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

  • Author(s): Soshnikov, Alexander
  • Fyodorov, Yan V.
  • et al.

Published Web Location

https://arxiv.org/pdf/math/0403425.pdf
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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.

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