Bijective enumeration of planar bipartite maps with three tight boundaries, or how to slice pairs of pants

TL;DR

This paper introduces a simplified bijective formula for counting tight pairs of pants (planar bipartite maps with three boundaries) using an extended slice decomposition approach with new fundamental building blocks.

Contribution

It provides a simpler formula for tight pairs of pants maps and extends the bijective approach with novel structures like bigeodesic triangles and diangles.

Findings

New simple formula for tight pairs of pants maps

Extended slice decomposition with bigeodesic structures

Asymptotic analysis of minimal separating loops

Abstract

We consider planar maps with three boundaries, colloquially called pairs of pants. In the case of bipartite maps with controlled face degrees, a simple expression for their generating function was found by Eynard and proved bijectively by Collet and Fusy. In this paper, we obtain an even simpler formula for \emph{tight} pairs of pants, namely for maps whose boundaries have minimal length in their homotopy class. We follow a bijective approach based on the slice decomposition, which we extend by introducing new fundamental building blocks called bigeodesic triangles and diangles, and by working on the universal cover of the triply punctured sphere. We also discuss the statistics of the lengths of minimal separating loops in (non necessarily tight) pairs of pants and annuli, and their asymptotics in the large volume limit.