Random matrices: The distribution of the smallest singular values

Abstract

Let $\a$ be a real-valued random variable of mean zero and variance 1. Let $M_n(\a)$ denote the $n \times n$ random matrix whose entries are iid copies of $\a$ and $\sigma_n(M_n(\a))$ denote the least singular value of $M_n(\a)$. ($\sigma_n(M_n(\a))^2$ is also usually interpreted as the least eigenvalue of the Wishart matrix $M_n M_n^{\ast}$.) We show that (under a finite moment assumption) the probability distribution $n \sigma_n(M_n(\a))^2$ is {\it universal} in the sense that it does not depend on the distribution of $\a$. In particular, it converges to the same limiting distribution as in the special case when $a$ 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 $M_n(\a)$ 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.