Spectral density of generalized Wishart matrices and free multiplicative convolution

We investigate the level density for several ensembles of positive random matrices of a Wishart-like structure, $W=X{X}^{†}$, where $X$ stands for a non-Hermitian random matrix. In particular, making use of the Cauchy transform, we study the free multiplicative powers of the Marchenko-Pastur (MP) distribution, ${\mathrm{MP}}^{⊠s}$, which for an integer $s$ yield Fuss-Catalan distributions corresponding to a product of $s$-independent square random matrices, $X=X_1\cdots X_s$. New formulas for the level densities are derived for $s=3$ and $s=1/3$. Moreover, the level density corresponding to the generalized Bures distribution, given by the free convolution of arcsine and MP distributions, is obtained. We also explain the reason of such a curious convolution. The technique proposed here allows for the derivation of the level densities for several other cases.

Figure 1

Generalized Fuss-Catalan distribution of order 2 $P_{2,c}\left(x\right)$ plotted for rectangularity parameter $c=1/2$.

Figure 2

The continuous part of the generalized Bures distribution $B_{1,c}\left(x\right)$ plotted for rectangularity parameter $c=2$, so the shaded area equals $1/2$.

Figure 3

As in Fig. 2 for $c=4$, so the area under the curve is $1/4$.

Figure 4

Fuss-Catalan distribution $P_3\left(x\right)={\left[P_1\left(x\right)\right]}^{⊠3}$ given in Eq. (28).