Critical edge behavior in the modified Jacobi ensemble and the Painlevé V transcendents
Critical edge behavior in the modified Jacobi ensemble and the Painlevé V transcendents
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DOI：
10.1063/1.4819244
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发表时间：
2013-08
影响因子：
1.3
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
研究了扰动Chebyshev权w(x)=(1−x2)−12(tn2−x2)α, x∈(−1,1),tn > 1所定义的Jacobi酉系综。代数奇点与权函数支持的端点，即硬边tn→1，合并为n→∞。我们使用Riemann-Hilbert方法（Deift和Zhou的方法）来获得关于w(x)的正交多项式的渐近性。我们证明了普适性是保留的：特征值相关性在大部分频谱中的渐近行为是用正弦核来描述的。主要的结果是在光谱边缘的局部行为。当tn−1 = O（n−2）时，边缘处的极限核用painlevel V方程的特解的ψ−函数来描述。我们还证明了当tn变化到一个固定的t bbb1时，painlev核跃迁到贝塞尔核J−12。另一方面，当tn接近1时，painlev核可以用另一个贝塞尔核Jα−12来近似。
We study the Jacobi unitary ensemble defined by the perturbed Chebyshev weight w(x)=(1−x2)−12(tn2−x2)α, x ∈ (−1, 1), tn > 1. The algebraic singularity coalesces with the end point of the support of the weight function, namely, tn → 1, the hard edge, as n → ∞. We use the Riemann-Hilbert approach (the method of Deift and Zhou) to obtain the asymptotics of the polynomials orthogonal with respect to w(x). We show that the universality property is preserved: the asymptotic behavior of the eigenvalue correlations in the bulk of the spectrum is described in terms of the sine kernel. The main result is on the local behavior at the edge of the spectrum. The limit kernel at the edge, when tn − 1 = O(n−2), is described by the ψ −function for a specific solution of the Painleve V equation. We also show that when tn varies to a fixed t > 1, the Painleve V kernel transits to the Bessel kernel J−12. On the other hand, when tn approaches 1, the Painleve V kernel can be approximated by another Bessel kernel Jα−12.