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 $\mathcal{K}_n=(\mathcal{V}_n,\mathcal{E}_n)$ be the complete graph with $n$ vertices and $o\in\mathcal{V}_n$ be a special vertex called the root. Initially, $1+\eta_o$ active particles are placed at the root and $\eta_v$ inactive particles are placed at each other vertex $v\in\mathcal{V}_n\setminus\{o\}$, where $\{\eta_v\}_{v\in \mathcal{V}_n}$ 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 $\mathcal{K}_n$ until the moment it dies, activating all inactive particles it hits along its path. Let $V_\infty(\mathcal{K}_n,p)$ 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 $V_\infty(\mathcal{K}_n,p_n)\geq (1-\epsilon)n$ with high probability for any fixed $\epsilon>0$ whenever $p_n\rightarrow 1$. Furthermore, we establish the critical growth rate of $p_n$ so that all vertices are visited. Specifically, we show that if $p_n=1-\frac{\alpha}{\log n}$, then $V_\infty(\mathcal{K}_n,p_n)=n$ with high probability whenever $0<\alpha<E(\eta)$ and $V_\infty(\mathcal{K}_n,p_n)<n$ with high probability whenever $\alpha>E(\eta)$.