Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of construction developed here involves a novel asymptotic symbol calculus for difference operators based on the relation between spectral asymptotics for Hermitian random matrices and asymptotics of orthogonal polynomials with exponential weights. These closed form expressions have a universal character in the sense that they are independent of the explicit valence distribution of the cellular networks within a broad class. Nevertheless the valence distributions may be recovered from the closed form generating functions by a remarkable unwinding identity in terms of Appell polynomials generated by Bessel functions. Our treatment reveals the generating functions to be solutions of nonlinear conservation laws and their prolongations. This characterization enables one to gain insights that go beyond more traditional methods that are purely combinatorial. Universality results are connected to stability results for characteristic singularities of conservation laws that were studied by Caflisch, Ercolani, Hou and Landis, Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems, Commun. Pure Appl. Math. 46 (1993) 453–499, as well as directly related to universality results for random matrix spectra.

地图是黎曼曲面上的多边形蜂窝网络。本文分析了与固定但任意属的映射相关的枚举生成函数的封闭形式一般表示的构造。这里开发的构造方法涉及一种新颖的差分算子的渐近符号演算，它基于 Hermitian 随机矩阵的谱渐近和具有指数权重的正交多项式的渐近之间的关系。这些封闭形式的表达式具有普遍性，因为它们独立于广泛类别中蜂窝网络的显式价分布。然而，化合价分布可以通过显着的展开恒等式从封闭形式的生成函数中恢复根据贝塞尔函数生成的 Appell 多项式。我们的处理揭示了生成函数是非线性守恒定律及其扩展的解。这种表征使人们能够获得超越纯粹组合的更传统方法的见解。普遍性结果与 Caflisch、Ercolani、Hou 和 Landis、非线性双曲或椭圆系统的多值解和分支点奇异性研究的守恒定律特征奇点的稳定性结果有关，Commun。纯应用。数学。 46 (1993) 453–499，以及与随机矩阵光谱的普遍性结果直接相关。