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

The Brownian triangulation is a random compact subset of the unit disk introduced by Aldous. For $ϵ>0$, let $N(ϵ)$ be the number of triangles whose sizes (measured in different ways) are greater than $ϵ$ in the Brownian triangulation. We determine the asymptotic behaviour of $N(ϵ)$ as $ϵ\to 0$. 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 $N(ϵ)$ 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.