Freezing Laguerre ensemble in the hard edge (Probability Symposium)

Abstract

The fluctuation structure of the freezing limit in finite-dimensional random matrix ensembles, in which the inverse temperature parameter β tends to infinity, has been a topic with several recent developments. It is known that in this regime, the joint probability density of the eigenvalues obeys a multivariate Gaussian distribution. Recently, it was found that the covariance matrix involved in this distribution shows a surprisingly regular structure, and a complete description of its eigenvalues and eigenvectors was given by Andraus, Hermann and Voit for the Hermite (or Gaussian), Laguerre (or Wishart) and Jacobi ensembles of random matrices. In this paper, we showcase an application of these results to the hard edge statistics of the Laguerre ensemble. We show that the eigenvalue variance in the hard edge region is given asymptotically by a specific integral involving Bessel functions, which is itself derived from asymptotics of the covariance eigenvector matrix.