Hankel determinants for a singular complex weight and the first and third Painleve transcendents
Hankel determinants for a singular complex weight and the first and third Painleve transcendents
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奇异复权重的汉克尔行列式以及第一和第三 Painleve 超越项
DOI：
10.1016/j.jat.2016.01.006
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发表时间：
2016
影响因子：
0.9
通讯作者：
Zhao Yu-Qiu
中科院分区：
文献类型：
--
作者：
Xu Shuai-Xia;Dai Dan;Zhao Yu-Qiu
在复平面的某些轮廓上，我们考虑了在t< 0和z时，多项式与变化的扰动拉盖尔权e−n （z−log z+ t/z）正交。当参数n、t和度数k固定时，奇异复权的Hankel行列式显示为painleveiii方程的等构τ函数。当阶数k= n， n较大且t接近临界值时，受量子输运中Wigner时间延迟研究的启发，我们证明了递归系数和Hankel行列式的双尺度渐近行为是用painlev<e:1> I方程的Boutroux tronque解来描述的。我们的方法是基于黎曼-希尔伯特问题的Deift-Zhou非线性最陡下降法。
In this paper, we consider polynomials orthogonal with respect to a varying perturbed Laguerre weight e− n (z− log z+ t/z) for t< 0 and z on certain contours in the complex plane. When the parameters n, t and the degree k are fixed, the Hankel determinant for the singular complex weight is shown to be the isomonodromy τ-function of the Painlevé III equation. When the degree k= n, n is large and t is close to a critical value, inspired by the study of the Wigner time delay in quantum transport, we show that the double scaling asymptotic behaviors of the recurrence coefficients and the Hankel determinant are described in terms of a Boutroux tronquée solution to the Painlevé I equation. Our approach is based on the Deift–Zhou nonlinear steepest descent method for Riemann–Hilbert problems.