Finite‐size corrections at the hard edge for the Laguerre β ensemble

A fundamental question in random matrix theory is to quantify the optimal rate of convergence to universal laws. We take up this problem for the Laguerre β ensemble, characterized by the Dyson parameter β, and the Laguerre weightx a e − β x / 2, x > 0 in the hard edge limit. The latter relates to the eigenvalues in the vicinity of the origin in the scaled variable x ↦ x / 4 N . Previous work has established the corresponding functional form of various statistical quantities—for example, the distribution of the smallest eigenvalue, provided that a ∈ Z ≥ 0. We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling x ↦ x / 4 ( N + a / β ) , the rate of convergence to the limiting distribution is O ( 1 / N 2 ) , which is optimal. In the case β = 2 , general a > − 1 the explicit functional form of the distribution of the smallest eigenvalue at this order can be computed, as it can for a = 1 and general β > 0 . An iterative scheme is presented to numerically approximate the functional form for general a ∈ Z ≥ 2.