RECURSIVE TRIANGULATIONS OF THE DISK VIA FRAGMENTATION THEORY

• 成果类型: Article
• 署名作者: Curien, Nicolas; Le Gall, Jean-Francois
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/10-AOP608
• 发表日期: 2011
• 页码: 2224-2270
• 关键词: laws

摘要

We introduce and study an infinite random triangulation of the unit disk that arises as the limit of several recursive models. This triangulation is generated by throwing chords uniformly at random in the unit disk and keeping only those chords that do not intersect the previous ones. After throwing infinitely many chords and taking the closure of the resulting set, one gets a random compact subset of the unit disk whose complement is a countable union of triangles. We show that this limiting random set has Hausdorff dimension beta* + 1, where beta* = (root 17- 3)/2, and that it can be described as the geodesic lamination coded by a random continuous function which is Holder continuous with exponent beta* - epsilon, for every epsilon > 0. We also discuss recursive constructions of triangulations of the n-gon that give rise to the same continuous limit when n tends to infinity.