On the Largest Singular Values of Random Matrices with Independent Cauchy Entries

We apply the method of determinants to study the distribution of the largest singular values of large $m \times n$ real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the (rescaled by a factor of $\frac{1}{m^2\*n^2}$)largest singular values agree in the limit with the statistics of the inhomogeneous Poisson random point process with the intensity $\frac{1}{\pi} x^{-3/2}$ 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 complex rectangular $m \times n$ standard Wishart ensemble and real rectangular $2m \times 2n$ standard Wishart ensemble.