On Smooth Mesoscopic Linear Statistics of the Eigenvalues of Random Permutation Matrices

Valentin Bahier,Joseph Najnudel

doi:10.1007/s10959-021-01106-4

Abstract

AbstractWe 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 $$f(0)\ne 0$$ f ( 0 ) ≠ 0 , we have a central limit theorem with a logarithmic variance; and if $$f(0) = 0$$ f ( 0 ) = 0 , the convergence holds without normalization and the limit involves a scale-invariant Poisson point process.

Highlights

The spectrum of random permutation matrices has been studied with much attention in the last few decades
For a random permutation matrix following one of the Ewens measures, the number of eigenvalues lying on a fixed arc of the unit circle has been studied in detail by Wieand [34] and satisfies a central limit theorem when the order n goes to infinity, with a variance growing like log n
This rate of growth is similar to what is obtained for the Circular Unitary Ensemble and random matrices on other compact groups, for which a central limit theorem occurs, as it can be seen in Costin and Lebowitz [11], Soshnikov [30] and Wieand [33]

Summary

Introduction

The spectrum of random permutation matrices has been studied with much attention in the last few decades. For a random permutation matrix following one of the Ewens measures, the number of eigenvalues lying on a fixed arc of the unit circle has been studied in detail by Wieand [34] and satisfies a central limit theorem when the order n goes to infinity, with a variance growing like log n. This rate of growth is similar to what is obtained for the Circular Unitary Ensemble and random matrices on other compact groups, for which a central limit theorem occurs, as it can be seen in Costin and Lebowitz [11], Soshnikov [30] and Wieand [33]. Theorem 1.1 (ii) does not extend to Xσn,δn

Outline of the Paper

Var W

We now estimate the sum over the positive indices