Triangulating stable laminations

TL;DR

This paper investigates the asymptotic properties of random noncrossing planar trees by triangulating stable laminations, revealing new insights into their structure and introducing iterative methods for laminations and trees.

Contribution

It introduces the concept of triangulating stable laminations to analyze the asymptotic behavior of random planar trees and explores iterative processes for laminations and trees.

Findings

Distributional limits via triangulation of faces in stable laminations

Introduction of iterative methods for laminations and trees

New connections between stable Lévy processes and noncrossing trees

Abstract

We study the asymptotic behavior of random simply generated noncrossing planar trees in the space of compact subsets of the unit disk, equipped with the Hausdorff distance. Their distributional limits are obtained by triangulating at random the faces of stable laminations, which are random compact subsets of the unit disk made of non-intersecting chords coded by stable L\'evy processes. We also study other ways to "fill-in" the faces of stable laminations, which leads us to introduce the iteration of laminations and of trees.