Hard edge tail asymptotics

2011/09/19 by José A. Ramı́rez, Ramirez, Jose A., Brian Rider +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1109.4121

openalex publication_date 2011/09/19 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

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.

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