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arXiv · 2103.07816

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

Mengkun Zhu·Chuanzhong Li·Yang Chen
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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.

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Mengkun Zhu, Chuanzhong Li, Yang Chen. 2021-03-13. Painlev\'e V for a Jacobi unitary ensemble with random singularities. https://arxiv.org/abs/2103.07816

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