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 beta ensemble, characterized by the Dyson parameter beta, and the Laguerre weight xae-beta 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 xx/4N. Previous work has established the corresponding functional form of various statistical quantities-for example, the distribution of the smallest eigenvalue, provided that a is an element of Z0. We show, using the theory of multidimensional hypergeometric functions based on Jack polynomials, that with the modified hard edge scaling xx/4(N+a/beta), the rate of convergence to the limiting distribution is O(1/N2), which is optimal. In the case beta=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 beta>0. An iterative scheme is presented to numerically approximate the functional form for general a is an element of Z2.