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

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.

Random permutation matrices under the generalized Ewens measure

“…In Section 3 we study the multiplicative class functions associated to a function f and obtain the asymptotic behavior of the joint moments. In particular, we extend earlier results of [5,28] and of [12] on the characteristic polynomial of uniformly chosen random permutation matrices.…”

“…Moments of multiplicative class functions. We extend in this section the results of [5,12] and [28] to the generalized Ewens measure P Θ . More precisely, we compute the asymptotic behavior of the moments of the characteristic polynomial Z N (x) and of multiplicative class functions W N (P ) with respect to P Θ using the methods of generating functions and singularity analysis introduced in the previous section.…”

“…Equation (1.1) has been used by Dehaye and Zeindler [5] to introduce multiplicative class functions (i.e., invariant under conjugation) associated to function f . These functions have the form…”

The characteristic polynomial of a random permutation matrix at different points

“…Then, the characteristic polynomial of M (σ, z) has the same zeros as Z n,z (x). We will study the characteristic polynomial by identifying it with Z n,z (x), following the convention of [7], [26] or [27]. By using that the random variables z i , 1 ≤ i ≤ n are i.i.d., a simple computation shows the following equality in law (see [7], Lemma 4.2):…”

“…Indeed, the methods allow us to prove CLT's for multiplicative class functions. Multiplicative class functions have been studied by Dehaye and Dehaye-Zeindler, [7], [27].…”