The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles

Abstract

We calculate the $k$-point generating function of the correlated Jacobi ensemble using supersymmetric methods. We use the result for complex matrices for $k=1$ to derive a closed-form expression for eigenvalue density. For real matrices we obtain the density in terms of a twofold integral that we evaluate numerically. For both expressions we find agreement when comparing with Monte Carlo simulations. Relations between these quantities for the Jacobi and the Cauchy-Lorentz ensemble are derived.

29 pages, 3 figures

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• Exact Multivariate Amplitude Distributions for Non-Stationary Gaussian or Algebraic Fluctuations of Covariances or Correlations
• Matrix Moments in a Real, Doubly Correlated Algebraic Generalization of the Wishart Model
• Stable distributions and domains of attraction for unitarily invariant Hermitian random matrix ensembles
• Hard edge asymptotics of correlation functions between singular values and eigenvalues