On The Eigenvalue Rigidity of the Laguerre Unitary Ensemble

Abstract

In this paper, we establish an optimal global rigidity estimate for the eigenvalues of the Laguerre unitary ensemble. Using the central limit theorem, we first construct a random measure via the eigenvalue counting function and then prove its convergence to a Gaussian multiplicative chaos measure, which yields the desired rigidity result. To prove this convergence, we apply a sufficient condition due to Claeys et al. [7] and carry out an asymptotic analysis of the corresponding exponential moments.