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

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<mark>Journal publication date</mark>2013
<mark>Journal</mark>Annales de L'Institut Fourier
Issue number4
Volume63
Number of pages36
Pages (from-to)1227-1262
Publication StatusPublished
<mark>Original language</mark>English

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.