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The Eigenvector Moment Flow and local Quantum Unique Ergodicity
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In this article, the authors proved that the distribution of eigenvectors of generalized Wigner matrices is universal both in the bulk and at the edge, using the eigenvector moment flow, a multiparticle random walk in a random environment.Abstract:
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 and asymptotic normality of the eigenvector entries. The proof relies on analyzing the eigenvector flow under the Dyson Brownian motion. The key new ideas are: (1) the introduction of the eigenvector moment flow, a multi-particle random walk in a random environment, (2) an effective estimate on the regularity of this flow based on maximum principle and (3) optimal finite speed of propagation holds for the eigenvector moment flow with very high probability.read more
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Cleaning large correlation matrices: tools from random matrix theory
TL;DR: In this article, a review of recent results concerning the estimation of large covariance matrices using tools from Random Matrix Theory (RMT) is presented, with an emphasis on the Marchenko-Pastur equation that provides information on the resolvent of multiplicatively corrupted noisy matrices.
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Fixed energy universality of Dyson Brownian motion
TL;DR: In this paper, it was shown that the local eigenvalue statistics coincide with the GOE/GUE in the fixed energy sense after time t ≳ 1 / N if the density of states of V is bounded above and below down to scales η ≪ t in a window of size L ≫ t.
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Lectures on the local semicircle law for Wigner matrices
TL;DR: The local semicircle law of Wigner matrices has been studied in this paper, which states that the eigenvalue distribution of a random matrix with independent upper-triangular entries with zero expectation and constant variance is very similar to Wigners' distribution, down to spectral scales containing slightly more than one eigen value.
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Local Semicircle Law for Random Regular Graphs
TL;DR: In this paper, it was shown that the Green's function of the adjacency matrix and the Stieltjes transform of its empirical spectral measure are well approximated by Wigner's semicircle law, down to the optimal scale given by the typical eigenvalue spacing.
References
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Characterization of chaotic quantum spectra and universality of level fluctuation laws
TL;DR: In this article, it was found that the level fluctuations of the quantum Sinai's billiard are consistent with the predictions of the Gaussian orthogonal ensemble of random matrices.
Book ChapterDOI
Introduction to Random Matrices
Craig A. Tracy,Harold Widom +1 more
TL;DR: In this paper, a simplified derivation of the system of nonlinear completely integrable equations (the aj's are the independent variables) that were first derived by Jimbo, Miwa, Mori, and Sato in 1980 was presented.
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Uniform distribution of eigenfunctions on compact hyperbolic surfaces
TL;DR: In this article, the fonctions propres {φ k } du laplacien sur une surface hyperbolique compacte X devient uniformement distribuee sur X quand k→∞
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Ergodicité et fonctions propres du laplacien
TL;DR: In this paper, the preuve d'une generalisation d'un resultat recent de S. Zelditch concernant la repartition asymptotique des fonctions propres du laplacien sur une variete compacte don le flot geodesique est ergodique.
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Random matrices: Universality of local eigenvalue statistics
Terence Tao,Van Vu +1 more
TL;DR: The universality of the local eigenvalue statistics of random matrices is studied in this article, where it is shown that these statistics are determined by the first four moments of the distribution of the entries.