第三十四类Painleve方程特解的渐近性
Asymptotics for a special solution of the thirty fourth Painleve equation
摘要：
在以前的文章中，我们研究了Z_{n,N} ^{-1}型酉随机矩阵系综的双尺度极限|\det M| ^{2α} e^{-N \Tr V（M）} dM，其中α> -1/2。因子|\det M| ^{2α}在原点附近诱导临界本征值行为。假设与V相关的极限平均特征值密度是正则的，并且原点是其支撑的右端点，我们计算了双标度极限中的极限特征值相关核为n，\to \infty使得n^{2/3}（n/N-1）= O（1），用Deift-Zhou最速下降方法求解Riemann-多项式在关于权|x| ^{2α} e^{-NV（x）}正交的直线上的Hilbert问题。我们的主要注意力是利用与Painleve XXXIV方程的一个特殊解u_α相联系的λ-函数构造原点附近的一个局部参数。这个解与Painleve II方程的一个特解有关，但它不同于通常的Hastings-McLeod解。本文将u_α（s）的渐近行为计算为s \to \pm \infty .我们猜想u_α的渐近性是由这一渐近性刻画的，并通过对包含u_α的Painleve XXXIV方程的一个单参数解族的渐近分析给出了支持性的论据.我们确定这个家庭作为家庭的tronquee解决方案的第三十四Painleve方程。
英文摘要：
In a previous paper we studied the double scaling limit of unitary random matrix ensembles of the form Z_{n,N}^{-1} |\det M|^{2α} e^{-N \Tr V(M)} dM with α> -1/2. The factor |\det M|^{2α} induces critical eigenvalue behavior near the origin. Under the assumption that the limiting mean eigenvalue density associated with V is regular, and that the origin is a right endpoint of its support, we computed the limiting eigenvalue correlation kernel in the double scaling limit as n, N \to \infty such that n^{2/3}(n/N-1) = O(1) by using the Deift-Zhou steepest descent method for the Riemann-Hilbert problem for polynomials on the line orthogonal with respect to the weight |x|^{2α} e^{-NV(x)}. Our main attention was on the construction of a local parametrix near the origin by means of the ψ-functions associated with a distinguished solution u_α of the Painleve XXXIV equation. This solution is related to a particular solution of the Painleve II equation, which however is different from the usual Hastings-McLeod solution. In this paper we compute the asymptotic behavior of u_α(s) as s \to \pm \infty. We conjecture that this asymptotics characterizes u_α and we present supporting arguments based on the asymptotic analysis of a one-parameter family of solutions of the Painleve XXXIV equation which includes u_α. We identify this family as the family of tronquee solutions of the thirty fourth Painleve equation.
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