The Evaluation of Random Subgraphs of the Cube

Let (Qt) M 0 be a random Q -process, that is let Q0 be the empty spanning subgraph of the cube Q n and, for 1 ≤ t ≤ M = nN/2 = n2n−1, let the graph Qt be obtained from Qt−1 by the random addition of an edge of Q not present in Qt−1. When t is about N/2, a typical Qt undergoes a certain ‘phase transition’: the component structure changes in a sudden and surprising way. Let t = (1 + )N/2 where is independent of n. Then all the components of a typical Qt have o(N) vertices if < 0, while if > 0 then, as proved by Ajtai, Komlós and Szemerédi, a typical Qt has a ‘giant’ component with at least α( )N vertices, where α( ) > 0. In this note we give essentially best possible results concerning the emergence of this giant component close to the time of phase transition. Our results imply that if η > 0 is fixed and t ≤ (1 − n−η)N/2 then all components of a typical Qt have at most n vertices, where β(η) > 0. More importantly, if 60(logn)/n ≤ = (n) = o(1) then the largest component of a typical Qt has about 2 N vertices, while the second largest component has order O(n −2). Loosely put, the evolution of a typical Q-process is such that shortly after time N/2 the appearance of each new edge results in the giant component acquiring 4 new vertices.