Hard edge tail asymptotics

Abstract

Let $Λ$ be the limiting smallest eigenvalue in the general (β, a)-Laguerre ensemble of random matrix theory. Here β>0, a >-1; for β=1,2,4 and integer a, this object governs the singular values of certain rank n Gaussian matrices. We prove that P(Λ> λ) = e^{- (β/2) λ+ 2 γλ^{1/2}} λ^{- (γ(γ+1))/(2β) + γ/4} E (β, a) (1+o(1)) as λgoes to infinity, in which γ= (β/2) (a+1)-1 and E(β, a) is a constant (which we do not determine). This estimate complements/extends various results previously available for special values of βand a.

Minor revision; to appear in Elec. Comm. Probability