Growing uniform planar maps face by face

TL;DR

This paper introduces growth schemes for uniformly generating random 2p-angulations and simple triangulations of the sphere, enabling incremental construction while preserving uniformity.

Contribution

It provides novel methods to insert faces into uniform maps, including a new bijective approach for simple triangulations, advancing the understanding of random planar map generation.

Findings

Provides explicit growth schemes for 2p-angulations

Introduces a new bijective presentation for simple triangulations

Enables incremental uniform map generation

Abstract

We provide "growth schemes" for inductively generating uniform random $2p$-angulations of the sphere with $n$ faces, as well as uniform random simple triangulations of the sphere with $2n$ faces. In the case of $2p$-angulations, we provide a way to insert a new face at a random location in a uniform $2p$-angulation with $n$ faces in such a way that the new map is precisely a uniform $2p$-angulation with $n+1$ faces. Similarly, given a uniform simple triangulation of the sphere with $2n$ faces, we describe a way to insert two new adjacent triangles so as to obtain a uniform simple triangulation of the sphere with $2n+2$ faces. The latter is based on a new bijective presentation of simple triangulations that relies on a construction by Poulalhon and Schaeffer.