On the largest singular values of random matrices with independent Cauchy entries

Abstract

We apply the method of determinants to study the distribution of the largest singular values of large m×n real rectangular random matrices with independent Cauchy entries. We show that for a special one-parametric class of statistics the properties of the largest singular values (rescaled by a factor 1∕m2n2) agree in the limit with the statistical properties of the Poisson random point process with the intensity (1∕π)x−3∕2 and, therefore, are different from the Tracy–Widom law. Among other corollaries of our method we show an interesting connection between the mathematical expectations of the determinants of the complex rectangular m×n standard Wishart ensemble and the real rectangular 2m×2n standard Wishart ensemble.