On smooth mesoscopic linear statistics of the eigenvalues of random permutation matrices

TL;DR

This paper investigates the asymptotic distribution of smooth linear eigenvalue statistics of random permutation matrices under Ewens measures, revealing different limiting behaviors depending on the test function's value at zero.

Contribution

It characterizes the mesoscopic limit laws of eigenvalue statistics for permutation matrices, identifying conditions for Gaussian or Poissonian limits.

Findings

Central limit theorem with logarithmic variance when f(0)≠0

Poisson point process limit when f(0)=0

Convergence in distribution for eigenvalue statistics as permutation size grows

Abstract

We study the limiting behavior of smooth linear statistics of the spectrum of random permutation matrices in the mesoscopic regime, when the permutation follows one of the Ewens measures on the symmetric group. If we apply a smooth enough test function $f$ to all the determinations of the eigenangles of the permutations, we get a convergence in distribution when the order of the permutation tends to infinity. Two distinct kinds of limit appear: if $f(0)\neq 0$, we have a central limit theorem with a logarithmic variance, and if $f(0) = 0$, the convergence holds without normalization and the limit involves a scale-invariant Poisson point process.