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Random Matrices: the Distribution of the Smallest Singular Values

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Abstract

Let ξ be a real-valued random variable of mean zero and variance 1. Let M n (ξ) denote the n × n random matrix whose entries are iid copies of ξ and σ n (M n (ξ)) denote the least singular value of M n (ξ). The quantity σ n (M n (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 n σ n (M n (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 M n (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.

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Authors and Affiliations

  1. Department of Mathematics, UCLA, Los Angeles, CA, 90095-1555, USA

    Terence Tao

  2. Department of Mathematics, Rutgers University, Piscataway, NJ, 08854-8019, USA

    Van Vu

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  1. Terence Tao
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  2. Van Vu
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Correspondence to Van Vu.

Additional information

T.T. is supported by a grant from the MacArthur Foundation, and by NSF grant DMS-0649473. VV is supported by grants DMS-0901216 and AFPRS-FA-9550-09-1-0167.

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Tao, T., Vu, V. Random Matrices: the Distribution of the Smallest Singular Values. Geom. Funct. Anal. 20, 260–297 (2010). https://doi.org/10.1007/s00039-010-0057-8

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