Asymptotic enumeration of constellations and related families of maps on orientable surfaces

TL;DR

This paper derives explicit asymptotic formulas for counting certain classes of maps on orientable surfaces, revealing probabilistic relationships and confirming previous conjectures.

Contribution

It extends the Bouttier-Di Francesco-Guitter bijection to orientable surfaces and provides new asymptotic enumeration formulas for m-hypermaps and m-constellations.

Findings

Asymptotic formulas for map counts with specified face degrees

Each fundamental cycle contributes a factor m in the enumeration

Large maps with even face degrees are bipartite with probability 1/2^{2g}

Abstract

We perform the asymptotic enumeration of two classes of rooted maps on orientable surfaces of genus g: m-hypermaps and m-constellations. For m=2, they correspond respectively to maps with even face degrees and bipartite maps. We obtain explicit asymptotic formulas for the number of such maps with any finite set of allowed face degrees. Our proofs rely on the generalisation to orientable surfaces of the Bouttier-Di Francesco-Guitter bijection, and on generating series methods. We show that each of the 2g fondamental cycles of the surface contributes a factor m between the numbers of m-hypermaps and m-constellations -- for example, large maps of genus g with even face degrees are bipartite with probability tending to 1/2^{2g}. A special case of our results implies former conjectures of Gao.