Random permutation matrices under the generalized Ewens measure

TL;DR

This paper explores a generalized Ewens measure for symmetric groups, analyzing moments of characteristic polynomials and the asymptotic behavior of linear statistics like traces of permutation matrices.

Contribution

It introduces a new generalized Ewens measure and studies its impact on moments and asymptotic properties of permutation-related statistics.

Findings

Calculated moments of characteristic polynomials under the new measure

Analyzed asymptotic behavior of linear permutation statistics

Provided insights into multiplicative statistics in symmetric groups

Abstract

We consider a generalization of the Ewens measure for the symmetric group, calculating moments of the characteristic polynomial and similar multiplicative statistics. In addition, we study the asymptotic behavior of linear statistics (such as the trace of a permutation matrix or of a wreath product) under this new measure.