Geodesics in large planar maps and in 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: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-010-0056-5
• 发表日期: 2010
• 页码: 287-360
• 关键词: scaling limits; CONVERGENCE; GROWTH; SPACES; snake; tour

摘要

We study geodesics in the random metric space called the Brownian map, which appears as the scaling limit of large planar maps. In particular, we completely describe geodesics starting from the distinguished point called the root, and we characterize the set S of all points that are connected to the root by more than one geodesic. The set S is dense in the Brownian map and homeomorphic to a non-compact real tree. Furthermore, for every x in S, the number of distinct geodesics from x to the root is equal to the number of connected components of S\{x}. In particular, points of the Brownian map can be connected to the root by at most three distinct geodesics. Our results have applications to the behavior of geodesics in large planar maps.