论文标题
地图枚举的多个尺度渐进性
Multiple Scale Asymptotics of Map Enumeration
论文作者
论文摘要
我们引入了一种系统的方法，以表达与平面表面相关的单个生成函数在高属表面上枚举地图枚举的生成功能。这项工作的核心是从两个不同的数学领域获得的两个渐近扩展的比较：正交多项式的Riemann-Hilbert分析和离散动力学系统的理论。通过将这些扩展在其参数中的共同有效性区域中等同，我们恢复已知的结果并提供了新的表达式，以生成与属0到7属表面上的图形枚举相关的功能的新表达式。尽管文章的主体集中在4个价值地图上，此处介绍的方法扩展到适用于详细范围的正常映射，以详细介绍一些奇数，以详细介绍一些奇怪的范围。
We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two asymptotic expansions obtained from two different fields of mathematics: the Riemann-Hilbert analysis of orthogonal polynomials and the theory of discrete dynamical systems. By equating the coefficients of these expansions in a common region of uniform validity in their parameters, we recover known results and provide new expressions for generating functions associated with graphical enumeration on surfaces of genera 0 through 7. Although the body of the article focuses on 4-valent maps, the methodology presented here extends to regular maps of arbitrary even valence and to some cases of odd valence, as detailed in the appendices.
