Evolution of the n-cube

Let Cⁿ denote the graph with vertices (ε{lunate}₁,...,ε{lunate}_n), ε{lunate}_i = 0, 1 and vertices adjacent if they differ in exactly one coordinate. We call Cⁿ the n-cube. Let G = G_n,p denote the random subgraph of Cⁿ defined by letting Prob({i,j} ∈ G) = p for all i, j ∈ Cⁿ and letting these probabilities be mutually independent. We wish to understand the "evolution" of G as a function of p. Section 1 consists of speculations, without proofs, involving this evolution. Set f{hook}_n = Prof(G_n,p is connected) We show in Section 2: Theorem Lim n f{hook}_n(p) = 0 if p<0.5e^-1 if p=0.51 if p>0.5. The first and last parts were shown by Yu. Burtin[1]. For completeness, we show all three parts.