Existence and sharpness of the phase transition for the frog model on transitive graphs

• 成果类型: Article; Early Access
• 署名作者: Angel, Omer; Riva, Daniel de la; Hermon, Jonathan; Shi, Yuliang
• 署名单位: University of British Columbia
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-026-01517-6
• 发表日期: 2026-08-14
• 关键词: Frog model; sharpness; EXISTENCE; Phase Transition; percolation; random walk; bernoulli percolation; random-walks; polynomial-growth; shape theorem; SPREAD; PROOF; transience; recurrence; driven

摘要

We consider a slight modification of the frog model. For a given graph, each vertex has Poisson(lambda) particles (or frogs). At time zero, only the particles at the origin are active, and all the other particles are sleeping. Each active particle performs an independent, continuous-time simple random walk up to a fixed lifetime t, after which the particle dies and is removed from the system. Once an active frog jumps to a vertex, it activates all of its particles. The survival of active particles can be studied as a dependent percolation model with two parameters lambda and t. In the present work, we establish the existence of a phase transition with respect to each parameter for non-amenable graphs of bounded degree and quasi-transitive graphs of superlinear polynomial growth, as well as prove the sharpness of the phase transition for transitive graphs.