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Journal of the London Mathematical Society 1991 s2-44(2):373-384; doi:10.1112/jlms/s2-44.2.373
© 1991 by London Mathematical Society
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© Oxford University Press

Percolation in High Dimensions

Daniel M. Gordon

Department of Computer Science, University of Georgia Athens, Georgia 30602, USA

Let pc(d) be the critical probability for percolation in Zd. It is shown that limd->{infty} 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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