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.
By sampling tessellations according to a Boltzmann weight depending on the degree of the faces, Kortchemski REF obtained limit laminations which are not triangulations and are encoded by excursions of stable spectrally positive Lévy processes (with Lévy measure concentrated on [0, ∞)).
Random stable laminations of the disk
Annals of Probability 2014, Vol. 42, No. 2, 725-759
Mathematics