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

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.

💡 Research Summary

This paper studies the uniform asymptotics of real double Hurwitz numbers with triple ramification, i.e. branched covers of the Riemann sphere whose ramification profiles over 0 and ∞ are arbitrary partitions λ and μ, while all other branch points are either simple (ramification type (2,1,…,1)) or triple (type (3,1,…,1)). The authors build on the “modified tropical correspondence theorem” introduced in the first part of the series, which allows one to express such real Hurwitz numbers as weighted counts of coloured tropical covers. In this tropical picture, positive and negative real branch points are distinguished by two colouring rules for even edges, and the presence of 4‑valent vertices (triple ramification) and genus‑one 2‑valent vertices makes the sign analysis considerably more intricate than in the ordinary (simple‑branch) case.

The central contribution is the definition of a new combinatorial invariant, a generalized zigzag number denoted (Z_g(\lambda,\mu;\Lambda_{s,t})). This invariant counts coloured tropical covers with a prescribed distribution of triple and simple branch points (encoded in the sequence (\Lambda_{s,t})). By analysing a gluing formula (Lemma 3.7) and a wall‑crossing phenomenon when a triple branch point passes a simple one, the authors introduce a “proper” zigzag number (Z_g(\lambda,\mu;s,t)) that is independent of the detailed ordering of the branch points. Theorem 5.3 proves that this proper zigzag number provides a lower bound for the original generalized zigzag number, which in turn bounds the real Hurwitz numbers from below: \