UNIQUENESS AND UNIVERSALITY OF THE BROWNIAN MAP

• 成果类型: Article
• 署名作者: Le Gall, Jean-Francois
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/12-AOP792
• 发表日期: 2013
• 页码: 2880-2960
• 关键词: scaling limits; INVARIANCE-PRINCIPLES; planar maps

摘要

We consider a random planar map M-n which is uniformly distributed over the class of all rooted q-angulations with n faces. We let m(n) be the vertex set of M-n, which is equipped with the graph distance d(gr). Both when q >= 4 is an even integer and when q = 3, there exists a positive constant c(q) such that the rescaled metric spaces (m(n), c(q)n(-1/4)d(gr)) converge in distribution in the Gromov-Hausdorff sense, toward a universal limit called the Brownian map. The particular case of triangulations solves a question of Schramm.