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, 1 is small and p ≥ p1( 1), then starting from all vertices occupied the fraction of occupied vertices is ≥ 1 − 2 1 up to time exp(γ1(r)n) with high probability. We also show that for p2 < 1 there is an 2(p2) > 0 so that if p ≤ p2 and the initial density is ≤ 2(p2)n, 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. AMS 2010 subject classifications: Primary 60K35; secondary 05C80.