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 d > λ n 1 / 2 , for some large λ > 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.