The Evolution of Random Subgraphs of the Cube


be a random Qn”‐process, that is let Q0 be the empty spanning subgraph of the cube Qn 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 Qn 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 + ϵ) N/2 where ϵ is independent of n. Then all the components of a typical Qt have o(N) vertices if ϵ < 0, while if ϵ > 0 then, as proved by Ajtai, Komlós, and Szemerédi, a typical Qt has a “giant” component with at least α(ϵ)N vertices, where α(ϵ) > 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 nβ(η) vertices, where β(η) > 0. More importantly, if 60(log n)3/n ⩽ ϵ = ϵn = o(1), then the largest component of a typical Qt has about 2ϵN vertices, while the second largest component has order O(nϵ−2). Loosely put, the evolution of a typical Qn process is such that shortly after time N/2 the appearance of each new edge results in the giant component acquiring 4 new vertices. Copyright © 1992 Wiley Periodicals, Inc., A Wiley Company

Publication Title

Random Structures & Algorithms