Martingales in self-similar growth-fragmentations and their connections with random planar maps

• 成果类型: Article
• 署名作者: Bertoin, Jean; Budd, Timothy; Curien, Nicolas; Kortchemski, Igor
• 署名单位: University of Zurich; University of Copenhagen; Niels Bohr Institute; CEA; Centre National de la Recherche Scientifique (CNRS); Universite Paris Saclay; Universite Paris Saclay; Centre National de la Recherche Scientifique (CNRS); Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0818-5
• 发表日期: 2018
• 页码: 663-724
• 关键词: exponential functionals; SCALING LIMITS; CONVERGENCE

摘要

The purpose of the present work is twofold. First, we develop the theory of general self-similar growth-fragmentation processes by focusing on martingales which appear naturally in this setting and by recasting classical results for branching random walks in this framework. In particular, we establish many-to-one formulas for growth-fragmentations and define the notion of intrinsic area of a growth-fragmentation. Second, we identify a distinguished family of growth-fragmentations closely related to stable Levy processes, which are then shown to arise as the scaling limit of the perimeter process in Markovian explorations of certain random planar maps with large degrees (which are, roughly speaking, the dual maps of the stable maps of Le Gall and Miermont in Ann Probab 39:1-69, 2011). As a consequence of this result, we are able to identify the law of the intrinsic area of these distinguished growth-fragmentations. This generalizes a geometric connection between large Boltzmann triangulations and a certain growth-fragmentation process, which was established in Bertoin et al. (Ann Probab, accepted).