Spectral density of products of Wishart dilute random matrices. Part I: the dense case

In this work we study the spectral density of products of Wishart diluted random matrices of the form $X(1)\cdots X(M)(X(1)\cdots X(M))^T$ using the Edwards-Jones trick to map this problem into a system of interacting particles with random couplings on a multipartite graph. We apply the cavity method to obtain recursive relations in typical instances from which to obtain the spectral density. As this problem is fairly rich, we start by reporting in part I a lengthy analysis for the case of dense matrices. Here we derive that the spectral density is a solution of a polynomial equation of degree $M+1$ and obtain exact expressions of it for $M=1$, $2$ and $3$. For general $M$, we are able to find the exact expression of the spectral density only when all the matrices $X(t)$ for $t=1,\ldots, M$ are square. We also make some observations for general $M$, based admittedly on some weak numerical evidence, which we expect to be correct.