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

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.