Le second auteur avait calculé explicitement les fonctions génératrices pour les moments de polynômes caractéristiques de matrices de permutations (sur points). Dans cet article, nous généralisons différents aspects de ces résultats. Nous introduisons des shifts aléatoires des valeurs propres de ces matrices, de deux manières différentes : indépendamment ou pas pour chacun des sous-ensembles de valeurs propres associées au même cycle. Nous considérons aussi des fonctions beaucoup plus générales que ces polynômes caractéristiques, en traduisant notre définition en termes de décompositions en cycles de la permutation. Nous regardons d’autres groupes que les groupes symétriques, tels que les groupes alternés ou d’autres groupes de Weyl. Enfin, nous calculons des résultats asymptotiques lorsque tend vers l’infini. Ce dernier résultat nécessite de nouvelles idées : nous utilisons l’accouplement de Feller, qui donne les lois asymptotiques pour les longueurs de cycles dans des permutations sur beaucoup de points.
The second author had previously obtained explicit generating functions for moments of characteristic polynomials of permutation matrices ( points). In this paper, we generalize many aspects of this situation. We introduce random shifts of the eigenvalues of the permutation matrices, in two different ways: independently or not for each subset of eigenvalues associated to the same cycle. We also consider vastly more general functions than the characteristic polynomial of a permutation matrix, by first finding an equivalent definition in terms of cycle-type of the permutation. We consider other groups than the symmetric group, for instance the alternating group and other Weyl groups. Finally, we compute some asymptotics results when tends to infinity. This last result requires additional ideas: it exploits properties of the Feller coupling, which gives asymptotics for the lengths of cycles in permutations of many points.
@article{AIF_2013__63_4_1227_0,
author = {Dehaye, Paul-Olivier and Zeindler, Dirk},
title = {On averages of randomized class functions on the symmetric groups and~their~asymptotics},
journal = {Annales de l'Institut Fourier},
volume = {63},
year = {2013},
pages = {1227-1262},
doi = {10.5802/aif.2802},
zbl = {06359588},
mrnumber = {3137354},
language = {en},
url = {http://dml.mathdoc.fr/item/AIF_2013__63_4_1227_0}
}
Dehaye, Paul-Olivier; Zeindler, Dirk. On averages of randomized class functions on the symmetric groups and their asymptotics. Annales de l'Institut Fourier, Tome 63 (2013) pp. 1227-1262. doi : 10.5802/aif.2802. http://gdmltest.u-ga.fr/item/AIF_2013__63_4_1227_0/
[1] An elementary view of Euler’s summation formula, Amer. Math. Monthly, Tome 106 (1999) no. 5, pp. 409-418 | Article | MR 1699259 | Zbl 1076.41509
[2] Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics (2003) | MR 2032426 | Zbl 1040.60001
[3] Convergence of probability measures, John Wiley & Sons Inc., New York, Wiley Series in Probability and Statistics: Probability and Statistics (1999) | MR 1700749 | Zbl 0944.60003 | Zbl 0172.21201
[4] Lie groups, Springer-Verlag, New York, Graduate Texts in Mathematics, Tome 225 (2004) | MR 2062813 | Zbl 1053.22001
[5] Analytic Combinatorics, Cambridge University Press, New York (2009) | MR 2483235 | Zbl 1165.05001
[6] Complex analysis, Springer-Verlag, Berlin, Universitext (2005) (Translated from the 2005 German edition by Dan Fulea) | MR 2172762 | Zbl 1085.30001
[7] Probability: a graduate course, Springer, New York, Springer Texts in Statistics (2005) | MR 2125120 | Zbl 1151.60300
[8] The characteristic polynomial of a random permutation matrix, Stochastic Process. Appl., Tome 90 (2000) no. 2, pp. 335-346 | Article | MR 1794543 | Zbl 1047.60013
[9] Symmetric functions and Hall polynomials, The Clarendon Press Oxford University Press, New York, Oxford Mathematical Monographs (1995) | MR 1354144 | Zbl 0487.20007
[10] An invitation to modern number theory, Princeton University Press, Princeton, (2006) | MR 2208019 | Zbl 1155.11001
[11] The distribution of eigenvalues of randomized permutation matrices (2010) http://arxiv.org/abs/1005.0402
[12] Eigenvalue distributions of random matrices in the permutation group and compact Lie groups, Harvard University (1998) (Ph. D. Thesis) | MR 2698352
[13] Eigenvalue distributions of random permutation matrices, Ann. Probab., Tome 28 (2000) no. 4, pp. 1563-1587 | Article | MR 1813834 | Zbl 1044.15017
[14] Permutation matrices, wreath products, and the distribution of eigenvalues, J. Theoret. Probab., Tome 16 (2003) no. 3, pp. 599-623 | Article | MR 2009195 | Zbl 1043.60007
[15] Associated Class Functions and Characteristic Polynomials on the Symmetric Group, University Zürich (2010) (Ph. D. Thesis) | MR 2659758
[16] Permutation matrices and the moments of their characteristics polynomials, Electronic Journal of Probability, Tome 15 (2010), pp. 1092-1118 http://www.math.washington.edu/~ejpecp/EjpVol15/paper34.abs.html | Article | MR 2659758 | Zbl 1225.15038
[17] Central limit theorem for multiplicative class functions on the symmetric group, Journal of Theoretical Probability, OnlineFirst (2011) (doi:10.1007/s10959-011-0382-3)