论文标题
高斯统一的合奏与跳跃不连续性以及耦合的PainlevéII和IV系统
Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems
论文作者
论文摘要
我们研究与高斯重量相关的正交多项式和两个跳跃不连续性相关的汉克尔决定因素。当$ n $是有限的程度时，正交多项式和汉克尔决定因素被证明连接到耦合的painlevéIV系统。在双缩放限制中，由于跳跃不连续性倾向于光谱的边缘，而$ n $的$ n $生长到无穷大，我们为Hankel决定簇和正交多项式建立了渐近扩展，这些扩展是用pachlevéII系统的解决方案表示的。作为应用，我们重新介绍了最近发现的tracy-widom类型表达式，因为差距概率是在有限的间隔中没有特征值在大型高斯单位合奏的极端特征值附近，并且通过考虑一个稀薄的过程，在高斯单位的特征分布的极限条件分布。
We study the orthogonal polynomials and the Hankel determinants associated with Gaussian weight with two jump discontinuities. When the degree $n$ is finite, the orthogonal polynomials and the Hankel determinants are shown to be connected to the coupled Painlevé 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 Painlevé 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 the extreme eigenvalue of large Gaussian unitary ensemble and the limiting conditional distribution of the largest eigenvalue in Gaussian unitary ensemble by considering a thinned process.
