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

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

2004-03-25

arxiv.org/abs/math/0403425v3

J. Math. Phys. 46, 033302 (2005), 15 pages

Mathematics

Probability

We have shown in the revised version that the statistics of the largest eigenavlues of a sample covariance random matrix with

Scientific paper

10.1063/1.1855932

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.

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