Random Subgraphs Of Finite Graphs: III. The Phase Transition For The n-Cube

We study random subgraphs of the n-cube {0, 1}n, where nearest-neighbor edges are occupied with probability p. Let pc(n) be the value of p for which the expected cluster size of a fixed vertex attains the value λ2n/3, where λ is a small positive constant. In two previous papers, we developed a general theory of random subgraphs of high-dimensional finite connected transitive graphs. In this paper, we summarize the results of the theory as it applies to give detailed estimates concerning the phase transition on the n-cube at pc(n), and prove that for p−pc(n) ≥ e−cn 1/3 the size of the largest cluster is Θ([p−pc(n)]n2). The proof is based on the method of “sprinkling” and relies heavily on the specific geometry of the n-cube.