  In this paper we study monotone cellular automata in $d$ dimensions. We
develop a general method for bounding the growth of the infected set when the
initial configuration is chosen randomly, and then use this method to prove a
lower bound on the critical probability for percolation that is sharp up to a
constant factor in the exponent for every 'critical' model. This is one of
three papers that together confirm the Universality Conjecture of Bollob\'as,
Duminil-Copin, Morris and Smith.
