Asymptotics of the largest eigenvalue distribution of the Laguerre unitary ensemble

We study the probability that all the eigenvalues of n x n Hermitian matrices, from the Laguerre unitary ensemble with the weight x(gamma)e(-)(4nx) x is an element of [0, infinity), 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, 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 Delft et al. when gamma = 0. It should be pointed out that our derivation is straightforward but not rigorous, and hence, the above results are stated as conjectures. Published under license by AIP Publishing.