2017/11/13 by Min Chen, Yang Chen, Chen, Min +3
Mathematics · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Fractional Differential Equations Solutions #Mathematical Physics (math-ph) #Mathematical functions and polynomials #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.1711.04383
openalex publication_date 2017/11/13 · openalex created_date 2017/12/04 · openalex updated_date 2026/07/28
In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of w(x,t)=(x+t)λxαe-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=4nt, 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\rm V) transcendent via a simple transformation. Moreover, this P\rm V transcendent is equivalent to a general Painlevé P\rm III transcendent. For large s, the P\rm V kernel reduces to the Bessel kernel Jα+λ. For small s, the P\rm V kernel reduces to another Bessel kernel Jα. At the soft edge, the limiting kernel is the Airy kernel as the classical Laguerre weight.