On a limiting point process related to modified permutation matrices

TL;DR

This paper studies the eigenvalue distribution of modified permutation matrices under Ewens' measures, showing that the number of eigenvalues in large intervals follows an asymptotic normal distribution.

Contribution

It introduces a new limiting point process for eigenvalues of modified permutation matrices and proves asymptotic normality of eigenvalue counts in large intervals.

Findings

Eigenvalues form a limiting point process as matrix size grows.

Eigenvalue counts in large intervals are asymptotically normal.

Results hold for both modified and unmodified permutation matrices.

Abstract

We consider random permutation matrices following a one-parameter family of deformations of the uniform distribution, called Ewens' measures, and modifications of these matrices where the entries equal to one are replaced by i.i.d uniform random variables on the unit circle. For each of these two ensembles of matrices, rescaling properly the eigenangles provides a limiting point process as the size of the matrices goes to infinity. If $J$ is an interval of $\mathbb{R}$, we show that, as the length of $J$ tends to infinity, the number of points lying in $J$ of the limiting point process related to modified permutation matrices is asymptotically normal. Moreover, for permutation matrices without modification, if $a$ and $a+b$ denote the endpoints of $J$, we still have an asymptotic normality for the number of points lying in $J$, in the two following cases: [$a$ fixed and $b \to \infty$] and [$a,b \to \infty$ with $b$ proportional to $a$].