The $\tau$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant

Abstract

$\tau$-functions of certain Painlevé equations (PVI,PV,PIII) can be expressed as a Fredholm determinant. Further, the minor expansion of these determinants provide an interesting connection to Random partitions. This paper is a step towards understanding whether the $\tau$-function of Painlevé II has a Fredholm determinant representation. The Ablowitz-Segur family of solutions are special one parameter solutions of Painlevé II and the corresponding $\tau$-function is known to be the Fredholm determinant of the Airy Kernel. We develop a formalism for open contour in parallel to the one formulated in \cite{CGL} in terms of the Widom constant and verify that the Widom constant for Ablowitz-Segur family of solutions is indeed the determinant of the Airy Kernel. Finally, we construct a suitable basis and obtain the minor expansion of the Ablowitz-Segur $\tau$-function.

References in corpus (4)

• Argyres-Douglas theories, Painlevé II and quantum mechanics
• Tau functions as Widom constants
• Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
• Asymptotics of a Fredholm determinant involving the second Painlevé transcendent

Cited by in corpus (3)

• Isomonodromic tau functions on a torus as Fredholm determinants, and charged partitions
• Fredholm determinant representation of the Painlevé II $\tau$-function
• Painlevé/CFT correspondence on a torus