随机矩阵模型中双标度极限的普适性
Universality of the double scaling limit in random matrix models
摘要：
本文研究了酉随机矩阵系综在极限平均特征值密度在支撑内点二次为零的临界情形。我们建立了双标度极限的特征值相关核在这样一个临界点的极限的普遍性。极限核由与第二个Painleve方程相关的函数构成。这推广了Bleher及其在临界四次势的特殊情况下的一个结果。我们使用的两个主要工具是均衡测度和Riemann-Hilbert问题。在我们处理平衡测度时，我们允许临界点附近有一个负密度，这使我们能够同时处理所有情况。Riemann-Hilbert问题的渐近分析是用Deift/Zhou最速下降分析来完成的。为了在临界点处构造一个局部参数，我们对Baik，Deift和Johansson的方法进行了修改，以便我们能够完全满足所需的跳跃特性。
英文摘要：
We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painleve equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann-Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann-Hilbert problem is done with the Deift/Zhou steepest descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly.
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