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

TL;DR

This paper extends the analysis of permutation matrices by introducing random eigenvalue shifts, generalizing functions beyond characteristic polynomials, exploring other groups, and deriving asymptotic behaviors using the Feller coupling.

Contribution

It introduces new models of random shifts in permutation matrices, generalizes functions based on cycle types, and computes asymptotics for large permutations across various groups.

Findings

Derived explicit generating functions for moments of generalized functions

Analyzed asymptotic behavior of these functions as permutation size grows

Extended results to other groups like the alternating and Weyl groups

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.