Spectral density of a Wishart model for nonsymmetric Correlation Matrices

The Wishart model for real symmetric correlation matrices is defined as $\mathsf{W}={\mathsf{A}\mathsf{A}}^t$, where matrix $\mathsf{A}$ is usually a rectangular Gaussian random matrix and ${\mathsf{A}}^t$ is the transpose of $\mathsf{A}$. Analogously, for nonsymmetric correlation matrices, a model may be defined for two statistically equivalent but different matrices $\mathsf{A}$ and $\mathsf{B}$ as ${\mathsf{A}\mathsf{B}}^t$. The corresponding Wishart model, thus, is defined as $\mathbf{C}={\mathsf{A}\mathsf{B}}^t{\mathsf{B}\mathsf{A}}^t$. We study the spectral density of $\mathbf{C}$ for the case when $\mathsf{A}$ and $\mathsf{B}$ are not statistically independent. The ensemble average of such nonsymmetric matrices, therefore, does not simply vanishes to a null matrix. In this paper we derive a Pastur self-consistent equation which describes spectral density of large $\mathbf{C}$. We complement our analytic results with numerics.