Resolution of a conjecture on majority dynamics: Rapid stabilization in dense random graphs

We study majority dynamics on the binomial random graph G(n, p) with p = d/n and (Formula presented.), for some large (Formula presented.). In this process, each vertex has a state in {− 1, + 1} and at each round every vertex adopts the state of the majority of its neighbors, retaining its state in the case of a tie. We show that with high probability the process reaches unanimity in at most four rounds. This confirms a conjecture of Benjamini et al.