Two Lax systems for the Painlev\'e II equation, and two related kernels in random matrix theory

TL;DR

This paper explores two Lax systems for the Painlevé II equation, establishing a connection between them and deriving kernels for determinantal processes relevant in random matrix theory.

Contribution

It demonstrates how solutions of a 4x4 Lax system can be obtained from a 2x2 system via an integral transform, linking their Stokes multipliers and related kernels.

Findings

Derived the 4x4 system solutions from the 2x2 system.

Expressed two determinantal kernels as contour integrals involving the Lax systems.

Connected the kernels to models of nonintersecting paths and two-matrix models.

Abstract

We consider two Lax systems for the homogeneous Painlev\'{e} II equation: one of size $2\times 2$ studied by Flaschka and Newell in the early 1980's, and one of size $4\times 4$ introduced by Delvaux-Kuijlaars-Zhang and Duits-Geudens in the early 2010's. We prove that solutions to the $4\times 4$ system can be derived from those to the $2\times 2$ system via an integral transform, and consequently relate the Stokes multipliers for the two systems. As corollaries we are able to express two kernels for determinantal processes as contour integrals involving the Flaschka-Newell Lax system: the tacnode kernel arising in models of nonintersecting paths, and a critical kernel arising in a two-matrix model.