Critical Edge Behavior and the Bessel to Airy Transition in the Singularly Perturbed Laguerre Unitary Ensemble

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journal

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Original languageEnglish
Pages (from-to)1257-1296
Journal / PublicationCommunications in Mathematical Physics
Volume332
Issue number3
Online published7 Aug 2014
Publication statusPublished - Dec 2014

Abstract

In this paper, we study the singularly perturbed Laguerre unitary ensemble(formula presentet.)>with(formula presentet.) and t > 0. Due to the effect of t/x for varying t, the eigenvalue correlation kernel has a new limit instead of the usual Bessel kernel at the hard edge 0. This limiting kernel involves ψ-functions associated with a special solution to a new third-order nonlinear differential equation, which is then shown to be equivalent to a particular Painlevé III equation. The transition of this limiting kernel to the Bessel and Airy kernels is also studied when the parameter t changes in a finite interval (0, d]. Our approach is based on Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.