Distribution of the Smallest Eigenvalue in Complex and Real Correlated Wishart Ensembles

Abstract

For the correlated Gaussian Wishart ensemble we compute the distribution of the smallest eigenvalue and a related gap probability.We obtain exact results for the complex (β=2) and for the real case (β=1). For a particular set of empirical correlation matrices we find universality in the spectral density, for both real and complex ensembles and all kinds of rectangularity. We calculate the asymptotic and universal results for the gap probability and the distribution of the smallest eigenvalue. We use the Supersymmetry method, in particular the generalized Hubbard-Stratonovich transformation and superbosonization.

30 pages, 2 figures, improving pictures, correcting typos

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Cited by in corpus (5)

• Completing the picture for the smallest eigenvalue of real Wishart matrices
• Eigenvalue Density of the Doubly Correlated Wishart Model: Exact Results
• Limiting Statistics of the Largest and Smallest Eigenvalues in the Correlated Wishart Model
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Asymptotic Coincidence of the Statistics for Degenerate and Non-Degenerate Correlated Real Wishart Ensembles