Painlevé tau-functions and Fredholm determinants

It is now known that the tau-functions associated to the generic solutions of the Painlevé equations III, V, VI can be expressed as a Fredholm determinants. The minor expansion of these determinants provide an interesting connection to random partitions. We show thatthe generic tau-function of the Painlevé II equation can be written as a Fredholm determinant of an integrable (Its-Izergin-Korepin-Slavnov) operator. The tau-function dependson the isomonodromic timetand two Stokes parameters, and the vanishing locus of the tau-function, called the Malgrange divisoris determined by the zeros of the Fredholm determinant. As a mid-step, we show that the Fredholm determinant of the Airy kernel which is also the tau-function of the Ablowitz-Segur family of solutions to Painlevé II, can be expressed as a combination of Toeplitz operators called the Widom constant. Furthermore, constructing a suitable basis, we obtain the minor expansion of the determinant of the Airy kernel labelled by colourless and chargeless Maya diagrams. We also generalise the techniques to study the tau-functions of Painlevé III, V, VI to the case of Fuchsian system with generic monodromies in GL(N,C) on a torus, and show that the associated tau-function can be written as a Fredholm determinant of Plemelj operators. We further show that the minor expansion of this Fredholm determinant is described by a serieslabeled by charged partitions. As an example, we show that in the case ofSL(2,C) this combinatorial expression takes the form of a dual Nekrasov-Okounkov partition function.