The uniform asymptotics for real double Hurwitz numbers with triple ramification II: lower bounds and asymptotics

TL;DR

This paper establishes uniform lower bounds and asymptotic equivalences for real double Hurwitz numbers with triple ramification, advancing understanding of their growth and relation to complex analogues in large-degree and large-genus regimes.

Contribution

It introduces a combinatorial invariant as a lower bound and proves asymptotic equivalences, partially answering an open question on Hurwitz number bounds.

Findings

Established uniform lower bounds for large-degree and large-genus asymptotics.

Proved logarithmic equivalence between real and complex Hurwitz numbers as degree increases.

Showed that the growth order of real and complex Hurwitz numbers matches in high genus limit.

Abstract

This is the second of two papers on the uniform asymptotics for real double Hurwitz numbers with triple ramification. Using the modified tropical correspondence theorem established in the first paper of this series, we introduce a combinatorial invariant that serves as a lower bound for real double Hurwitz numbers with triple ramification. We derive a uniform lower bound for the large-degree and large-genus logarithmic asymptotics of these combinatorial invariants. This uniform lower bound yields the following results: (1) We establish a uniform lower bound for the large-degree and large-genus logarithmic asymptotics of real double Hurwitz numbers with triple ramification and their complex analogues. In particular, we provide a partial answer to an open question proposed by Dubrovin, Yang and Zagier on the uniform bound for simple Hurwitz numbers. (2) We prove logarithmic equivalence between real double Hurwitz numbers with triple ramification and their complex analogues as the degree tends to infinity and only simple branch points are added. (3) As the genus tends to infinity and only simple branch points are added, we show that the logarithms of real double Hurwitz numbers with triple ramification and their complex analogues are of the same order.