The Characteristic Polynomial of a Random Permutation Matrix at Different Points

TL;DR

This paper investigates the asymptotic behavior of the logarithm of the characteristic polynomial of random permutation matrices at multiple points, revealing independence and normality in the limit, with broader applications to related matrices.

Contribution

It introduces a novel analysis of the characteristic polynomial at multiple points for permutation matrices under Ewens distribution, establishing independence and asymptotic normality.

Findings

Behavior at different points is asymptotically independent

Logarithm of the characteristic polynomial is asymptotically normal

Method extends to related matrices and class functions

Abstract

We consider the logarithm of the characteristic polynomial of random permutation matrices, evaluated on a finite set of different points. The permutations are chosen with respect to the Ewens distribution on the symmetric group. We show that the behavior at different points is independent in the limit and are asymptotically normal. Our methods enables us to study more general matrices, closely related to permutation matrices, and multiplicative class functions.