On averages of randomized class functions on the symmetric groups and their asymptotics

Dehaye, Paul-Olivier; Zeindler, Dirk

Annales de l'Institut Fourier, Tome 63 (2013) no. 4, pp. 1227-1262.

Le second auteur avait calculé explicitement les fonctions génératrices pour les moments de polynômes caractéristiques de matrices de permutations (sur $n$ 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 $n$ 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 ($n$ 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 $n$ 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.

[1] Apostol, Tom M. An elementary view of Euler’s summation formula, Amer. Math. Monthly, Volume 106 (1999) no. 5, pp. 409-418 | DOI | MR | Zbl

[2] Arratia, Richard; Barbour, A.D.; Tavaré, Simon Logarithmic combinatorial structures: a probabilistic approach, EMS Monographs in Mathematics, 2003 | 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 | MR | Zbl

[4] Bump, Daniel Lie groups, Graduate Texts in Mathematics, 225, Springer-Verlag, New York, 2004 | MR | Zbl

[5] Flajolet, Philippe; Sedgewick, Robert Analytic Combinatorics, Cambridge University Press, New York, 2009 | MR | Zbl

[6] Freitag, Eberhard; Busam, Rolf Complex analysis, Universitext, Springer-Verlag, Berlin, 2005 (Translated from the 2005 German edition by Dan Fulea) | MR | Zbl

[7] Gut, Allan Probability: a graduate course, Springer Texts in Statistics, Springer, New York, 2005 | 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] Macdonald, I. G. Symmetric functions and Hall polynomials, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1995 | MR | Zbl

[10] Miller, Steven J.; Takloo-Bighash, Ramin An invitation to modern number theory, Princeton University Press, Princeton,, 2006 | MR | Zbl

[11] Najnudel, Joseph; Nikeghbali, Ashkan The distribution of eigenvalues of randomized permutation matrices (2010) | arXiv

[12] Wieand, Kelly Eigenvalue distributions of random matrices in the permutation group and compact Lie groups, Harvard University (1998) (Ph. D. Thesis) | MR

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

[14] 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

[15] Zeindler, Dirk Associated Class Functions and Characteristic Polynomials on the Symmetric Group, University Zürich (2010) (Ph. D. Thesis) | MR

[16] Zeindler, Dirk Permutation matrices and the moments of their characteristics polynomials, Electronic Journal of Probability, Volume 15 (2010), pp. 1092-1118 http://www.math.washington.edu/~ejpecp/EjpVol15/paper34.abs.html | DOI | MR | Zbl

[17] Zeindler, Dirk Central limit theorem for multiplicative class functions on the symmetric group, Journal of Theoretical Probability, OnlineFirst (2011) (doi:10.1007/s10959-011-0382-3)