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

The τ-functions of certain Painlevé equations (PVI, PV, and PIII) can be expressed as Fredholm determinants. Furthermore, the minor expansion of these determinants provides an interesting connection to random partitions. This paper is a step toward understanding whether the τ-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 τ-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 terms of a suitable combination of Toeplitz operators called the Widom constant and verify that the Widom constant for the 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 τ-function.