Voter Model: Coarsening Dynamics

Each site copies a random neighbor's opinion. In 1D, interface density ρ(t) ~ 1/√t — slow coarsening toward consensus. In 2D, domains grow as R(t) ~ √(t/ln t), slower than Ising.

Parameters

Update rule: pick site i at random,
copy spin of neighbor i±1.

Interface density:
ρ(t) = (# sign changes) / N

Theory: ρ(t) ~ (πt)^{−1/2}

1D voter model = annihilating
random walks
of domain walls.

At t → ∞: absorbing consensus
(all +1 or all −1).
t = 0
ρ(t) =
ρ·√t =