Central limit theorem for multiplicative class functions on the symmetric group

TL;DR

This paper extends a central limit theorem for permutation matrices to multiplicative class functions under Ewens measure, providing covariance calculations and convergence rate estimates.

Contribution

It generalizes previous results to a broader class of functions and measures, including covariance analysis and convergence rate estimation.

Findings

Central limit theorem established for multiplicative class functions under Ewens measure.

Covariance of real and imaginary parts computed in the limit.

Rate of convergence estimated using Wasserstein distance.

Abstract

Hambly, Keevash, O'Connell and Stark have proven a central limit theorem for the characteristic polynomial of a permutation matrix with respect to the uniform measure on the symmetric group. We generalize this result in several ways. We prove here a central limit theorem for multiplicative class functions on symmetric group with respect to the Ewens measure and compute the covariance of the real and the imaginary part in the limit. We also estimate the rate of convergence with the Wasserstein distance.