标题： 随机矩阵，图的计数和Toda格子的连续极限 显示英文标题

标题： Random Matrices, Graphical Enumeration and the Continuum Limit of Toda Lattices

摘要： 在本文中，我们推导出一组微分方程，这些方程唯一确定了在大$N$时，$N \times N$酉随机矩阵的划分函数对数渐近展开式的系数。 这些系数是黎曼面上图形计数的生成函数。 我们特别考虑的情况是，底层测度与高斯权重仅在次数为$2\nu$的单项式项上有所不同。 该项的耦合参数在微分方程中起着独立动力变量的作用。 从这些方程可以得出渐近展开式中系数的功能分析特征。 此外，这个常微分方程组可以递归求解，以显式构造这些系数作为耦合参数的函数。 对渐近系数精细结构的分析可以扩展到多个耦合参数，并且我们针对两个参数的情况提供了一个有限的说明。 显示英文摘要

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 $2\nu$. 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.