A first order phase transition in the threshold θ≥2θ≥2 contact process on random rr-regular graphs and rr-trees

We consider the discrete time threshold-θθ contact process on a random rr-regular graph. We show that if θ≥2θ≥2, r≥θ+2r≥θ+2, ϵ1ϵ1 is small and p≥p1(ϵ1)p≥p1(ϵ1), then starting from all vertices occupied the fraction of occupied vertices is ≥1−2ϵ1≥1−2ϵ1 up to time exp(γ1(r)n)exp(γ1(r)n) with high probability. We also show that for p2<1p2<1 there is an ϵ2(p2)>0ϵ2(p2)>0 so that if p≤p2p≤p2 and the initial density is ≤ϵ2(p2)≤ϵ2(p2), then the process dies out in time O(logn)O(logn). These results imply that the process on the rr-tree has a first-order phase transition.