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 polynomiality, 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 et al. (2016) [18]. 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.

中文翻译：

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实单调双 Hurwitz 数的渐近


近年来，单调的双 Hurwitz 数作为双 Hurwitz 数的自然组合修饰而被引入。单调双 Hurwitz 数与经典数有许多结构特性，例如分段多项式，而这两个数的定量特性则大不相同。我们考虑单调双 Hurwitz 数的实模拟，并研究这些真实模拟的渐近。关键要素是对真实热带覆盖的解释，其中任意分裂作为对称群中的因式分解，这概括了 Guay-Paquet 等人（2016）[18] 的结果。通过使用上述解释，我们考虑了单调双 Hurwitz 数的三种实数类似物：相对于简单分裂的实单调双 Hurwitz 数、相对于任意分裂的实混合双 Hurwitz 数和实混合双 Hurwitz 数。在某些条件下，我们找到这些实数模拟的下限，并获得相对于任意分裂和实混合双 Hurwitz 数的实单调双 Hurwitz 数的对数渐近。特别是，在给定条件下，实混合双 Hurwitz 数在对数上等价于复数双 Hurwitz 数。我们构建了一个真实的热带覆盖层族，并使用它们来证明相对于简单分裂的真实单调双 Hurwitz 数在对数上等价于具有特定条件的单调双 Hurwitz 数。这与实双 Hurwitz 数和复数双 Hurwitz 数的对数等价一致。