Largest random component of a k-cube

Let C ^k denote the graph with vertices (e{open} ₁, ..., e{open} _k ), e{open} _i =0,1 and vertices adjacent if they differ in exactly one coordinate. We call C ^k the k-cube. Let G=G _{k, p} denote the random subgraph of C ^k defined by letting {Mathematical expression} for all i, j ∈ C ^k and letting these probabilities be mutually independent. We show that for p=λ/k, λ>1, G _{k, p} almost surely contains a connected component of size c2 ^k, c=c(λ). It is also true that the second largest component is of size o(2 ^k ).