随机矩阵、图计数与Toda格的连续极限
Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices
摘要：
本文导出了一类微分方程，它唯一地确定了$N$ Hermitian随机矩阵的配分函数的对数的渐近展开式中的系数。这些系数是黎曼曲面上图形计数的生成函数。我们特别考虑的情况是一个潜在的措施，从高斯权重不同的程度$2ν$一个单项。这一项的耦合参数在微分方程中起着独立动力学变量的作用。从这些方程中可以推导出渐近展开式中系数的泛函分析特征。此外，该方程组可以递归求解，以显式地将这些系数构造为耦合参数的函数。这种精细结构的渐近系数的分析可以扩展到多个耦合参数，我们提出了一个有限的说明，这两个参数的情况下。
英文摘要：
In this paper we derive a hierarchy of differential equations which uniquely determine the coefficients in the asymptotic expansion, for large $N$, of the logarithm of the partition function of $N \times N$ Hermitian random matrices. These coefficients are generating functions for graphical enumeration on Riemann surfaces. The case that we particularly consider is for an underlying measure that differs from the Gaussian weight by a single monomial term of degree $2ν$. The coupling parameter for this term plays the role of the independent dynamical variable in the differential equations. From these equations one may deduce functional analytic characterizations of the coefficients in the asymptotic expansion. Moreover, this ode system can be solved recursively to explicitly construct these coefficients as functions of the coupling parameter. This analysis of the fine structure of the asymptotic coefficients can be extended to multiple coupling parameters and we present a limited illustration of this for the case of two parameters.
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