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 Lévy process of index $\theta$. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely $2-1/\theta$.
"Random stable laminations of the disk." Ann. Probab. 42 (2) 725 - 759, March 2014. https://doi.org/10.1214/12-AOP799