The asymptotics a Bessel-kernel determinant which arises in Random Matrix Theory

TL;DR

This paper analyzes the asymptotic behavior of Bessel-kernel determinants related to local correlations in Random Matrix Theory, providing new insights into their large-interval limits using operator theory.

Contribution

It introduces a novel operator theoretic approach to compute the asymptotics of Bessel-kernel determinants in the hard edge limit, extending previous results.

Findings

Asymptotic formulas for Bessel-kernel determinants as R approaches infinity

Characterization of the influence of parameter α on gap probabilities

Extension of asymptotic analysis techniques in Random Matrix Theory

Abstract

In Random Matrix Theory the local correlations of the Laguerre and Jacobi Unitary Ensemble in the hard edge scaling limit can be described in terms of the Bessel kernel (containing a parameter $\alpha$). In particular, the so-called hard edge gap probabilities can be expressed as the Fredholm determinants of the corresponding integral operator restricted to the finite interval [0, R]. Using operator theoretic methods we are going to compute their asymptotics as R goes to infinity under certain assumption on the parameter $\alpha$.