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.

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