Asymptotics of the largest eigenvalue distribution of the Laguerre unitary ensemble

Abstract

We study the probability that all the eigenvalues of n × n Hermitian matrices, from the Laguerre unitary ensemble with the weight $x^{\gamma }{\mathrm{e}}^{-4nx},x\in \left[0,\infty \right),\gamma >-1$ , lie in the interval [0, α]. 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 α 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, 151–174 (1994)] and later proved by Deift, Its, and Krasovsky [Commun. Math. Phys. 278, 643–678 (2008)]. Our conclusions are reduced to those of Deift et al. when γ = 0. It should be pointed out that our derivation is straightforward but not rigorous, and hence, the above results are stated as conjectures.