Hard edge tail asymptotics

Details Hard edge tail asymptotics Hard edge tail asymptotics

2011-09-19

arxiv.org/abs/1109.4121v2

Mathematics

Probability

Minor revision; to appear in Elec. Comm. Probability

Scientific paper

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.

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