RANDOM STABLE LAMINATIONS OF THE DISK

• 成果类型: Article
• 署名作者: Kortchemski, Igor
• 署名单位: Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP799
• 发表日期: 2014
• 页码: 725-759
• 关键词: scaling limits; random trees; planar maps; triangulations; number

摘要

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 is an element of (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 is an element of (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 Levy process of index theta. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely 2 - 1/theta.