论文标题
真正的单调双Hurwitz编号的渐近学
Asymptotics for real monotone double Hurwitz numbers
论文作者
论文摘要
近年来，单调双Hurwitz数字被引入了自然组合的双Hurwitz数字。单调双Hurwitz数字具有许多结构性属性，其经典属性（例如分段多项式索引），而这两个数字的定量属性却大不相同。我们考虑了单调双Hurwitz数量的真实类似物，并研究了这些实际类似物的渐近学。关键要素是对真正的热带覆盖物的解释，并用任意分裂为对称组的因素化，该群体概括了Guay-Paquet，Markwig和Rau（Int。Math。Res。Not。Imrn，2016（1）（1）：258-293，2016）。通过使用上述解释，我们考虑了单调双Hurwitz数字的三种类型的实际类似物：相对于任意分裂和实际混合双Hurwitz数字，相对于简单的分离，实际单调双Hurwitz数字。在某些条件下，我们找到了这些实际类似物的下限，并获得对数的渐近造型，用于实际单调双Hurwitz数量相对于任意分裂和实际混合双Hurwitz数字。特别是，在给定条件下，实际混合双Hurwitz数量在对数上等同于复杂的双Hurwitz数字。我们构建了一个真正的热带覆盖物的家族，并使用它们来表明相对于简单分裂的真实单调双hurwitz数字在对数上等同于具有特定条件的单调双hurwitz数字。这与真实双Hurwitz数字和复杂的双Hurwitz数字的对数等效性是一致的。
In recent years, monotone double Hurwitz numbers were introduced as a naturally combinatorial modification of double Hurwitz numbers. Monotone double Hurwitz numbers share many structural properties with their classical counterparts, such as piecewise polynomaility, while the quantitative properties of these two numbers are quite different. We consider real analogues of monotone double Hurwitz numbers and study the asymptotics for these real analogues. The key ingredient is an interpretation of real tropical covers with arbitrary splittings as factorizations in the symmetric group which generalizes the result from Guay-Paquet, Markwig, and Rau (Int. Math. Res. Not. IMRN, 2016(1):258-293, 2016). By using the above interpretation, we consider three types of real analogues of monotone double Hurwitz numbers: real monotone double Hurwitz numbers relative to simple splittings, relative to arbitrary splittings and real mixed double Hurwitz numbers. Under certain conditions, we find lower bounds for these real analogues, and obtain logarithmic asymptotics for real monotone double Hurwitz numbers relative to arbitrary splittings and real mixed double Hurwitz numbers. In particular, under given conditions real mixed double Hurwitz numbers are logarithmically equivalent to complex double Hurwitz numbers. We construct a family of real tropical covers and use them to show that real monotone double Hurwitz numbers relative to simple splittings are logarithmically equivalent to monotone double Hurwitz numbers with specific conditions. This is consistent with the logarithmic equivalence of real double Hurwitz numbers and complex double Hurwitz numbers.
