The authors study a family of random matrix ensembles which are obtained from random permutations matrices (chosen at random according to the Ewens measure of a 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 S. Kerov, A. Olshanski and A. Vershik [C. R. Acad. Sci., Paris, Sér. I 316, 773–778 (1993; Zbl 0796.43005)], and studied with a probabilistic point of view by N. V. Tsilevich [J. Math. Sci., New York 87, No. 6, 4072–4081 (1997); translation from Zap. Nauchn. Semin. POMI 223, 148–161 (1995; Zbl 0909.60011)], the authors 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. Depending on the precise model which is considered, 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) were obtained.