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

TL;DR

This work advances the understanding of uniform asymptotics for real double Hurwitz numbers with triple ramification and their complex counterparts by developing a robust combinatorial framework based on generalized zigzag numbers and a refined tropical viewpoint. Central to the approach is the modified tropical correspondence theorem, which expresses real RDH numbers as weighted sums over effectively coloured resolving tropical covers, with a multiplicity that captures edge weights and sign data; this underpins the construction of generalized zigzag numbers $Z_g(\\lambda,\\mu;\\Lambda_{s,t})$ and their distribution-independent lower bounds. The authors establish non-vanishing results and asymptotic growth rates for zigzag counts, define proper zigzag numbers via wall-crossing, and prove injection-based lower bounds that render these quantities uniform in the degree and genus. These combinatorial bounds translate into uniform logarithmic lower bounds for real RDH numbers and yield logarithmic equivalence with the complex analogues in several asymptotic regimes, providing partial answers to the open question of Dubrovin–Yang–Zagier on uniform bounds for simple Hurwitz numbers. Overall, the paper delivers a cohesive real-tropical strategy to obtain sharp, distribution-insensitive lower bounds and to compare real and complex Hurwitz numbers in a unified asymptotic framework.

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.

Figures (13)

• Figure 1: Pairs of vertices of resolving tropical covers.
• Figure 2: Positive and negative vertices: even edges are coloured, odd edges are in black, and edges in $I_\rho$ are dotted.
• Figure 3: Signed pairs: even edges are drawn in colours, odd edges in black. Dotted edges are the symmetric cycles or forks contained in $I_\rho$.
• Figure 4: Tails for generalized zigzag covers. The number of symmetric cycles can be arbitrary.
• Figure 6: generalized zigzag cover $\varphi_i:C_i\to T\mathbb{P}^1$ of type $(0,1^4,1^4,\underline x_i)$.

Theorems & Definitions (58)

• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Remark 2.5: rau2019