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

We study majority dynamics on the binomial random graph G(n,p) with p = d/n and d > c n^{1/2}, for some large c>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 neighbours, 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, Chan, O'Donnel, Tamuz and Tan.