Asymptotics of Spacing Distributions at the Hard Edge for β-Ensembles

Abstract

In a previous work [J. Math. Phys.35 (1994) 2539-2551], generalized hypergeometric functions have been used to a give a rigorous derivation of the large s asymptotic form of the general β > 0 gap probability Eβhard(0; (0, s); βa/2), provided both βa/2 Z≥0 and 2/β Z+. It is shown how the details of this method can be extended to remove the requirement that 2/β Z+. Furthermore, a large deviation formula for the gap probability Eβ(n; (0, x); MEβ, N(λaβ/2e-2βNλ)) is deduced by writing it in terms of the characteristic function of a certain linear statistic. By scaling x = s/(4N)2 and taking N → ∞, this is shown to reproduce a recent conjectured formula for Eβhard(n; (0, s); β a/2), βa/2 Z≥0, and..