Painlevé V for a Jacobi unitary ensemble with random singularities

Abstract

In this paper, we focus on the relationship between the fifth Painlevé equation and a Jacobi weight perturbed with random singularities, \begin{equation*} w(z)=\left(1-z^2\right)^α{\rm e}^{-\frac{t}{z^2-k^2}},~~~z,k\in[-1,1],~α,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 Painlevé V equation.

Applied Mathematics Letters,2021