Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances

TL;DR

This paper derives exact spectral correlation functions for the sum of two independent complex Wishart matrices with different covariances, using advanced supersymmetric methods, and provides finite-size and asymptotic results.

Contribution

It introduces a comprehensive method to compute spectral statistics of the sum of correlated Wishart matrices with arbitrary covariances, including explicit kernels and finite-N solutions.

Findings

Explicit spectral kernel for the half-degenerate case.

Finite-N spectral density obtained through supersymmetric evaluation.

Numerical simulations confirm theoretical predictions and suggest large-N asymptotics.

Abstract

We compute the spectral statistics of the sum H of two independent complex Wishart matrices, each of which is correlated with a different covariance matrix. Random matrix theory enjoys many applications including sums and products of random matrices. Typically ensembles with correlations among the matrix elements are much more difficult to solve. Using a combination of supersymmetry, superbosonisation and bi-orthogonal functions we are able to determine all spectral k-point density correlation functions of H for arbitrary matrix size N. In the half-degenerate case, when one of the covariance matrices is proportional to the identity, the recent results by Kumar for the joint eigenvalue distribution of H serve as our starting point. In this case the ensemble has a bi-orthogonal structure and we explicitly determine its kernel, providing its exact solution for finite N. The kernel follows from computing the expectation value of a single characteristic polynomial. In the general non-degenerate case the generating function for the k-point resolvent is determined from a supersymmetric evaluation of the expectation value of k ratios of characteristic polynomials. Numerical simulations illustrate our findings for the spectral density at finite N and we also give indications how to do the asymptotic large-N analysis.