SCALING LIMIT OF DYNAMICAL PERCOLATION ON CRITICAL ERDOS-RENYI RANDOM GRAPHS

• 成果类型: Article
• 署名作者: Rossignol, Raphael
• 署名单位: Centre National de la Recherche Scientifique (CNRS); Communaute Universite Grenoble Alpes; Universite Grenoble Alpes (UGA)
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/20-AOP1472
• 发表日期: 2021
• 页码: 322-399
• 关键词: multiplicative coalescent; trees

摘要

Consider a critical Erdos-Renyi random graph: n is the number of vertices, each one of the ((n)(2)) possible edges is kept in the graph independently from the others with probability n(-1) + lambda n (-4/3), lambda being a fixed real number. When n goes to infinity, Addario-Berry, Broutin and Goldschmidt (Probab. Theory Related Fields 152 (2012) 367-406) have shown that the collection of connected components, viewed as suitably normalized measured compact metric spaces, converges in distribution to a continuous limit G(lambda) made of random real graphs. In this paper, we consider notably the dynamical percolation on critical Erdos-Renyi random graphs. To each pair of vertices is attached a Poisson process of intensity n(-1/3), and every time it rings, one resamples the corresponding edge. Under this process, the collection of connected components undergoes coalescence and fragmentation. We prove that this process converges in distribution, as n goes to infinity, toward a fragmentation-coalescence process on the continuous limit G(lambda). We also prove convergence of discrete coalescence and fragmentation processes and provide general Feller-type properties associated to fragmentation and coalescence.