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

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