论文标题
Laguerre统一的合奏与跳跃不连续，PDE和耦合Painlevév System
Laguerre Unitary Ensembles with Jump Discontinuities, PDEs and the Coupled Painlevé V System
论文作者
论文摘要
我们研究了Laguerre重量产生的Hankel决定因素，而跳跃不连续性为$ t_k，k = 1，\ cdots，m $。通过采用梯子操作员方法来建立Riccati方程，我们表明$σ_n（t_1，\ cdots，t_m）$，$ n $ dimensional-demensional hankel的对数衍生品，可满足$σ$ - form formentive ofPachlevéVequination。通过研究相关正交多项式的Riemann-Hilbert问题，通过LAX对，我们就耦合的PainlevéV系统的解决方案表示$σ_n$。我们还建立了上述两种方法中引入的辅助数量之间的关系，该方法提供了Riccati方程与LAX对之间的连接。此外，当每个$ t_k $都趋向于频谱的硬边缘，而$ n $则为$ \ infty $时，显示了缩放的$σ_n$，以满足广义的painlevéIIII方程。
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 $σ_n(t_1,\cdots,t_m)$, the logarithmic derivative of the $n$-dimensional Hankel determinant, satisfies a generalization of the $σ$-from of Painlevé V equation. Through investigating the Riemann-Hilbert problem for the associated orthogonal polynomials and via Lax pair, we express $σ_n$ 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 $σ_n$ is shown to satisfy a generalized Painlevé III equation.
