Majority Dynamics on Resampled Sparse Erd\H{o}s--R\'enyi Graphs: Gaussian Winner Selection and Pace to Unanimity

Abstract

We study the two-opinion majority dynamics process: at each time step, every vertex adopts the majority opinion among its neighbors, retaining its current opinion if there is a tie. Independently at each step, the interaction graph is resampled from the sparse Erd\H{o}s--R\'enyi model $\mathbb G(N,p)$ with $p=b\log N/N$ and fixed $b>1$. Our results identify three regimes governed by the initial advantage $\Delta_0=|B_0|-|R_0|$, where $|B_0|$ and $|R_0|$ denote the initial blue and red camps, respectively. First, an initial blue advantage above an explicit constant multiple of $N/\sqrt{\log N}$ leads to blue unanimity within two updates with high probability. Second, throughout the intermediate regime $\sqrt{N/\log N}\ll\Delta_0\lesssim N/\sqrt{\log N}$, we obtain explicit high-probability upper and lower bounds on the blue-unanimity time. Finally, uniformly in the critical window $\Delta_0\sqrt p=O(1)$, the blue- and red-unanimity probabilities equal $\Phi(\sqrt{2/\pi}\,\Delta_0\sqrt p)+o(1)$ and $\Phi(-\sqrt{2/\pi}\,\Delta_0\sqrt p)+o(1)$, respectively, and unanimity is reached within $(1+o(1))\log N/\log\log N$ many updates with high probability. This resolves the resampled version of the \emph{optimal power-of-few} conjecture raised by Tran and Vu (2025).