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