Let pc(d) be the critical probability for percolation in Zd. It is shown that limd→∞ 2dpc(d) = 1. The proof uses the properties of a random subgraph of an m-ary d-dimensional cube. If each edge in this cube is included with probability greater than 1/2d(1−3/m), then, for large d, the cube will have a connected component of size cmd for some c > 0. This generalizes a result of Ajtai, Komlós and Szemerédi.
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Daniel M. Gordon (1991) studied this question.