A note on the expansion of the smallest eigenvalue distribution of the LUE at the hard edge

TL;DR

This paper proves a refined correction term for the smallest eigenvalue distribution of the Laguerre unitary ensemble at the hard edge, improving convergence rates and connecting to Bessel kernel determinants.

Contribution

It provides a short proof of the $n^{-1}$ correction term in the smallest eigenvalue distribution, achieving an optimal $O(n^{-2})$ convergence rate and relating it to Bessel kernel determinants.

Findings

Established an $O(n^{-2})$ convergence rate for the distribution

Identified the correction term as the derivative of the limit distribution

Connected the correction to the logarithmic derivative of the Bessel kernel Fredholm determinant

Abstract

In a recent paper, Edelman, Guionnet and P\'{e}ch\'{e} conjectured a particular $n^{-1}$ correction term of the smallest eigenvalue distribution of the Laguerre unitary ensemble (LUE) of order $n$ in the hard-edge scaling limit: specifically, the derivative of the limit distribution, that is, the density, shows up in that correction term. We give a short proof by modifying the hard-edge scaling to achieve an optimal $O(n^{-2})$ rate of convergence of the smallest eigenvalue distribution. The appearance of the derivative follows then by a Taylor expansion of the less optimal, standard hard-edge scaling. We relate the $n^{-1}$ correction term further to the logarithmic derivative of the Bessel kernel Fredholm determinant in the work of Tracy and Widom.