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

Abstract: $\tau$-functions of certain Painlev\'e 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 Painlev\'e II has a Fredholm determinant representation. The Ablowitz-Segur family of solutions are special one parameter solutions of Painlev\'e 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.