Abstract
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.
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.
| Original language | English |
|---|---|
| Journal | Journal Of Physics A-Mathematical And Theoretical |
| Volume | 49 |
| Early online date | 13 Jan 2016 |
| DOIs | |
| Publication status | E-pub ahead of print - 13 Jan 2016 |