THE TOPOLOGY OF SCALING LIMITS OF POSITIVE GENUS RANDOM QUADRANGULATIONS

• 成果类型: Article
• 署名作者: Bettinelli, Jeremie
• 署名单位: Universite Paris Saclay
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/11-AOP675
• 发表日期: 2012
• 页码: 1897-1944
• 关键词: planar maps; trees

摘要

We discuss scaling limits of large bipartite quadrangulations of positive genus. For a given g, we consider, for every n >= 1, a random quadrangulation q(n) uniformly distributed over the set of all rooted bipartite quadrangulations of genus g with n faces. We view it as a metric space by endowing its set of vertices with the graph metric. As n tends to infinity, this metric space, with distances rescaled by the factor n(-1/4), converges in distribution, at least along some subsequence, toward a limiting random metric space. This convergence holds in the sense of the Gromov-Hausdorff topology on compact metric spaces. We show that, regardless of the choice of the subsequence, the limiting space is almost surely homeomorphic to the genus g-torus.