (for similar results, see also [14,24]), we ﬁnd for all ℓ∈ Nand all bounded and continuous ftha”
The Eigenvector Moment Flow and Local Quantum Unique Ergodicity (2013, methodology): “For eigenvectors near the edges, the matching condition can be reduced to only the first two moments [21].”
A Dynamical Approach to Random Matrix Theory (2017, result): “2 of [28] (a similar result was also obtained in [87, 132] under the condition that the first four moments match with those of the Gaussian): Theorem 18.”
Random matrices: Universality of local spectral statistics of non-Hermitian matrices (2012, methodology): “g of one or two of the coeﬃcients of that ensemble. The original argument in [55] was based on a swapping analysis of individual eigenvalues λi(Mn), which was somewhat complicated technically; but in [21, 31] it was observed that one could work instead with the simpler UNIVERSALITY FOR NON-HERMITIAN”
Universality of local spectral statistics of random matrices (2011, methodology): “oincides with the Tracy-Widom distribution apart from the Hermitian case, when the method of [47] can be applied. The extension of Theorem 9.1 to eigenvectors was recently obtained by Knowles and Yin [48], i.e., under the assumption (9.2), the distributions for the largest eigenvectors coincide. Sim”
Universality of covariance matrices (2011, methodology): “For proving bulk universality of eigenvalues, we follow the general approach for the universality of Gaussian divisible ensembles [17, 18, 10, 11, 25, 26] by embedding the covariance matrix into a stochastic flow of matrices and so that the eigenvalues evolve according to a distinguished coupled sys”
On the principal components of sample covariance matrices (2014, methodology): “8) from [29], and its proof is omitted.”
Eigenvector Statistics of Sparse Random Matrices (2016, background): “For Wigner type random matrices, it is proved in a series of papers [5,13–20,25,26,37] for the bulk and [21,32,36] for the edge, that the eigenvalue statistics are universal; it is proved in [6, 24, 38] that the eigenvectors are asymptotically normal.”