Spectral properties of the generalized diluted Wishart ensemble

The celebrated Mar\v{c}enko-Pastur law, that considers the asymptotic spectral density of random covariance matrices, has found a great number of applications in physics, biology, economics, engineering, among others. Here, using techniques from statistical mechanics of spin glasses, we derive simple formulas concerning the spectral density of generalized diluted Wishart matrices. These are defined as $\bm{F}\equiv \frac{1}{2d}\left( \bm{X}\bm{Y}^T+ \bm{Y}\bm{X}^T\right)$, where $\bm{X}$ and $\bm{Y}$ are diluted $N\times P$ rectangular matrices, whose entries correspond to the links of doubly-weighted random bipartite Poissonian graphs following the distribution $P(x_{i}^\mu,y_{i}^{\mu})=\frac{d}{N}\varrho(x_{i}^{\mu},y_{i}^{\mu})+\left(1-\frac{d}{N}\right)\delta_{x_{i}^{\mu},0}\delta_{y_{i}^{\mu},0}$, with the probability density $\varrho(x,y)$ controlling the correlation between the matrices entries of $\bm{X}$ and $\bm{Y}$. Our results cover several interesting cases by varying the parameters of the matrix ensemble, namely, the dilution of the graph $d$, the rectangularity of the matrices $\alpha=N/P$, and the degree of correlation of the matrix entries via the density $\varrho(x,y)$. Finally, we compare our findings to numerical diagonalisation showing excellent agreement.