The hull process of the Brownian plane

• 成果类型: Article
• 署名作者: Curien, Nicolas; Le Gall, Jean-Francois
• 署名单位: Universite Paris Saclay
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-015-0652-6
• 发表日期: 2016
• 页码: 187-231
• 关键词: scaling limits; trees; snake; MAPS

摘要

We study the random metric space called the Brownian plane, which is closely related to the Brownian map and is conjectured to be the universal scaling limit of many discrete random lattices such as the uniform infinite planar triangulation. We obtain a number of explicit distributions for the Brownian plane. In particular, we consider, for every , the hull of radius r, which is obtained by filling in the holes in the ball of radius r centered at the root. We introduce a quantity which is interpreted as the (generalized) length of the boundary of the hull of radius r. We identify the law of the process as the time-reversal of a continuous-state branching process starting from at time and conditioned to hit 0 at time 0, and we give an explicit description of the process of hull volumes given the process . We obtain an explicit formula for the Laplace transform of the volume of the hull of radius r, and we also determine the conditional distribution of this volume given the length of the boundary. Our proofs involve certain new formulas for super-Brownian motion and the Brownian snake in dimension one, which are of independent interest.