Fredholm determinant representation of the Painlevé II $\tau$-function

Abstract

We formulate the generic $\tau$-function of the Painlevé II equation as a Fredholm determinant of an integrable (Its-Izergin-Korepin-Slavnov) operator. The $\tau$-function depends on the isomonodromic time $t$ and two Stokes' parameters, and the vanishing locus of the $\tau$-function, called the Malgrange divisor is determined by the zeros of the Fredholm determinant.

29 pages, 8 figures. V3: some signs corrected; v4: There was an error with the t-derivative that is fixed in this version. This changes a factor of 2 in the subleasing term of the main result

References in corpus (3)

• Tau functions as Widom constants
• Painlevé IV Critical Asymptotics for Orthogonal Polynomials in the Complex Plane
• The $\tau$-function of the Ablowitz-Segur family of solutions to Painlevé II as a Widom constant

Cited by in corpus (2)

• Painlevé/CFT correspondence on a torus
• Painlevé Kernels and Surface Defects at Strong Coupling