On the number of eigenvalues of modified permutation matrices in mesoscopic intervals

TL;DR

This paper studies the eigenvalue distribution of permutation matrices under Ewens' distribution and their modifications, showing Gaussian fluctuations for eigenvalue counts in fixed arcs and analyzing spacing between eigenvalues.

Contribution

It introduces new results on eigenvalue fluctuations and spacing for permutation-based random matrices, extending to small arcs and modified ensembles.

Findings

Eigenvalue counts in fixed arcs are asymptotically Gaussian.

Fluctuations extend to small arcs with slow length decay.

Largest and smallest eigenvalue spacings are characterized.

Abstract

We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens' distribution of a given parameter $\theta >0$, and its modification where entries equal to $1$ in the matrices are replaced by independent random variables uniformly distributed on the unit circle. For the elements of each ensemble, we focus on the random numbers of eigenvalues lying in some specified arcs of the unit circle. We show that for a finite number of fixed arcs, the fluctuation of the numbers of eigenvalues belonging to them is asymptotically Gaussian. Moreover, for a single arc, we extend this result to the case where the length goes to zero sufficiently slowly when the size of the matrix goes to infinity. Finally, we investigate the behaviour of the largest and smallest spacing between two distinct consecutive eigenvalues.