Resolution of a conjecture on majority dynamics: rapid stabilisation in dense random graphs

We study majority dynamics on the binomial random graphG(n, p) withp = d/nandd>lambda n1/2, for some large lambda>0. 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.