论文标题
高斯单位合奏，有两个跳跃不连续性，PDE和耦合的PainlevéII和IV系统
Gaussian unitary ensemble with two jump discontinuities, PDEs and the coupled Painlevé II and IV systems
论文作者
论文摘要
我们认为高斯体重产生的汉克尔决定因素，并有两个跳跃不连续性。利用[C. Min and Y. Chen，数学。冰毒。应用。科学。 {\ bf 42}（2019），301--321]，其中通过使用适合正交多项式的梯子算子来推导hankel决定符的二阶PDE，我们得出了[x。 Wu和S. Xu，Arxiv：2002.11240V2]通过对正交多项式的Riemann-Hilbert问题的研究。在双重缩放下，我们表明，作为$ n \ rightarrow \ infty $，在缩放变量中，汉克尔决定因素的对数衍生物趋向于耦合的painlevéII系统的哈密顿式，并且满足了二阶pde pde。此外，我们获得了正交多项式复发系数的渐近系数，该系数与耦合的PainlevéII系统的溶液有关。
We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of [C. Min and Y. Chen, Math. Meth. Appl. Sci. {\bf 42} (2019), 301--321] where a second order PDE was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlevé IV system which was established in [X. Wu and S. Xu, arXiv: 2002.11240v2] by a study of the Riemann-Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as $n\rightarrow\infty$, the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlevé II system and it satisfies a second order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painlevé II system.
