Critical edge behavior in the modified Jacobi ensemble and Painlevé equations
Critical edge behavior in the modified Jacobi ensemble and Painlevé equations
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DOI：
10.1088/0951-7715/28/6/1633
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发表时间：
2014-04
期刊：
影响因子：
1.7
通讯作者：
Shuai‐Xia Xu;Yuqiu Zhao
中科院分区：
文献类型：
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作者：
Shuai‐Xia Xu;Yuqiu Zhao
我们研究了t > 1时受代数奇点扰动的Jacobi酉系综.对于固定的t，这是Kuijlaars等人研究的修正的Jacobi系综。然而，这里的主要焦点是代数奇点接近硬边的情况，即t → 1+。在双尺度极限的情况下，当t − 1是1/n2的数量级，n是矩阵的大小，特征值相关核被证明在硬边1处有一个新的极限核，由某个二阶非线性方程的λ-函数描述。该方程通过莫比乌斯变换与Painlevé III方程相关。它也是Painlevé V方程的一个推广，在特殊情况下，通过Bäcklund变换，它可以化为一个特殊的Painlevé V方程.还研究了极限核到贝塞尔核的过渡，n2（t-1）是大还是小。在本文中，该方法是基于Deift-Zhou非线性最速下降分析的Riemann-Hilbert问题。
We study the Jacobi unitary ensemble perturbed by an algebraic singularity at t > 1. For fixed t, this is the modified Jacobi ensemble studied by Kuijlaars et al. The main focus here, however, is the case when the algebraic singularity approaches the hard edge, namely t → 1+. In the double scaling limit case when t − 1 is of the order of magnitude of 1/n2, n being the size of the matrix, the eigenvalue correlation kernel is shown to have a new limiting kernel at the hard edge 1, described by the ψ-functions for a certain second-order nonlinear equation. The equation is related to the Painlevé III equation by a Möbius transformation. It also furnishes a generalization of the Painlevé V equation, and can be reduced to a particular Painlevé V equation via the Bäcklund transformations in special cases. The transitions of the limiting kernel to Bessel kernels are also investigated, with n2(t − 1) being large or small. In the present paper, the approach is based on the Deift–Zhou nonlinear steepest descent analysis for Riemann–Hilbert problems.