2024/07/26 by de Carvalho, Gustavo O., Machado, Fábio P.
#05C81 #60K35 #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2407.19027
We consider a system of interacting random walks known as the frog model. Let Kn=(Vn,En) be the complete graph with n vertices and o\inVn be a special vertex called the root. Initially, 1+ηo active particles are placed at the root and ηv inactive particles are placed at each other vertex v\inVn∖\o\, where \ηv\v∈ Vn are i.i.d. random variables. At each instant of time, each active particle may die with probability 1-p. Every active particle performs a simple random walk on Kn until the moment it dies, activating all inactive particles it hits along its path. Let V_∞(Kn,p) be the total number of visited vertices by some active particle up to the end of the process, after all active particles have died. In this paper, we show that V_∞(Kn,pn)≥ (1-ε)n with high probability for any fixed ε>0 whenever pn→ 1. Furthermore, we establish the critical growth rate of pn so that all vertices are visited. Specifically, we show that if pn=1-\fracαlog n, then V_∞(Kn,pn)=n with high probability whenever 0<αE(η).