Hard edge tail asymptotics
نویسندگان
چکیده
Let Λ be the limiting smallest eigenvalue in the general (β, a)-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, 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+2γ √ λ − γ(γ+1) 2β + 1 4 γ e(β, a)(1 + o(1)) as λ ↑ ∞ in which γ = β2 (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 β and a.
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