发表机构
Grinnell College; University of California, Santa Cruz; KTH Royal Institute of Technology（格林内尔学院; 加州大学圣克鲁兹分校; 瑞典皇家理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究具有任意度序列的连通、带标记、亏格为 \(g\) 的欧拉地图数量渐近性，结合正交多项式递推系数分析与多变量解析组合学理论，得出通用渐近性及亏格为 \(1\) 时的精确公式，是 \(g\geq 1\) 非正则地图计数问题的首个结果。
AI 中文摘要
我们计算了具有任意度序列的连通、带标记、亏格为 \(g\) 的欧拉地图数量在顶点总数趋于无穷时的渐近性。此渐近性是通用的，其主阶项仅取决于有限多个地图特征，公式中的常数因子与Painlevé I 方程相关。我们的方法结合了对特定正交多项式族相关递推系数的分析以及多变量解析组合学理论。还推导了连通、带标记、亏格为 \(1\) 的欧拉地图数量的精确公式。这些是关于 \(g\geq 1\) 的非正则（混合价）地图此类计数问题的首个结果。
英文摘要
We calculate the asymptotics of the number of connected, labeled, genus $g$ Eulerian maps with an arbitrary degree sequence, in the limit as the total number of vertices tends to infinity. This asymptotic is universal, in the sense that the leading-order term depends on only three map characteristics, regardless of the choice of the degree sequence. The constant factor in this formula is related to the Painlevé I equation. Our methods combine for the first time the analysis of the recurrence coefficients associated to a particular family of orthogonal polynomials, and the theory of analytic combinatorics of several variables. We also derive an exact formula for the number of connected, labeled, genus $1$ Eulerian maps. These are the first results on this kind of enumeration problem for $g\geq 1$, non-regular (mixed-valence) maps.
CommentsCorrected some minor typos, and some slight changes in notation
