Lower bounds and asymptotics of real double Hurwitz numbers

TL;DR

This paper investigates the properties of real double Hurwitz numbers, establishing lower bounds and asymptotic behavior, and demonstrating conditions for the existence of real Hurwitz covers using tropical geometry techniques.

Contribution

It introduces a lower bound for real double Hurwitz numbers based on tropical computations and explores their asymptotics and relation to classical Hurwitz numbers.

Findings

Established a lower bound for real double Hurwitz numbers.

Proved existence conditions for real Hurwitz covers.

Showed logarithmic equivalence between real and classical Hurwitz numbers.

Abstract

We study the real counterpart of double Hurwitz numbers, called real double Hurwitz numbers here. We establish a lower bound for these numbers with respect to their dependence on the distribution of branch points. We use it to prove, under certain conditions, existence of real Hurwitz covers as well as logarithmic equivalence of real and classical Hurwitz numbers. The lower bound is based on the \enquote{tropical} computation of real Hurwitz numbers in arXiv:1412.4235.