Asymptotics for real monotone double Hurwitz numbers

TL;DR

This paper investigates the asymptotic behavior of real monotone double Hurwitz numbers, revealing their logarithmic equivalence to complex double Hurwitz numbers through tropical geometry and group factorizations.

Contribution

It introduces a framework for analyzing real analogues of monotone double Hurwitz numbers and establishes their asymptotic relationships with classical counterparts.

Findings

Lower bounds for real monotone double Hurwitz numbers under certain conditions

Logarithmic asymptotics for real monotone double Hurwitz numbers relative to arbitrary splittings

Logarithmic equivalence between real mixed and complex double Hurwitz numbers

Abstract

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.