Voter-model duality (${\xi }_t(x)={\xi }_0(X_t^x)$)

Independent Poisson process copying arrows construct the voter model. A backward ancestral line from a site follows a continuous-time random walk; lines that meet then coincide. Thus a finite collection of Ising spins at time $t$ equals initial Ising spins at the positions of coalescing random walks. In particular $E_{\eta }{\xi }_t(x)={\sum }_z{p}_t(x,z)\eta (z)$, and the probability of disagreement at two sites is bounded by the probability their ancestral walks have not yet coalesced.

If the initial one-spin set $F$ is finite on the square lattice, voter-model duality gives $P({\xi }_t(x)=1)={\sum }_{z\in F}{p}_t(x,z)\to 0$. The transition probabilities tend to zero on this infinite lattice; their Fourier transform also proves the limit of a sequence by dominated convergence. A union bound on any finite set of observation sites then proves weak convergence of probability measures to the all-zero point mass in the product topology.

In a nearest-neighbor voter model on the square lattice, any two ancestral coalescing random walks meet almost surely. For an invariant probability measure of a Markov process, the disagreement probability is constant in time but bounded by the vanishing probability of noncoalescence. Hence every pair of sites agrees almost surely. Countability concentrates the measure on the two constant configurations; both are absorbing state, so every mixture of their Dirac measures is invariant. Translation invariance need not be assumed.