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

Paul-Olivier Dehaye[1]; Dirk Zeindler[2]

  • [1] Institut für Mathematik Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich
  • [2] Universität Bielefeld SFB 701 Postfach: 100 131 33501 Bielefeld Deutschland

Annales de l’institut Fourier (2013)

  • Volume: 63, Issue: 4, page 1227-1262
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Dehaye, Paul-Olivier, and Zeindler, Dirk. "On averages of randomized class functions on the symmetric groups and their asymptotics." Annales de l’institut Fourier 63.4 (2013): 1227-1262. <http://eudml.org/doc/275640>.

@article{Dehaye2013,
abstract = {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.},
affiliation = {Institut für Mathematik Universität Zürich Winterthurerstrasse 190 CH-8057 Zürich; Universität Bielefeld SFB 701 Postfach: 100 131 33501 Bielefeld Deutschland},
author = {Dehaye, Paul-Olivier, Zeindler, Dirk},
journal = {Annales de l’institut Fourier},
keywords = {symmetric group; characteristic polynomial; associated class functions; generating functions; Feller coupling; asymptotics of moments},
language = {eng},
number = {4},
pages = {1227-1262},
publisher = {Association des Annales de l’institut Fourier},
title = {On averages of randomized class functions on the symmetric groups and their asymptotics},
url = {http://eudml.org/doc/275640},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Dehaye, Paul-Olivier
AU - Zeindler, Dirk
TI - On averages of randomized class functions on the symmetric groups and their asymptotics
JO - Annales de l’institut Fourier
PY - 2013
PB - Association des Annales de l’institut Fourier
VL - 63
IS - 4
SP - 1227
EP - 1262
AB - 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.
LA - eng
KW - symmetric group; characteristic polynomial; associated class functions; generating functions; Feller coupling; asymptotics of moments
UR - http://eudml.org/doc/275640
ER -

References

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  1. Tom M. Apostol, An elementary view of Euler’s summation formula, Amer. Math. Monthly 106 (1999), 409-418 Zbl1076.41509MR1699259
  2. Richard Arratia, A.D. Barbour, Simon Tavaré, Logarithmic combinatorial structures: a probabilistic approach, (2003) Zbl1040.60001MR2032426
  3. Patrick Billingsley, Convergence of probability measures, (1999), John Wiley & Sons Inc., New York Zbl0172.21201MR1700749
  4. Daniel Bump, Lie groups, 225 (2004), Springer-Verlag, New York Zbl1053.22001MR2062813
  5. Philippe Flajolet, Robert Sedgewick, Analytic Combinatorics, (2009), Cambridge University Press, New York Zbl1165.05001MR2483235
  6. Eberhard Freitag, Rolf Busam, Complex analysis, (2005), Springer-Verlag, Berlin Zbl1085.30001MR2172762
  7. Allan Gut, Probability: a graduate course, (2005), Springer, New York Zbl1151.60300MR2125120
  8. B.M. Hambly, P. Keevash, N. O’Connell, D. Stark, The characteristic polynomial of a random permutation matrix, Stochastic Process. Appl. 90 (2000), 335-346 Zbl1047.60013MR1794543
  9. I. G. Macdonald, Symmetric functions and Hall polynomials, (1995), The Clarendon Press Oxford University Press, New York Zbl0487.20007MR1354144
  10. Steven J. Miller, Ramin Takloo-Bighash, An invitation to modern number theory, (2006), Princeton University Press, Princeton, Zbl1155.11001MR2208019
  11. Joseph Najnudel, Ashkan Nikeghbali, The distribution of eigenvalues of randomized permutation matrices, (2010) Zbl1188.60021
  12. Kelly Wieand, Eigenvalue distributions of random matrices in the permutation group and compact Lie groups, (1998) Zbl1044.15017MR2698352
  13. Kelly Wieand, Eigenvalue distributions of random permutation matrices, Ann. Probab. 28 (2000), 1563-1587 Zbl1044.15017MR1813834
  14. Kelly Wieand, Permutation matrices, wreath products, and the distribution of eigenvalues, J. Theoret. Probab. 16 (2003), 599-623 Zbl1043.60007MR2009195
  15. Dirk Zeindler, Associated Class Functions and Characteristic Polynomials on the Symmetric Group, (2010) Zbl1225.15038MR2659758
  16. Dirk Zeindler, Permutation matrices and the moments of their characteristics polynomials, Electronic Journal of Probability 15 (2010), 1092-1118 Zbl1225.15038MR2659758
  17. Dirk Zeindler, Central limit theorem for multiplicative class functions on the symmetric group, Journal of Theoretical Probability, OnlineFirst (2011) Zbl1286.60030

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