Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems
Gaussian unitary ensemble with jump discontinuities and the coupled Painlevé II and IV systems
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DOI：
10.1088/1361-6544/abc598
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发表时间：
2020-02
期刊：
影响因子：
1.7
通讯作者：
Xiao-Bo Wu;Shuai‐Xia Xu
中科院分区：
文献类型：
--
作者：
Xiao-Bo Wu;Shuai‐Xia Xu
本文研究了具有两个跳跃间断的高斯权的正交多项式和Hankel行列式。当次数n是有限的，正交多项式和Hankel行列式被证明是与耦合Painlevé IV系统。利用这种联系，我们得到了一系列特殊的功能解决方案的耦合Painlevé IV系统。在双标度极限下，当跳跃间断趋于谱的边缘且阶数n趋于无穷大时，我们建立了Hankel行列式和正交多项式的渐近展开式，并用耦合Painlevé II系统的解来表示.作为应用，我们考虑稀疏化过程，重新推导了最近发现的关于高斯酉系综（GUE）的大厄米特矩阵极值特征值附近有限区间内无特征值的差距概率的Tracy-Widom型表达式，以及GUE中最大特征值的极限条件分布.
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 Painlevé IV system. Using this connection, we obtain a sequence of special function solutions 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 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.