Painlev\'{e} VI, Painlev\'{e} III and the Hankel Determinant Associated with a Degenerate Jacobi Unitary Ensemble

TL;DR

This paper links the Hankel determinant of a perturbed Jacobi weight to Painlevé equations, revealing their role in eigenvalue distributions of a degenerate Jacobi ensemble and analyzing asymptotic behaviors.

Contribution

It establishes a connection between Hankel determinants and Painlevé VI and III equations using ladder operators, providing new insights into eigenvalue distributions.

Findings

Hankel determinant's derivative satisfies Painlevé VI equation.

Double scaling reduces the problem to Painlevé III.

Asymptotic behaviors are derived for key limits.

Abstract

This paper studies the Hankel determinant generated by a perturbed Jacobi weight, which is closely related to the largest and smallest eigenvalue distribution of the degenerate Jacobi unitary ensemble. By using the ladder operator approach for the orthogonal polynomials, we find that the logarithmic derivative of the Hankel determinant satisfies a nonlinear second-order differential equation, which turns out to be the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlev\'{e} VI equation by a translation transformation. We also show that, after a suitable double scaling, the differential equation is reduced to the Jimbo-Miwa-Okamoto $\sigma$-form of the Painlev\'{e} III. In the end, we obtain the asymptotic behavior of the Hankel determinant as $t\rightarrow1^{-}$ and $t\rightarrow0^{+}$ in two important cases, respectively.