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

with Shirshendu Chatterjee

Abstract. 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 ≥ p_1(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>0 so that if p ≤ p2 and the initial density is ≤ ε2(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.

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