Explicit enumeration and large-valence asymptotics of even-valent maps

Abstract

Let ${\mathcal{N}}_g(2\nu ,j)$ denote the number of connected labelled $2\nu$-valent maps of genus $g$ with $j$ vertices. Explicit bivariate formulae for ${\mathcal{N}}_g(2\nu ,j)$ have been available only in the planar and toroidal cases. Using a structural formula of Ercolani et al (2023), we translate the problem of determining an explicit bivariate formula for ${\mathcal{N}}_g(2\nu ,j)$, $g\ge 2$, to finding finitely many counts with a fixed number of vertices. For $g=2,3$ and $4$ we determine these counts using the associated orthogonal polynomials, yielding explicit bivariate formulae for ${\mathcal{N}}_g(2\nu ,j)$ in these genera. Furthermore, the same method applies for every $g\ge 5$ at the cost of additional computation. From these formulae we obtain the leading-order asymptotics of ${\mathcal{N}}_g(2\nu ,j)$ as $\nu \to \infty$ for $g=2,3,4$, and we conjecture the structure of these formulae in general genus. In addition, we establish an analogous reduction to finitely many counts, derive explicit formulae and large-valence asymptotics, and formulate corresponding conjectures for two-legged even-valent maps.

arXiv comment51 pages, 7 figures