Painlev\'e V for a Jacobi unitary ensemble with random singularities

TL;DR

This paper explores the connection between the fifth Painlevé equation and a Jacobi weight with random singularities, revealing that an auxiliary quantity related to orthogonal polynomials satisfies a Painlevé V equation.

Contribution

It establishes a novel link between Painlevé V and orthogonal polynomials with a perturbed Jacobi weight using the ladder operator approach.

Findings

Auxiliary quantity satisfies Painlevé V equation.

Connection between random singularities and Painlevé equations.

Method applicable to other orthogonal polynomial systems.

Abstract

In this paper, we focus on the relationship between the fifth Painlev\'{e} equation and a Jacobi weight perturbed with random singularities, \begin{equation*} w(z)=\left(1-z^2\right)^{\alpha}{\rm e}^{-\frac{t}{z^2-k^2}},~~~z,k\in[-1,1],~\alpha,t>0. \end{equation*} By using the ladder operator approach, we obtain that an auxiliary quantity $R_n(t)$, which is closely related to the recurrence coefficients of monic polynomials orthogonal with $w(z)$, satisfies a particular Painlev\'{e} V equation.