CONDENSATION IN CRITICAL CAUCHY BIENAYME-GALTON-WATSON TREES

• 成果类型: Article
• 署名作者: Kortchemski, Igor; Richier, Loic
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Institut Polytechnique de Paris; Ecole Polytechnique
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/18-AAP1447
• 发表日期: 2019
• 页码: 1837-1877
• 关键词: random planar maps; SCALING LIMITS; THEOREMS

摘要

We are interested in the structure of large Bienayme-Galton-Watson random trees whose offspring distribution is critical and falls within the domain of attraction of a stable law of index alpha = 1. In stark contrast to the case alpha is an element of (1, 2], we show that a condensation phenomenon occurs: in such trees, one vertex with macroscopic degree emerges (see Figure 1). To this end, we establish limit theorems for centered downwards skip-free random walks whose steps are in the domain of attraction of a Cauchy distribution, when conditioned on a late entrance in the negative real line. These results are of independent interest. As an application, we study the geometry of the boundary of random planar maps in a specific regime (called nongeneric of parameter 3/2). This supports the conjecture that faces in Le Gall and Miermont's 3/2-stable maps are self-avoiding.