Critical edge behavior in the perturbed Laguerre unitary ensemble and the Painlevé V transcendent

Abstract

In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of w ( x , t ) = ( x + t ) λ x α e − x with x ≥ 0 , t > 0 , α > 0 , α + λ + 1 > 0 . The Deift–Zhou nonlinear steepest descent approach is used to analyze the limit of the eigenvalue correlation kernel. It was found that under the double scaling s = 4 n t , n → ∞ , t → 0 such that s is positive and finite, at the hard edge, the limiting kernel can be described by the φ -function related to a third-order nonlinear differential equation, which is equivalent to a particular Painlevé V (shorted as P V ) transcendent via a simple transformation. Moreover, this P V transcendent is equivalent to a general Painlevé III transcendent. For large s , the P V kernel reduces to the Bessel kernel J α . For small s , the P V kernel reduces to another Bessel kernel J α + λ . At the soft edge, the limiting kernel is the Airy kernel as the classical Laguerre weight.