Representations of stack triangulations in the plane

TL;DR

This paper studies the geometric and probabilistic properties of random stack triangulations in the plane, analyzing their asymptotic shapes and vertex distribution limits under two probability models.

Contribution

It introduces two probability distributions for stack triangulations, provides their planar representations, and investigates the asymptotic behavior of these random structures and their vertex measures.

Findings

Convergence of the vertex occupation measure to a limit measure.

Asymptotic geometric properties of the random triangulations.

Differences between the two probability models in structure and distribution.

Abstract

Stack triangulations appear as natural objects when defining an increasing family of triangulations by successive additions of vertices. We consider two different probability distributions for such objects. We represent, or "draw" these random stack triangulations in the plane $\R^2$ and study the asymptotic properties of these drawings, viewed as random compact metric spaces. We also look at the occupation measure of the vertices, and show that for these two distributions it converges to some random limit measure.