From gap probabilities in random matrix theory to eigenvalue expansions

We present a method to derive asymptotics of eigenvalues for trace-class integral operators K LJ : ;d 2 ( )⥀l , acting on a single interval J Ì , which belongs to the ring of integrable operators (Its et al 1990 Int. J. Mod. Phys. B 4 1003–37). Our emphasis lies on the behavior of the spectrum i J { i 0 l ( )} = ¥ of K as ∣J∣  ¥ and i is fixed. We show that this behavior is intimately linked to the analysis of the Fredholm determinant det I K L J ( )∣ 2 - g ( ) as ∣J∣  ¥ and g  1 in a Stokes type scaling regime. Concrete asymptotic formulæ are obtained for the eigenvalues of Airy and Bessel kernels in random matrix theory.