The voter model is a paradigm for opinion dynamics and coarsening. Each site holds opinion ±1. At each step, a random site copies a random neighbor's opinion. Despite this simplicity, it has rich behavior:
In 1D: density of interfaces ρ(t) ~ t^{−1/2} (diffusive coarsening). In 2D: ρ(t) ~ 1/ln(t) — logarithmically slow (annihilating random walks of domain walls). On a complete graph: consensus in O(N) steps, fluctuation-driven. The noisy voter (ε > 0) adds spontaneous flips, creating a non-equilibrium steady state with finite interface density. In d ≤ 2, the voter model is critical — both opinions coexist at all scales without a phase transition. This connects to coalescing Brownian motions (dual process) and SLE₂ curves at the boundary.