THE SCALING LIMIT OF RANDOM SIMPLE TRIANGULATIONS AND RANDOM SIMPLE QUADRANGULATIONS

• 成果类型: Article
• 署名作者: Addario-Berry, Louigi; Albenque, Marie
• 署名单位: McGill University; Institut Polytechnique de Paris; Ecole Polytechnique; Centre National de la Recherche Scientifique (CNRS)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/16-AOP1124
• 发表日期: 2017
• 页码: 2767-2825
• 关键词: bipartite planar maps; brownian map; discrete snakes

摘要

Let M-n be a simple triangulation of the sphere S-2, drawn uniformly at random from all such triangulations with n vertices. Endow M-n with the uniform probability measure on its vertices. After rescaling graph distance by (3/(4n))(1/4), the resulting random measured metric space converges in distribution, in the Gromov-Hausdorff-Prokhorov sense, to the Brownian map. In proving the preceding fact, we introduce a labelling function for the vertices of M-n. Under this labelling, distances to a distinguished point are essentially given by vertex labels, with an error given by the winding number of an associated closed loop in the map. We establish similar results for simple quadrangulations.