Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System

Abstract

We study the Hankel determinant generated by the Laguerre weight with jump discontinuities at $t_k,k=1,\cdots ,m$. By employing the ladder operator approach to establish Riccati equations, we show that ${\sigma }_n(t_1,\cdots ,t_m)$, the logarithmic derivative of the $n$-dimensional Hankel determinant, satisfies a generalization of the $\sigma$-from of Painlevé V equation. Through investigating the Riemann-Hilbert problem for the associated orthogonal polynomials and via Lax pair, we express ${\sigma }_n$ in terms of solutions of a coupled Painlevé V system. We also build relations between the auxiliary quantities introduced in the above two methods, which provides connections between the Riccati equations and Lax pair. In addition, when each $t_k$ tends to the hard edge of the spectrum and $n$ goes to $\infty$, the scaled ${\sigma }_n$ is shown to satisfy a generalized Painlevé III equation.

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