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

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.

32 pages, 2 figures

References in corpus (11)

• Superbosonization of invariant random matrix ensembles
• On the Eigenvalue Density of Real and Complex Wishart Correlation Matrices
• Superbosonization formula and its application to random matrix theory
• The supersymmetry method of random matrix theory
• Eigenvalue statistics for the sum of two complex Wishart matrices
• Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure
• Integration of Grassmann variables over invariant functions on flat superspaces
• Eigenvalue Density of the Doubly Correlated Wishart Model: Exact Results
• The Correlated Jacobi and the Correlated Cauchy-Lorentz ensembles
• Hyperbolic Hubbard-Stratonovich transformation made rigorous
• Supersymmetry for Products of Random Matrices

Cited by in corpus (8)

• Multiplicative Convolution of Real Asymmetric and Real Antisymmetric Matrices
• On Products of Random Matrices
• Additive Matrix Convolutions of Pólya Ensembles and Polynomial Ensembles
• Hurwitz numbers from Feynman diagrams
• Spectral statistics for the difference of two Wishart matrices
• Finite rank perturbations in products of coupled random matrices: From one correlated to two Wishart ensembles
• Integrals of tau functions
• Universal Broadening of Zero Modes: A General Framework and Identification