  In this article we study in detail a family of random matrix ensembles which
are obtained from random permutations matrices (chosen at random according to
the Ewens measure of parameter $\theta>0$) by replacing the entries equal to
one by more general non-vanishing complex random variables. For these
ensembles, in contrast with more classical models as the Gaussian Unitary
Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very
explicitly computed by using the cycle structure of the permutations. Moreover,
by using the so-called virtual permutations, first introduced by Kerov,
Olshanski and Vershik, and studied with a probabilistic point of view by
Tsilevich, we are able to define, on the same probability space, a model for
each dimension greater than or equal to one, which gives a meaning to the
notion of almost sure convergence when the dimension tends to infinity. In the
present paper, depending on the precise model which is considered, we obtain a
number of different results of convergence for the point measure of the
eigenvalues, some of these results giving a strong convergence, which is not
common in random matrix theory.
