Critical conditions for the coverage of complete graphs with the frog model

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∈Vn 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∈Vn∖{o}, where η is a random variable taking values in {0,1,2,...} and {ηv}v∈Vn are i.i.d. random variables with the same law as η. 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−α logn, then V∞(Kn,pn)=n with high probability whenever 0<α<E(η) and V∞(Kn,pn)<n with high probability whenever α>E(η).