Majority dynamics on sparse random graphs

Majority dynamics on a graph G$$ G $$ is a deterministic process such that every vertex updates its ±1$$ \pm 1 $$ ‐assignment according to the majority assignment on its neighbor simultaneously at each step. Benjamini, Chan, O'Donnell, Tamuz and Tan conjectured that, in the Erdős–Rényi random graph G(n,p)$$ G\left(n,p\right) $$ , the random initial ±1$$ \pm 1 $$ ‐assignment converges to a 99%$$ 99\% $$ ‐agreement with high probability whenever p=ω(1/n)$$ p=\omega \left(1/n\right) $$ . This conjecture was first confirmed for p≥λn−1/2$$ p\ge \lambda {n}^{-1/2} $$ for a large constant λ$$ \lambda $$ 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 p0$$ {\lambda}^{\prime }>0 $$ .