Asymptotic behavior of spacing distributions for the eigenvalues of random matrices

It is known that the probability E_β(0, S) that an arbitrary interval of length S contains none of the eigenvalues of a random matrix chosen from the orthogonal (β = 1), unitary (β = 2) or symplectic (β = 4) ensemble can be expressed in terms if infinite products $∏n=0∞[1−λ2n(S)]$ and $∏n=0∞[1=λ2n+1(S)]$⁠, where λ_n(S) is an eigenvalue of a certain integral equation. Using values of λ_n(S), valid for S large, obtained in connection with a recent study of spheroidal functions, we derive asymptotic expressions (S ≫ 1) for E_β(0, S).