Majority dynamics on sparse random graphs

Majority dynamics on a graph $G$ is a deterministic process such that every vertex updates its $\pm 1$-assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnel, Tamuz and Tan conjectured that, in the Erdős--Rényi random graph $G(n,p)$, the random initial $\pm 1$-assignment converges to a $99\%$-agreement with high probability whenever $p=ω(1/n)$. This conjecture was first confirmed for $p\geqλn^{-1/2}$ for a large constant $λ$ by Fountoulakis, Kang and Makai. Although this result has been reproved recently by Tran and Vu and by Berkowitz and Devlin, it was unknown whether the conjecture holds for $p< λn^{-1/2}$. We break this $Ω(n^{-1/2})$-barrier by proving the conjecture for sparser random graphs $G(n,p)$, where $λ' n^{-3/5}\log n \leq p \leq λn^{-1/2}$ with a large constant $λ'>0$.