On the number of large triangles in the Brownian triangulation and fragmentation processes

TL;DR

This paper analyzes the asymptotic number of large triangles in the Brownian triangulation and introduces a novel approach to study large dislocations in self-similar fragmentation processes, with applications to geodesic stable laminations.

Contribution

It introduces a new method for analyzing large dislocations in fragmentation processes and applies it to understand the structure of the Brownian triangulation.

Findings

Asymptotic behavior of N(ε) as ε approaches 0 is determined.

A new concept of 'large' dislocations is proposed and developed.

The method also yields the distribution of the longest chord in the triangulation.

Abstract

The Brownian triangulation is a random compact subset of the unit disk introduced by Aldous. For $\epsilon>0$, let $N(\epsilon)$ be the number of triangles whose sizes (measured in different ways) are greater than $\epsilon$ in the Brownian triangulation. We determine the asymptotic behaviour of $N(\epsilon)$ as $\epsilon \to 0$. To obtain this result, a novel concept of "large" dislocations in fragmentations has been proposed. We develop an approach to study the number of large dislocations which is widely applicable to general self-similar fragmentation processes. This technique enables us to study $N(\epsilon)$ because of a bijection between the triangles in the Brownian triangulation and the dislocations of a certain self-similar fragmentation process. Our method also provides a new way to obtain the law of the length of the longest chord in the Brownian triangulation. We further extend our results to the more general class of geodesic stable laminations introduced by Kortchemski.