Critical edge behavior in the perturbed Laguerre ensemble and the Painleve V transcendent

TL;DR

This paper analyzes the edge behavior of the perturbed Laguerre ensemble, revealing a connection to Painleve V transcendent and showing how the kernel transitions between Bessel and Airy kernels under different scaling limits.

Contribution

It introduces a novel analysis of the perturbed Laguerre ensemble's edge behavior, linking the limiting kernel to a Painleve V transcendent and describing its transition properties.

Findings

At the hard edge, the kernel is described by a Painleve V related function.

The P_V kernel transitions to Bessel kernels for large and small scaling parameters.

At the soft edge, the kernel converges to the classical Airy kernel.

Abstract

In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of $$w(x,t)=(x+t)^{\lambda}x^{\alpha}e^{-x}$$ with $ x\geq 0,\ t>0,\ \alpha>0,\ \alpha+\lambda+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 $s=4nt,$ $n\to \infty,$ $t\to 0 $ such that $s$ is positive and finite, at the hard edge, the limiting kernel can be described by the $\varphi$-function related to a third-order nonlinear differential equation, which is equivalent to a particular Painlev\'e V (shorted as P$_{\rm V}$) transcendent via a simple transformation. Moreover, this P$_{\rm V}$ transcendent is equivalent to a general Painlev\'e P$_{\rm III}$ transcendent. For large $s,$ the P$_{\rm V}$ kernel reduces to the Bessel kernel $\mathbf{J}_{\alpha+\lambda}.$ For small $s,$ the P$_{\rm V}$ kernel reduces to another Bessel kernel $\mathbf{J}_\alpha.$ At the soft edge, the limiting kernel is the Airy kernel as the classical Laguerre weight.