The distribution of eigenvalues of randomized permutation matrices

Najnudel, Joseph; Nikeghbali, Ashkan

Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 773-838.

Dans cet article, nous étudions en détail une famille d’ensembles de matrices aléatoires qui sont obtenues à partir de matrices de permutation aléatoires en remplaçant les coefficients égaux à un par des variables aléatoires complexes non nulles plus générales. Pour ces ensembles, les valeurs propres peuvent être calculées très explicitement en utilisant la structure en cycles des permutations. De plus, en utilisant les permutations virtuelles, étudiées par Kerov, Olshanski, Vershik et Tsilevich, nous sommes capables de définir, sur le même espace de probabilité, un modèle pour chaque dimension supérieure ou égale à un, ce qui donne un sens à la notion de convergence presque sûre quand la dimension tend vers l’infini. Dans le présent article, selon le modèle précis qui est étudié, nous obtenons différents résultats de convergence pour la mesure ponctuelle des valeurs propres, certains de ces résultats donnant une convergence forte.

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.

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