Characteristic polynomials of modified permutation matrices at microscopic scale

TL;DR

This paper investigates the asymptotic behavior of characteristic polynomials of random permutation matrices and their modifications, revealing convergence to explicit limiting functions and connecting to Circular Unitary Ensemble results.

Contribution

It introduces new results on the limiting distribution of characteristic polynomials for modified permutation matrices under various measures, including Ewens' measures.

Findings

Characteristic polynomial converges to an explicit entire function

Results apply to matrices with entries replaced by i.i.d uniform variables

Connections established with Circular Unitary Ensemble

Abstract

We study the characteristic polynomial of random permutation matrices following some measures which are invariant by conjugation, including Ewens' measures which are one-parameter deformations of the uniform distribution on the permutation group. We also look at some modifications of permutation matrices where the entries equal to one are replaced by i.i.d uniform variables on the unit circle. Once appropriately normalized and scaled, we show that the characteristic polynomial converges in distribution on every compact subset of $\mathbb{C}$ to an explicit limiting entire function, when the size of the matrices goes to infinity. Our findings can be related to results by Chhaibi, Najnudel and Nikeghbali on the limiting characteristic polynomial of the Circular Unitary Ensemble.