no. 3
The distribution of eigenvalues of randomized permutation matrices
[Sur la distribution des valeurs propres de matrices de permutation randomisées]
Najnudel, Joseph1 ; Nikeghbali, Ashkan1
1 Universität Zürich Institut für Mathematik Winterthurerstrasse 190 8057-Zürich( Switzerland)
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 θ>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.

Reçu le :
Accepté le :
DOI : 10.5802/aif.2777
Classification : 15A18, 15A52, 20B30
Keywords: Random matrix, permutation matrix, virtual permutation, convergence of eigenvalues
Mot clés : matrice aléatoire, matrice de permutation, permutation virtuelle, convergence des valeurs propres

Najnudel, Joseph 1 ; Nikeghbali, Ashkan 1

1 Universität Zürich Institut für Mathematik Winterthurerstrasse 190 8057-Zürich( Switzerland) 
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Najnudel, Joseph; Nikeghbali, Ashkan. The distribution of eigenvalues of randomized permutation matrices. Annales de l'Institut Fourier, Tome 63 (2013) no. 3, pp. 773-838. doi : 10.5802/aif.2777. https://aif.centre-mersenne.org/articles/10.5802/aif.2777/

[1] Anderson, Greg W.; Guionnet, Alice; Zeitouni, Ofer An introduction to random matrices, Cambridge Studies in Advanced Mathematics, 118, Cambridge University Press, Cambridge, 2010, xiv+492 pages | MR | Zbl

[2] Arratia, Richard; Barbour, A. D.; Tavaré, Simon Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2003, xii+363 pages | MR | Zbl

[3] Billingsley, Patrick Convergence of probability measures, Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons Inc., New York, 1999, x+277 pages (A Wiley-Interscience Publication) | MR | Zbl

[4] Dehaye, P-O.; Zeindler, D. On averages of randomized class functions on the symmetric group and their asymptotics (2009) (http://arxiv.org/pdf/0911.4038)

[5] Diaconis, Persi Patterns in eigenvalues: the 70th Josiah Willard Gibbs lecture, Bull. Amer. Math. Soc. (N.S.), Volume 40 (2003) no. 2, pp. 155-178 | DOI | MR | Zbl

[6] Diaconis, Persi; Shahshahani, Mehrdad On the eigenvalues of random matrices, J. Appl. Probab., Volume 31A (1994), pp. 49-62 (Studies in applied probability) | DOI | MR | Zbl

[7] Evans, Steven N. Eigenvalues of random wreath products, Electron. J. Probab., Volume 7 (2002), pp. 1-15 | DOI | MR | Zbl

[8] Hambly, B. M.; Keevash, P.; O’Connell, N.; Stark, D. The characteristic polynomial of a random permutation matrix, Stochastic Process. Appl., Volume 90 (2000) no. 2, pp. 335-346 | DOI | MR | Zbl

[9] Kerov, Serguei; Olshanski, Grigori; Vershik, Anatoli Harmonic analysis on the infinite symmetric group. A deformation of the regular representation, C. R. Acad. Sci. Paris Sér. I Math., Volume 316 (1993) no. 8, pp. 773-778 | MR | Zbl

[10] Mehta, Madan Lal Random matrices, Pure and Applied Mathematics (Amsterdam), 142, Elsevier/Academic Press, Amsterdam, 2004, xviii+688 pages | MR | Zbl

[11] Mezzadri, F.; Snaith, N.-C. (eds.) Recent perspectives in random matrix theory and number theory, London Mathematical Society Lecture Note Series, 322, Cambridge University Press, 2005 | MR | Zbl

[12] Olshanski, G. Asymptotic combinatorics with applications to mathematical physics, Lecture Notes in Mathematics, 1815, Springer, 2003 | MR

[13] Pitman, J. Combinatorial stochastic processes, Lecture Notes in Mathematics, 1875, Springer-Verlag, Berlin, 2006, x+256 pages (Lectures from the 32nd Summer School on Probability Theory held in Saint-Flour, July 7–24, 2002, With a foreword by Jean Picard) | MR | Zbl

[14] Tsilevich, N. V. Distribution of cycle lengths of infinite permutations, J. Math. Sci., Volume 87 (1997) no. 6, pp. 4072-4081 | DOI | MR | Zbl

[15] Tsilevich, N. V. Stationary measures on the space of virtual permutations for an action of the infinite symmetric group (1998) (PDMI Preprint)

[16] Wieand, Kelly Eigenvalue distributions of random permutation matrices, Ann. Probab., Volume 28 (2000) no. 4, pp. 1563-1587 | DOI | MR | Zbl

[17] Wieand, Kelly Permutation matrices, wreath products, and the distribution of eigenvalues, J. Theoret. Probab., Volume 16 (2003) no. 3, pp. 599-623 | DOI | MR | Zbl

[18] Zeindler, Dirk Permutation matrices and the moments of their characteristic polynomial, Electron. J. Probab., Volume 15 (2010) no. 34, pp. 1092-1118 | MR | Zbl

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