The uniform asymptotics for real double Hurwitz numbers with triple ramification

TL;DR

This work develops a comprehensive tropical framework to study real double Hurwitz numbers with triple ramification, introducing generalized zigzag numbers as uniform lower bounds and connecting them to tropical counts via Markwig–Rau. It establishes uniform logarithmic lower bounds and logarithmic equivalence between real and complex Hurwitz numbers as degree and ramification data grow, and analyzes wall-crossing phenomena to define proper zigzag numbers with invariant lower bounds. The methods combine circle representations, simple resolutions, and refined tropical geometry to translate real counting problems into structured combinatorial counts, yielding asymptotic growth results that address long-standing questions about uniform bounds for Hurwitz numbers. The results have implications for understanding the relationship between real and complex enumerative counts, and provide partial answers to the Dubrovin–Yang–Zagier questions on uniform bounds for simple Hurwitz numbers in large-genus/degree limits.

Abstract

We consider the problem of counting real ramified coverings of the complex projective line $\mathbb{C}\mathbb{P}^1$ by real Riemann surfaces of genus $g$, where the ramification profiles over $0$ and $\infty$ are $λ$ and $μ$ respectively, and the ramification profiles over other real branch points consist of either $(3,1,\ldots,1)$ or $(2,1,\ldots,1)$. The solutions to this problem are called real double Hurwitz numbers with triple ramification and are denoted by $H^{\mathbb{R}}_g(λ,μ;Λ^-_{s,t},Λ^+_{s,t})$. We apply a modified version of the tropical computaton developed by Markwig and Rau for real Hurwitz numbers to compute $H^{\mathbb{R}}_g(λ,μ;Λ^-_{s,t},Λ^+_{s,t})$. Based on a generalization of Rau's zigzag number, we introduce a combinatorial invariant that serves as the lower bound for $H^{\mathbb{R}}_g(λ,μ;Λ^-_{s,t},Λ^+_{s,t})$. We obtain a uniform lower bound for the large degree and large genus logarithmic asymptotics of these combinatorial invariants. Our uniform lower bound implies the following results. (1). We establish a uniform lower bound for the large degree and large genus logarithmic asymptotics of $H^{\mathbb{R}}_g(λ,μ;Λ^-_{s,t},Λ^+_{s,t})$ and its complex counterparts. In particular, we give a partial answer to an open question proposed by Dubrovin, Yang and Zagier on the uniform bound for simple Hurwitz numbers. (2). We obtain the logarithmic equivalence between $H^{\mathbb{R}}_g(λ,μ;Λ^-_{s,t},Λ^+_{s,t})$ and its complex counterparts 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 $H^{\mathbb{R}}_g(λ,μ;Λ^-_{s,t},Λ^+_{s,t})$ and its complex counterparts are of the same order.

Figures (41)

• Figure 1: Circle representing $\sigma$.
• Figure 3: Symmetric structures of real cycles.
• Figure 4: Real cut of the real cycle $\sigma$.
• Figure 5: Generators of $\pi_1(\mathbb{C} P^1\setminus\{0,p,\infty\},x_0)$.
• Figure 6: Two monodromy graphs. When $e_1=e_2$, the number of 3-cycles is in the bracket.

Theorems & Definitions (103)

• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Definition 2.1
• Definition 2.2
• Definition 2.3