THE WINNER TAKES IT ALL

• 成果类型: Article
• 署名作者: Deijfen, Maria; van der Hofstad, Remco
• 署名单位: Stockholm University; Eindhoven University of Technology
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/15-AAP1151
• 发表日期: 2016
• 页码: 2419-2453
• 关键词: 1st passage percolation; competing spatial growth; 1st-passage percolation; giant component; degree sequence; Random graphs; coexistence; models

摘要

We study competing first passage percolation on graphs generated by the configuration model. At time 0, vertex 1 and vertex 2 are infected with the type 1 and the type 2 infection, respectively, and an uninfected vertex then becomes type 1 (2) infected at rate lambda(1) (lambda(2)) times the number of edges connecting it to a type 1 (2) infected neighbor. Our main result is that, if the degree distribution is a power-law with exponent tau is an element of (2, 3), then as the number of vertices tends to infinity and with high probability, one of the infection types will occupy all but a finite number of vertices. Furthermore, which one of the infections wins is random and both infections have a positive probability of winning regardless of the values of lambda(1) and lambda(2). The picture is similar with multiple starting points for the infections.