Laguerre Unitary Ensembles with Multiple Discontinuities, PDE, and the Coupled Painlevé V System

Abstract: We study the Hankl generated by the Laguerre weight with jump discontinuities at $t_k$, $k=1,2,\ldots,m$. By employing the ladder operator approach we establish (multi-time) Riccati equations, to show that $\sigma_n(t_1, ...,t_m)$, the log derivative of the $n\times n$ Hankel determinant, satisfies a generalization of the $\sigma$ of a Painlev\'e V equation. Through investigating the Riemann-Hibert problem (or Homogenous Hilbert Problem ) for the orthogonal polynomials generated by the LUEMD and via Lax pair, we express $\sigma_n$ in terms of solutions of a coupled Painlev\'e V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provide connections between the Riccati equations and the Lax Pair.

In addition, when each $t_k$ tends to the hard edge of the spectrum and $n$ goes to infinity, the scaled $\sigma_n$ is shown to satisfy a generalized Painlev\'e III system.

Yang Chen (University of Macau, Macau), Shulin Lyu (Qilu University of Technology, Shandong Academy of Science), Shuai-Xia Xu (Institut Franco-Chinois de l'Energie Nculearie, Sun Yat-sen University, Guangzhou, China

complex variablesdynamical systems

Audience: researchers in the topic

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