The uniform asymptotics for real double Hurwitz numbers with triple ramification I: the tropical correspondence

This is the first of two papers on the uniform asymptotics for real double Hurwitz numbers with triple ramification. Real double Hurwitz numbers with triple ramification count the number of 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 $\lambda$ and $\mu$ respectively, and the ramification profiles over the remaining real branch points consist of either $(3,1,\ldots,1)$ or $(2,1,\ldots,1)$. We apply a modified version of the tropical computation framework developed by Markwig and Rau for real Hurwitz numbers (Math. Z. 281 (2015), no. 1-2, 501-522) to compute the real double Hurwitz numbers with triple ramification. The new ingredient in our computation is the application of real simple resolution, a technique that enables us to resolve a triple branch point into a pair of simple branch points. Using real simple resolution, we establish a correspondence between real double Hurwitz numbers with triple ramification and weighted counts of tropical covers. This modified tropical correspondence simplifies the asymptotic analysis of real double Hurwitz numbers with triple ramification.