For a broad class of bi-unitarily invariant random matrix ensembles, the large-n limit of the joint density of one eigenradius and k singular values at the hard edge is expressed through the limiting kernel of the singular value process.
The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
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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.
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Hard edge asymptotics of correlation functions between singular values and eigenvalues
For a broad class of bi-unitarily invariant random matrix ensembles, the large-n limit of the joint density of one eigenradius and k singular values at the hard edge is expressed through the limiting kernel of the singular value process.