On the lower bounds for real double Hurwitz numbers

TL;DR

This paper establishes a lower bound for real double Hurwitz numbers using tropical geometry and demonstrates their asymptotic growth is logarithmically equivalent to complex Hurwitz numbers, advancing understanding of their asymptotic behavior.

Contribution

It introduces a new lower bound for real double Hurwitz numbers and proves their asymptotic growth matches that of complex Hurwitz numbers.

Findings

Lower bound derived from tropical computation

Asymptotic growth of real and complex Hurwitz numbers are logarithmically equivalent

Provides insights into the asymptotic behavior of real double Hurwitz numbers

Abstract

As the real counterpart of double Hurwitz number, the real double Hurwitz number depends on the distribution of real branch points. We consider the problem of asymptotic growth of real and complex double Hurwitz numbers. We provide a lower bound for real double Hurwitz numbers based on the tropical computation of real double Hurwitz numbers. By using this lower bound and J. Rau's result ( Math. Ann. 375(1-2): 895-915, 2019), we prove the logarithmic equivalence of real and complex Hurwitz numbers.