We study the threshold θ contact process on ℤd with infection parameter λ. We show that the critical point λc, defined as the threshold for survival starting from every site occupied, vanishes as d→∞. This implies that the threshold θ voter model on ℤd has a nondegenerate extremal invariant measure, when d is large.
"The survival of large dimensional threshold contact processes." Ann. Probab. 37 (4) 1483 - 1501, July 2009. https://doi.org/10.1214/08-AOP440