Random recursive triangulations of the disk via fragmentation theory

TL;DR

This paper introduces a new model of infinite random triangulation of the disk generated by recursive chord insertions, analyzes its fractal properties, and connects it to a continuous geodesic lamination coded by a Hölder continuous function.

Contribution

It provides a novel recursive construction of random disk triangulations and characterizes their fractal and geometric properties in the limit.

Findings

Hausdorff dimension of the limit set is $eta^*+1$

Limit set can be described by a Hölder continuous coding function

Recursive models converge to the same continuous limit

Abstract

We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact subset of the unit disk whose complement is a countable union of triangles. We show that this limiting random set has Hausdorff dimension $\beta^*+1$, where $\beta^*=(\sqrt{17}-3)/2$, and that it can be described as the geodesic lamination coded by a random continuous function which is H\"{o}lder continuous with exponent $\beta^*-\varepsilon$, for every $\varepsilon>0$. We also discuss recursive constructions of triangulations of the $n$-gon that give rise to the same continuous limit when $n$ tends to infinity.