Spectral density of the noncentral correlated Wishart ensembles

Wishart ensembles of random matrix theory have been useful in modeling positive definite matrices encountered in classical and quantum chaotic systems. We consider nonzero means for the entries of the constituting matrix $\mathbb{A}$ which defines the correlated Wishart matrix as $\mathbb{W}={\mathbb{AA}}^{†}$, and refer to the ensemble of such Wishart matrices as the noncentral correlated Wishart ensemble (nc-CWE). We derive the Pastur self-consistent equation which describes the spectral density of nc-CWE at large matrix dimension.

Figure 1

Spectral density of the nc-WOE where $B_{j\nu }={\delta }_{j\nu }\mu \sqrt{j},N=1024,T=2N,{\sigma }^2=1$, and $\mu =0,1,3,$ and 10. Solid lines in this figure represent the theory obtained from the numerical solution of Eq. (44) and open circles represent the histogram data obtained from the Monte Carlo simulation of $\mathsf{C}$. Dashed line represents the Marčenko-Pastur density, i.e., the $\mu =0$ case. In the inset (a) we show the density for $\mu =10$ and in (b) we show the density for $\lambda \le 1$ for the same $\mu$.

Figure 2

Spectral density for nc-CWOE where ${\xi }_{jk}={\delta }_{jk}+(1-{\delta }_{jk}){\mu }_0^{\left(\right|j-k\left|\right)}$ and ${\mathsf{B}}_{j\nu }={\mu }^{|j-\nu |}$ with $\mu =0.5$ and ${\sigma }^2=0.25$. In this figure we have used $N=512$ and $T=2N$. In the main figure we show the spectral density on semilog plot for ${\mu }_0=0.1,0.3$, and $0.5$, respectively with black, orange, and brown colors. Solids lines represent the theory (55) for nc-CWE and dashed lines represent the corresponding $\mu =0$ cases. We use the same color code for ${\mu }_0$ in the inset where we compare only theory for $\mu =0$ and $\mu =0.5$ on the log-log scale.