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

Abstract

τ -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 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 [11] 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 τ -function.

Fredholm determinant representation of the Painlevé II $τ$-function

“…Series representation of the τ -function (1.8) can be obtained from the minor expansion of the Fredholm determinant on an appropriate basis. A similar computation for the case of the Ablowitz-Segur solutions of Painlevé II is worked out in [17]. Furthermore, we expect that the methods developed in this manuscript can be applied to some solutions of Painlevé I and IV equations.…”

“…We then reduce the RHP to a RHP with a discontinuity on the imaginary axis in Section 3. However, we will see that the jump on the imaginary axis does not admit Birkhoff factorization and hence the technique to construct Fredholm determinants in [17] is not applicable to our case. Instead, we use a variation of the formalism in [2] namely, a lower, diagonal, upper triangular (LDU) factorization of the jump matrix to construct the τ -function as a Fredholm determinant of an integrable (IIKS) [14,28] operator with the parabolic cylinder functions acting as the 'building blocks'.…”

“…4. The limit from the general τ -function of Painlevé II in (1.8) to the τ -function of the Ablowitz-Segur family of solutions (determinant of the Airy kernel) in [17] is singular because s 3 s 1 = 1 (ν in (1.7) goes to infinity).…”

Fredholm determinant representation of the homogeneous Painlevé II τ-function

“…A natural question then is whether the τ -functions of Painlevé I, II, IV admit a Fredholm determinant representation. In a first step to answer this question in the case of Painlevé II, the present author recently showed that the RHP corresponding to the special one-parameter (Ablowitz-Segur) family of solutions to the Painlevé II equation [17] can be recast as a RHP on the imaginary axis as opposed to the unit circle in [15], hinting at a similar structure for the general RHP of Painlevé II [18]. As a consequence, the corresponding τ -function (which is known to be the determinant of the Airy kernel [12] and appears in random matrix theory as the Tracy-Widom distribution [14,19,20]), can be formulated as a Widom constant.…”

“…In the case of the RHP of Painlevé II, it is known that under particular transformations that facilitate asymptotic analysis of the Painlevé II transcendent at x → −∞, the local parametrices are described by parabolic cylinder functions D ν (z) [21,22] which we recall in section 2. Refer to [18,[23][24][25][26] for further details on the asymptotic analysis of Painleve II. We then reduce the RHP to a RHP with a discontinuity on the imaginary axis in section 3.…”