On the triviality of the shocked map

• 成果类型: Article
• 署名作者: Fredes, Luis; Sepulveda, Avelio
• 署名单位: Universite de Bordeaux; Centre National de la Recherche Scientifique (CNRS); CNRS - National Institute for Mathematical Sciences (INSMI); Universidad de Chile
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-025-01369-6
• 发表日期: 2025
• 页码: 431-469
• 关键词: quadrangulations; CONVERGENCE; walk

摘要

The (non-spanning) tree decorated quadrangulation is a random pair consisting of a quadrangulation and a subtree chosen uniformly over the set of pairs with a prescribed size. In this paper, we study the tree decorated quadrangulation in the critical regime: when the number f of faces of the map is proportional to the square of the size of the tree. We show that with high probability in this regime, the diameter of the tree lies between o(f(1/4)) and f(1/4)/log(alpha)(f), for all alpha > 1. Thus after scaling distances by f(-1/4), the critical tree decorated quadrangulation converges to a Brownian disk whose boundary has been identified to a point. These results imply the triviality of the shocked map: the metric space generated by gluing a Brownian disk with a continuous random tree.