The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles

TL;DR

This paper derives explicit formulas for the eigenvalue density of correlated Jacobi ensembles using supersymmetric methods, providing numerical evaluations and establishing relations with Cauchy-Lorentz ensembles.

Contribution

It introduces a supersymmetric approach to compute eigenvalue densities for correlated Jacobi ensembles and relates these results to Cauchy-Lorentz ensembles.

Findings

Closed-form eigenvalue density for complex matrices

Numerical density evaluation for real matrices

Agreement with Monte Carlo simulations

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.