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

Dehaye, Paul-Olivier and Zeindler, Dirk (2013) On averages of randomized class functions on the symmetric groups and their asymptotics. Annales de L'Institut Fourier, 63 (4). pp. 1227-1262. ISSN 1777-5310

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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.

Item Type:
Journal Article
Journal or Publication Title:
Annales de L'Institut Fourier
Uncontrolled Keywords:
/dk/atira/pure/subjectarea/asjc/2600/2608
Subjects:
ID Code:
70745
Deposited By:
Deposited On:
12 Sep 2014 08:41
Refereed?:
Yes
Published?:
Published
Last Modified:
01 Jan 2020 08:58