Multi-critical unitary random matrix ensembles and the general Painlevé II equation
Multi-critical unitary random matrix ensembles and the general Painlevé II equation
复制标题
DOI：
10.4007/annals.2008.168.601
复制
发表时间：
2005-08
影响因子：
4.9
通讯作者：
T. Claeys;A. Kuijlaars;M. Vanlessen
中科院分区：
文献类型：
--
作者：
T. Claeys;A. Kuijlaars;M. Vanlessen
本文研究了形式为Z-′ N\detM\2 ~ e-NTrVWdM的酉随机矩阵系综，其中a > -1/2，V使得n，N - > α和n/N -ω 1的极限平均特征值密度在原点二次为零.为了计算特征值相关核在原点附近的双标度极限，我们使用Deift/Zhou最速下降法，该方法应用于关于权重\x\2ae ~NV^x\的真实的直线上的正交多项式的Riemann-Hilbert问题。与Painleve II方程q”= sq +2 q3- a的特解qa相关联的函数。通过证明相应的Riemann-Hilbert问题的可解性，证明了当a > -1/2时，qa没有真实的极点.我们还证明了正交多项式的递归系数的渐近性可以表示为qa的双标度极限。
We study unitary random matrix ensembles of the form Z-'N\detM\2«e-NTrVWdM, where a > -1/2 and V is such that the limiting mean eigenvalue density for n, N - > oc and n/N - ► 1 vanishes quadratically at the origin. In order to compute the double scaling limits of the eigenvalue correlation kernel near the origin, we use the Deift/Zhou steepest descent method applied to the Riemann-Hilbert problem for orthogonal polynomials on the real line with respect to the weight \x\2ae~NV^x\ Here the main focus is on the construction of a local parametrix near the origin with ^-functions associated with a special solution qa of the Painleve II equation q" = sq + 2q3 - a. We show that qa has no real poles for a > -1/2, by proving the solvability of the corresponding Riemann-Hilbert problem. We also show that the asymptotics of the recurrence coefficients of the orthogonal polynomials can be expressed in terms of qa in the double scaling limit.