A simple derivation of the Tracy–Widom distribution of the maximal eigenvalue of aGaussian unitary random matrix

Abstract

In this paper, we first briefly review some recent results on the distribution of the maximal eigenvalue ofan (N × N) random matrix drawn from Gaussian ensembles. Next we focus on the Gaussian unitaryensemble (GUE) and by suitably adapting a method of orthogonal polynomials developedby Gross and Matytsin in the context of Yang–Mills theory in two dimensions, we provide arather simple derivation of the Tracy–Widom law for GUE. Our derivation is based on theelementary asymptotic scaling analysis of a pair of coupled nonlinear recursion relations. Asan added bonus, this method also allows us to compute the precise subleading termsdescribing the right large deviation tail of the maximal eigenvalue distribution. In theYang–Mills language, these subleading terms correspond to non-perturbative (in1/N expansion) corrections to the two-dimensional partition function in the so called ‘weak’coupling regime.