  Random tensors can be used to produce random matrices. This idea is, for
instance, very natural when one studies random quantum states with the aim of
exploring properties that are generically true, or true with some probability.
We hereby study the moments generating function, in the sense of the Stieltjes
transform - i.e. the resolvent -, of a random matrix defined as a product of
two different marginals of the same random tensor. We study the resolvent in
two different asymptotical regimes.
  In the first regime, the resolvent is easily computed thanks to freeness
results for the two different marginals and straightforward application of free
harmonic analysis. In the second regime, we show that the resolvent satisfies
an algebraic equation of degree six. This algebraic equation possesses roots
whose expressions can be given explicitly in terms of radicals. We obtain this
result by using an enumerative combinatorics approach. One of the interesting
aspects of the second regime is that the corresponding probability density
function interpolates between the square of a Marchenko-Pastur and the free
multiplicative square of a Marchenko-Pastur law.
