= Finite-seed local extinction in the voter model
If the initial one-spin set $F$ is finite on the <square lattice>, <voter-model duality> gives $P(\xi_t(x)=1)=\sum_{z\in F}p_t(x,z)\to0$. The <transition probabilities> tend to zero on this infinite <square lattice>; their <Fourier transform> also proves this decay 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>.
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