Compact Brownian surfaces I: Brownian disks

• 成果类型: Article
• 署名作者: Bettinelli, Jeremie; Miermont, Gregory
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Institut Polytechnique de Paris; Ecole Polytechnique; Institut Polytechnique de Paris; Ecole Polytechnique; Ecole Normale Superieure de Lyon (ENS de LYON); Institut Universitaire de France
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-016-0752-y
• 发表日期: 2017
• 页码: 555-614
• 关键词: bipartite planar maps; SCALING LIMITS; INVARIANCE-PRINCIPLES; quadrangulations; geodesics

摘要

We show that, under certain natural assumptions, large random plane bipartite maps with a boundary converge after rescaling to a one-parameter family of random metric spaces homeomorphic to the closed unit disk of , the space being called the Brownian disk of perimeter L and unit area. These results can be seen as an extension of the convergence of uniform plane quadrangulations to the Brownian map, which intuitively corresponds to the limit case where . Similar results are obtained for maps following a Boltzmann distribution, in which the perimeter is fixed but the area is random.