Percolation in high dimensions

Let pc(d) be the critical probability for percolation in Z d. In this paper it is shown that limd→ ∞ 2d pc(d) = 1. The proof uses the prop-erties of a random subgraph of an m-ary d-dimensional cube. If each edge in this cube is included with probability>, then for 1 2d(1−3/m) large d, the cube will have a connected component of size cm d for some c> 0. This generalizes a result of Ajtai, Komlós and Szemerédi