Random matrices: Universality of local eigenvalue statistics

By Terence Tao, Van H. Vu. 2009

532 citations recorded by the source corpus.

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 distribution of the entries. As a consequence, we derive the universality of eigenvalue gap distribution and k-point correlation, and many other statistics (under some mild assumptions) for both Wigner Hermitian matrices and Wigner real symmetric matrices.

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

View Paper: https://www.semanticscholar.org/paper/1961250a1b27286eb29055456aa25996fa844451

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In Discrete Mathematics and Combinatorics

In Mathematical Physics

In Statistics and Probability

Citations in the corpus

Random matrices: Universal properties of eigenvectors (2011, methodology): “.,pk,q1,...,qk. The deduction of Theorem 7 from Corollary 4 and Theorem 8 is routine and is performed in Section 2. Theorem 8 is an extension of the four moment theorem for eigenvalues established in [17], [18], [19]. Indeed, the latter theorem is essentially the special case of Theorem 8 in which k”
RANDOM COVARIANCE MATRICES: UNIVERSALITY OF LOCAL STATISTICS OF EIGENVALUES UP TO THE EDGE (2011, methodology): “We assume substantial familiarity with the materials in [13], [12], and will cite from them repeatedly.”
Central Limit Theorems for Linear Statistics of Heavy Tailed Random Matrices (2013, background): “The absence of normalization by √ N shows that the eigenvalues of A ﬂuctuate very little, as precisely studied by Erd¨os, Schlein, Yau, Tao, Vu and their co-authors, who analyzed their rigidity in e.g. [ 25, 26, 46].”
Spectral Gaps of Random Graphs and Applications (2012, background): “tance from the spectral edge, and edge statistics which pertain to extremal eigenvalues. In the case of Wigner matrices, many of these questions have been resolved in very great generality (see, e.g. [32]). Among the edge statistics, the most pertinent is the distribution of the largest eigenvalue, ”
Bulk universality for Wigner matrices (2009, result): “timate the ﬂuctuations of the empirical density of eigenvalues in short scales. Shortly after this manuscript appeared on the arXiv, we learned that our main result was also obtained by Tao and Vu in [18] under essentially no regularity conditions on the initial distribution ν provided the third mom”
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.”
Alternative asymptotics for cointegration tests in large VARs (2016, unknown): “Given that Conjecture 9 holds, limp,T→c∞ LR0,p,T p limT→∞,p→∞LR0,p,T = 1 + c 2c2 ln (1 + c)− 1− c 2c2 ln (1− c) + 1− 2c 2c2 ln (1− 2c) .”
The Elliptic Law (2012, methodology): “Recently, this technique has also been applied to study random matrices with exchangeable or independent entries (see, for example, [5, 47] and references therein).”
Localization and delocalization of eigenvectors for heavy-tailed random matrices (2012, background): “[3–5,8,15,28] for reviews, or [20,21,23,27,29] for recent results on universality.”
Universality for Random Tensors (2011, background): “The spacing of eigenvalues for instance is determined only by the first four moments of the distribution of the matrix entries [36] and follows Dyson’s sine law [11,12].”