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 ∈ (1, 2]. As n goes to infinity, we prove that these random dissections converge in distribution towards a random compact set, called the random stable lamination. If θ = 2, we recover Aldous ’ Brownian triangulation. However, if θ ∈ (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