Hard edge tail asymptotics

Abstract

Let Λ be the limiting smallest eigenvalue in the general (β, α)-Laguerre ensemble of random matrix theory. That is, Λ is the n ↑ ∞ distributional limit of the (scaled) minimal point drawn from the density proportional to ∏1≤i 0, α > −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/2λ+2γ√λ λ−λ(λ+1-β/2)/2β e(β, α)(1 + o(1)) as λ ↑ ∞ in which γ = β/2(a + 1) − 1 and e(β, a) > 0 is a constant (which we do not determine). This estimate complements/extends various results previously available for special values of β and a. © 2011 Association for Symbolic Logic.