Critical edge behavior in the perturbed Laguerre unitary ensemble and the Painlevé V transcendent
Critical edge behavior in the perturbed Laguerre unitary ensemble and the Painlevé V transcendent
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DOI：
10.1016/j.jmaa.2019.01.064
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发表时间：
2017-11
影响因子：
1.3
通讯作者：
Min Chen;Yang Chen;E. Fan
中科院分区：
文献类型：
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作者：
Min Chen;Yang Chen;E. Fan
本文考虑权函数w（x，t）=（x+ t）λ x α e− x（x≥ 0，t> 0，α> 0，α+ λ+ 1> 0）描述的扰动Laguerre酉系综.采用Deift-Zhou非线性最速下降法分析了特征值相关核的极限。结果表明，在双尺度s= 4 nt，n→∞，t→ 0，使得s为正且有限的情况下，在硬边处，极限核可以用一个与三阶非线性微分方程相关的φ函数来描述，通过简单的变换，它等价于一个特殊的Painlevé V（简称PV）超越.此外，这个P V超越性等同于一般的Painlevé III超越性。对于较大的s，P-V核简化为贝塞尔核J α。对于小的s，P-V核简化为另一个贝塞尔核J α+ λ。在软边缘处，极限核是作为经典Laguerre权重的Airy核。
In this paper, we consider the perturbed Laguerre unitary ensemble described by the weight function of w (x, t)=(x+ t) λ x α e− x with x≥ 0, t> 0, α> 0, α+ λ+ 1> 0. The Deift–Zhou nonlinear steepest descent approach is used to analyze the limit of the eigenvalue correlation kernel. It was found that under the double scaling s= 4 n t, n→∞, t→ 0 such that s is positive and finite, at the hard edge, the limiting kernel can be described by the φ-function related to a third-order nonlinear differential equation, which is equivalent to a particular Painlevé V (shorted as P V) transcendent via a simple transformation. Moreover, this P V transcendent is equivalent to a general Painlevé III transcendent. For large s, the P V kernel reduces to the Bessel kernel J α. For small s, the P V kernel reduces to another Bessel kernel J α+ λ. At the soft edge, the limiting kernel is the Airy kernel as the classical Laguerre weight.