The Critical Contact Process Dies Out

Abstract

By producing a finite-box criterion for survival in a slab, we show that the critical contact process dies out and that, in two and more dimensions, the critical parameter value coincides with the limit of slab thresholds. Using the techniques developed in this paper, one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates. Our results apply also to the discrete-time analogue of the contact process, viz., oriented percolation.