THE DUAL TREE OF A RECURSIVE TRIANGULATION OF THE DISK

• 成果类型: Article
• 署名作者: Broutin, Nicolas; Sulzbach, Henning
• 署名单位: McGill University
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/13-AOP894
• 发表日期: 2015
• 页码: 738-781
• 关键词: partial match queries; SCALING LIMITS; galton-watson; quadtrees; THEOREM; circle

摘要

In the recursive lamination of the disk, one tries to add chords one after another at random; a chord is kept and inserted if it does not intersect any of the previously inserted ones. Curien and Le Gall [Ann. Probab. 39 (2011) 2224-2270] have proved that the set of chords converges to a limit triangulation of the disk encoded by a continuous process M. Based on a new approach resembling ideas from the so-called contraction method in function spaces, we prove that, when properly rescaled, the planar dual of the discrete lamination converges almost surely in the Gromov-Hausdorff sense to a limit real tree J, which is encoded by M. This confirms a conjecture of Curien and Le Gall.