Random Matrices: Universality of Local Eigenvalue Statistics up to the Edge

By Terence Tao, Van H. Vu. 2009

289 citations recorded by the source corpus.

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 this paper are an extension of the results in Tao and Vu (http://arxiv.org/abs/0908.1982v4[math.PR], 2010) from the bulk of the spectrum up to the edge. In particular, we prove a variant of the universality results of Soshnikov (Commun Math Phys 207(3):697–733, 1999) for the largest eigenvalues, assuming moment conditions rather than symmetry conditions. The main new technical observation is that there is a significant bias in the Cauchy interlacing law near the edge of the spectrum which allows one to continue ensuring the delocalization of eigenvectors.

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

View Paper: https://www.semanticscholar.org/paper/ea2a86064f49801fa0b226f6179e3fb43b91bcf9

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Citations in the corpus

Random matrices: Sharp concentration of eigenvalues (2012, background): “…for N I ( W n ) (and for closely related objects, such as the Stieltjes transform s W n ( z ) := 1 n trace( W n − z ) − 1 of W n ) for short intervals I , due to the applicability of such results to establishing various universality properties of such matrices; see [11, 12, 13, 30, 31, 14, 16, 17].”
Random matrix theory in statistics: A review (2014, background): “Tao and Vu (2010b, 2011) extended the result of Gustavsson (2005) for the GUE to a class of Hermitian Wigner matrices with nonGaussian entries whose first four moments match the Gaussian moments, which is an instance of the universality phenomena discussed in detail in Section 3.3.”
Rigidity of eigenvalues of generalized Wigner matrices (2010, methodology): “Using this result [27], one can remove the vanishing third moment condition in [41] for Hermitian Wigner ensembles.”
Spectral Statistics of Erdős-Rényi Graphs II: Eigenvalue Spacing and the Extreme Eigenvalues (2011, background): “) The Tracy-Widom law for the largest eigenvalue of Wigner matrices was first proved in [33] under a Gaussian decay assumption, and was proved later in [29, 35, 19, 24] under various weaker restrictions on the distributions of the matrix elements.”
Spectral statistics of Erdős–Rényi graphs I: Local semicircle law (2011, background): “We remark that the universality of Wigner matrices, under certain restrictions on the distribution of the matrix entries, was also established in [36, 37].”
Eigenvector distribution of Wigner matrices (2011, background): “Another approach to both bulk and edge universality was developed in [26, 27, 20].”
Localization and delocalization of eigenvectors for heavy-tailed random matrices (2012, background): “by λn(X) ≤ ··· ≤ λ1(X). In the large nlimit, the spectral properties of this matrix are now well understood as soon as Xij has at least two ﬁnite moments see e.g. [4, 5, 8, 3, 15, 28] for reviews, or [23, 20, 21, 27, 29] for recent results on universality. The starting point of this analysis is the ”
Edge Universality of Beta Ensembles (2013, background): “In the consequent works, after partial results in [47, 54], the symmetry condition was completely eliminated [30].”
Random matrices: Universal properties of eigenvectors (2011, methodology): “Unsurprisingly, the proof of Theorem 8 will rely heavily on the machinery developed in [17], [18], [19].”
Eigenvector Statistics of Sparse Random Matrices (2016, background): “This quantity plays an important role in [36, 37], where it was observed that Qi(A) captures quantitatively the derivatives of the eigenvalues λi of A.”
Universality of covariance matrices (2011, unknown): “There are also related results by [37, 36].”
Transition from Tracy–Widom to Gaussian fluctuations of extremal eigenvalues of sparse Erdős–Rényi graphs (2017, background): “A different approach to edge universality for Wigner matrices based on the direct comparison with corresponding Gaussian ensembles was developed in [28, 59].”
The Isotropic Semicircle Law and Deformation of Wigner Matrices (2011, methodology): “It uses a Lindeberg replacement strategy, which previously appeared in the context of random matrix theory in [14, 33, 34].”
Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population (2014, methodology): “However, as for the deformed Wigner matrices considered in [28], a direct application of the conventional Green function comparison via Lindeberg’s replacement strategy [37] does not work for the sample covariance matrices in the non-null case.”
Universality of local spectral statistics of random matrices (2011, background): “The edge universality without any symmetry assumption was proved in [68] under the condition that the distribution of matrix elements is subexponential decay and the first three moments match those of a Gaussian distribution.”