Linear Statistics of Random Matrix Ensembles at the Spectrum Edge Associated with the Airy Kernel

TL;DR

This paper analyzes the asymptotic behavior of linear statistics in large random matrix ensembles at the spectrum edge, revealing statistical equivalences and employing Coulomb fluid methods for new results.

Contribution

It derives large N asymptotics of the moment-generating function for linear statistics at the spectrum edge in Gaussian and Laguerre ensembles, establishing their statistical equivalence.

Findings

Asymptotics of MGF linked to Airy kernel

Statistical equivalence between Gaussian and Laguerre ensembles

Coulomb fluid method reproduces known results

Abstract

In this paper, we study the large $N$ behavior of the moment-generating function (MGF) of the linear statistics of $N\times N$ Hermitian matrices in the Gaussian unitary, symplectic, orthogonal ensembles (GUE, GSE, GOE) and Laguerre unitary, symplectic, orthogonal ensembles (LUE, LSE, LOE) at the edge of the spectrum. From the finite $N$ Fredholm determinant expression of the MGF of the linear statistics, we find the large $N$ asymptotics of the MGF associated with the Airy kernel in these Gaussian and Laguerre ensembles. Then we obtain the mean and variance of the suitably scaled linear statistics. We show that there is an equivalence between the large $N$ behavior of the MGF of the scaled linear statistics in Gaussian and Laguerre ensembles, which leads to the statistical equivalence between the mean and variance of suitably scaled linear statistics in Gaussian and Laguerre ensembles. In the end, we use the Coulomb fluid method to obtain the mean and variance of another type of linear statistics in GUE, which reproduces the result of Basor and Widom.