Bijections for planar maps with boundaries

TL;DR

This paper develops bijections for planar maps with boundaries, providing new and existing enumerative formulas for triangulations and quadrangulations, and applies these results to solve the dimer model exactly.

Contribution

It introduces a unified bijective approach to enumerate planar maps with boundaries, including new formulas for quadrangulations and generalizations for various face degrees.

Findings

Recovered Krikun's formula for triangulations

Derived new formula for quadrangulations

Provided exact solutions for the dimer model

Abstract

We present bijections for planar maps with boundaries. In particular, we obtain bijections for triangulations and quadrangulations of the sphere with boundaries of prescribed lengths. For triangulations we recover the beautiful factorized formula obtained by Krikun using a (technically involved) generating function approach. The analogous formula for quadrangulations is new. We also obtain a far-reaching generalization for other face-degrees. In fact, all the known enumerative formulas for maps with boundaries are proved bijectively in the present article (and several new formulas are obtained). Our method is to show that maps with boundaries can be endowed with certain "canonical" orientations, making them amenable to the master bijection approach we developed in previous articles. As an application of our enumerative formulas, we note that they provide an exact solution of the dimer model on rooted triangulations and quadrangulations.