A simple derivation of the Tracy-Widom distribution of the maximal eigenvalue of a Gaussian unitary random matrix

TL;DR

This paper presents a straightforward derivation of the Tracy-Widom distribution for the largest eigenvalue of Gaussian Unitary Matrices using orthogonal polynomial methods, also capturing subleading tail corrections.

Contribution

It introduces a simple asymptotic analysis approach to derive the Tracy-Widom law for GUE and computes detailed tail correction terms.

Findings

Derived the Tracy-Widom distribution for GUE eigenvalues.

Provided explicit subleading tail correction terms.

Connected eigenvalue distribution results to Yang-Mills theory insights.

Abstract

In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue of a $(N\times N)$ random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian Unitary Ensemble (GUE) and by suitably adapting a method of orthogonal polynomials developed by Gross and Matytsin in the context of Yang-Mills theory in two dimensions, we provide a rather simple derivation of the Tracy-Widom law for GUE. Our derivation is based on the elementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations. As an added bonus, this method also allows us to compute the precise subleading terms describing the right large deviation tail of the maximal eigenvalue distribution. In the Yang-Mills language, these subleading terms correspond to non-perturbative (in $1/N$ expansion) corrections to the two-dimensional partition function in the so called `weak' coupling regime.