RANDOM MATRICES: THE DISTRIBUTION OF THE SMALLEST SINGULAR VALUES

Let xi be a real-valued random variable of mean zero and variance 1. Let M-n(xi) denote the n x n random matrix whose entries are iid copies of xi and sigma(n)(M-n(xi)) denote the least singular value of M-n(xi). The quantity sigma(n)(M-n(xi))(2) is thus the least eigenvalue of the Wishart matrix MnMn*. We show that (under a finite moment assumption) the probability distribution n sigma(n)(M-n(xi))(2) is universal in the sense that it does not depend on the distribution of xi. In particular, it converges to the same limiting distribution as in the special case when xi 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(xi) 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.