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 $m\times n$ 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.

We have shown in the revised version that the statistics of the largest eigenavlues of a sample covariance random matrix with i.i.d. Cauchy entries agree in the limit with the statistics of the inhomogeneous Poisson random point process with the intensity $\frac{1}{\pi }{x}^{-3/2}.$

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Cited by in corpus (3)

• Spectral Density of Sparse Sample Covariance Matrices
• Random Matrices in 2D, Laplacian Growth and Operator Theory
• Large deviations of the top eigenvalue of large Cauchy random matrices