Random stable laminations of the disk

TL;DR

This paper investigates the scaling limits of random polygon dissections weighted by stable laws, revealing a new class of stable laminations with fractal properties and connecting them to Lévy processes.

Contribution

It introduces the concept of random stable laminations for polygon dissections in the domain of attraction of stable laws, extending previous work on Brownian triangulations.

Findings

Convergence of random dissections to stable laminations as size grows

Identification of the stable lamination coding via Lévy process excursions

Determination of the Hausdorff dimension as 2 - 1/θ

Abstract

We study large random dissections of polygons. We consider random dissections of a regular polygon with $n$ sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index $\theta\in(1,2]$. As $n$ goes to infinity, we prove that these random dissections converge in distribution toward a random compact set, called the random stable lamination. If $\theta=2$, we recover Aldous' Brownian triangulation. However, if $\theta\in(1,2)$, large faces remain in the limit and a different random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive L\'{e}vy process of index $\theta$. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely $2-1/\theta$.