We study the orthogonal polynomials and the Hankel determinants associated with the Gaussian weight with two jump discontinuities. When the degree n is finite, the orthogonal polynomials and the Hankel determinants are shown to be connected with the coupled Painlev IV system. Using this connection, we obtain a sequence of special function solutions to the coupled Painlev IV system. In the double scaling limit as the jump discontinuities tend to the edge of the spectrum and the degree n grows to infinity, we establish the asymptotic expansions for the Hankel determinants and the orthogonal polynomials, which are expressed in terms of solutions of the coupled Painlev II system. As applications, we re-derive the recently found Tracy–Widom type expressions for the gap probability of there being no eigenvalues in a finite interval near the extreme eigenvalue of large Hermitian matrix from the Gaussian unitary ensemble (GUE) and the limiting conditional distribution of the largest eigenvalue in the GUE by considering a thinned process.

我们研究了与具有两个跳跃不连续性的高斯权重相关的正交多项式和 Hankel 行列式。当次数n是有限的时，正交多项式和 Hankel 行列式显示为与耦合的 Painlev IV 系统连接。使用这种联系，我们获得了耦合 Painlev IV 系统的一系列特殊函数解。在双标度限制中，由于跳跃不连续性趋向于谱的边缘和度数n增长到无穷大，我们建立 Hankel 行列式和正交多项式的渐近展开式，它们以耦合 Painlev II 系统的解表示。作为应用，我们重新推导了最近发现的 Tracy-Widom 类型表达式，用于在来自高斯酉系综 (GUE) 的大 Hermitian 矩阵的极值特征值附近的有限区间内没有特征值的间隙概率和通过考虑细化过程，GUE 中的最大特征值。