Random Matrices: the Distribution of the Smallest Singular Values

By Terence Tao, Van H. Vu. 2009

192 citations recorded by the source corpus.

Let ξ be a real-valued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ). The quantity σn(Mn(ξ))2 is thus the least eigenvalue of the Wishart matrix $${M_nM_n^\ast}$$.We show that (under a finite moment assumption) the probability distribution nσn(Mn(ξ))2 is universal in the sense that it does not depend on the distribution of ξ. In particular, it converges to the same limiting distribution as in the special case when ξ is real gaussian. (The limiting distribution was computed explicitly in this case by Edelman.)We also proved a similar result for complex-valued random variables of mean zero, with real and imaginary parts having variance 1/2 and covariance zero. Similar results are also obtained for the joint distribution of the bottom k singular values of Mn(ξ) for any fixed k (or even for k growing as a small power of n) and for rectangular matrices.Our approach is motivated by the general idea of “property testing” from combinatorics and theoretical computer science. This seems to be a new approach in the study of spectra of random matrices and combines tools from various areas of mathematics.

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

View Paper: https://www.semanticscholar.org/paper/4345ac99fcc0d58eea2de480fc4d599126b2bf7d

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Citations in the corpus

Statistical Windows in Testing for the Initial Distribution of a Reversible Markov Chain (2018, background): “Understanding the spectra of random (symmetric) matrices has been intensely studied in recent years (see e.g. Tao and Vu (2010); Chafäı (2009); Rudelson and Vershynin (2008); Vershynin (2014)).”
RANDOM COVARIANCE MATRICES: UNIVERSALITY OF LOCAL STATISTICS OF EIGENVALUES UP TO THE EDGE (2011, background): “Recently a universality result has been established by Tao and Vu [11] for the entries with bounded sufficiently high moments.”
Small Ball Probability, Inverse Theorems, and Applications (2012, background): “Edelman’s conjecture has been proved by Tao and the second author in [64].”
Random matrix theory in statistics: A review (2014, background): “More complete and self-containing treatments of a number of “core” topics of RMT not covered in this paper, including the Riemann–Hilbert approach to the asymptotics of orthogonal polynomials and random matrices, the distribution of spacings and correlation functions of eigenvalues and their connect”
Lower bounds for the smallest singular value of structured random matrices (2016, methodology): “The limiting distribution of √ nsn(X) was obtained by Edelman for the case of Gaussian entries [11], and this law was shown by Tao and Vu to hold for all iid matrices with entries ξij having a sufficiently large finite moment [35].”
Spectral norm of products of random and deterministic matrices (2008, background): “This will quickly yield the main Theorem 1.1 for almost square matrices, see Corollary 4.11 below.”
Around the circular law (2011, methodology): “This was almost solved by Rudelson and Vershynin [127] and Tao and Vu [148].”
The smallest singular value of inhomogeneous square random matrices (2019, background): “The behavior of the smallest singular values of random matrices have been extensively studied [2, 4, 5, 11, 15, 16, 17, 20, 22, 24, 25, 26, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43].”
Random matrices: Law of the determinant (2011, methodology): “n ≤ k ≤ n/2, there exist 2k singular values of An in the interval [0,ck/ √ n], for some absolute constant c, with probability at least 1−O(n−101). We can prove Lemma 4.2 by following the arguments in [26], Lemma 4.1, almost word by word. By the interlacing law and Lemma 4.2, we conclude that B′ has ”
Concentration inequalities for random tensors (2019, unknown): “For d = 1, this problem has been extensively studied in random matrix theory [16, 31, 23, 32, 25, 3, 26, 34, 27, 13, 10, 35, 36].”
Universality of covariance matrices (2011, background): “(2) For the ‘square case’ (N = M), in [35] the authors proceeded via analyzing”