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Abstract We study monotone cellular automata (also known as ‐bootstrap percolation) in with random initial configurations. Confirming a conjecture of Balister, Bollobás, Przykucki and Smith, who proved the corresponding result in two dimensions, we show that the critical probability is non‐zero for all subcritical models.
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We consider a dynamical process on a graphG, in which vertices are infected (randomly) at a rate which depends on the number of their neighbors that are already infected. This model includes bootstrap percolation and first‐passage percolation as its extreme points. We give a precise description of the evolution of this process on the graph, significantly sharpening results of Dehghanpour and Schonmann. In particular, we determine the typical infection time up to a constant factor for almost all natural values of the parameters, and in a large range we obtain a stronger, sharp threshold.more » « less
