Asymptotics of the Largest Eigenvalue Distribution of the Laguerre Unitary Ensemble

TL;DR

This paper derives the large $n$ asymptotics for the largest eigenvalue distribution in the Laguerre Unitary Ensemble, extending previous finite $n$ results and confirming the Tracy-Widom constant at the soft edge.

Contribution

It provides the asymptotic behavior of the largest eigenvalue distribution for the Laguerre Unitary Ensemble, including the constant at the soft edge, generalizing prior results for $eta=0$.

Findings

Asymptotic formulas for the largest eigenvalue distribution as $n$ grows large.

Confirmation of the Tracy-Widom constant at the soft edge.

Extension of results to general $eta$ in the Laguerre ensemble.

Abstract

We study the probability that all the eigenvalues of $n\times n$ Hermitian matrices, from the Laguerre unitary ensemble with the weight $x^{\gamma}\mathrm{e}^{-4nx},\;x\in[0,\infty),\;\gamma>-1$, lie in the interval $[0,\alpha]$. By using previous results for finite $n$ obtained by the ladder operator approach of orthogonal polynomials, we derive the large $n$ asymptotics of the largest eigenvalue distribution function with $\alpha$ ranging from 0 to the soft edge. In addition, at the soft edge, we compute the constant conjectured by Tracy and Widom [Commun. Math. Phys. 159 (1994), 151-174], later proved by Deift, Its and Krasovsky [Commun. Math. Phys. 278 (2008), 643-678]. Our results are reduced to those of Deift et al. when $\gamma=0$.