Absence of percolation for infinite Poissonian systems of stopped paths

• 成果类型: Article
• 署名作者: Coupier, David; Dereudre, David; Gouere, Jean-Baptiste
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Universite de Lille; CNRS - National Institute for Mathematical Sciences (INSMI); Centre National de la Recherche Scientifique (CNRS); Universite de Tours
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-025-01453-x
• 发表日期: 2026-08
• 页码: 1559-1624
• 关键词: Germ-grain model; point process; Stochastic geometry; brownian path; EXISTENCE; models

摘要

The state space of our model is the Euclidean space in dimension d=2. Simultaneously, from all points of a homogeneous Poisson point process, we let grow independent and identically distributed random continuum paths. Each path stops growing at time t>0 if it hits the trace of the other curves realized up until timet. Such a dynamic is well-defined as long as the distribution of paths has a finite second moment at each time t>0. Letting the time run until infinity so that each path reaches its stopping curve, we study the connected property of the graph formed by all stopped curves. Our main result states the absence of percolation in this graph, meaning that each cluster consists of a finite number of curves. The assumptions on the distribution of paths are very mild, with the main one being the so-called 'loop assumption' which ensures that finite clusters (necessarily containing a loop) occur with positive prob-ability. The main issue in this model comes from the long-range dependence arising from long sequences of causalities in the hitting/stopping procedure. Most methods based on block approaches fail to effectively address the question of percolation in this setting.