RANDOM PLANAR MAPS AND GROWTH-FRAGMENTATIONS

• 成果类型: Article
• 署名作者: Bertoin, Jean; Curien, Nicolas; Kortchemski, Igor
• 署名单位: University of Zurich; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/17-AOP1183
• 发表日期: 2018
• 页码: 207-260
• 关键词: self-similar fragmentations; SCALING LIMITS; brownian plane; quadrangulations; triangulations; percolation

摘要

We are interested in the cycles obtained by slicing at all heights random Boltzmann triangulations with a simple boundary. We establish a functional invariance principle for the lengths of these cycles, appropriately rescaled, as the size of the boundary grows. The limiting process is described using a self-similar growth-fragmentation process with explicit parameters. To this end, we introduce a branching peeling exploration of Boltzmann triangulations, which allows us to identify a crucial martingale involving the perimeters of cycles at given heights. We also use a recent result concerning self-similar scaling limits of Markov chains on the nonnegative integers. A motivation for this work is to give a new construction of the Brownian map from a growth-fragmentation process.