Hard edge tail asymptotics

TL;DR

This paper derives the asymptotic tail distribution of the smallest eigenvalue in the (eta, a)-Laguerre ensemble, extending known results for special cases and providing a general formula for large or the probability tail.

Contribution

It provides a general asymptotic formula for the tail distribution of the smallest eigenvalue in the (eta, a)-Laguerre ensemble, broadening previous special case results.

Findings

Asymptotic tail probability formula for rom the Laguerre ensemble

Extension of previous results to general or eta>0 and a>-1

Identification of the dominant exponential and polynomial factors in the tail

Abstract

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