Random stable laminations of the disk

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 $θ\in(1,2]$. 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 $θ=2$, we recover Aldous' Brownian triangulation. However, if $θ\in(1,2)$, 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 $θ$. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely $2-1/θ$.

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