Coarsening dynamics on $\mathbb{Z}^d$ with frozen vertices

We study Markov processes in which $\pm 1$-valued random variables $σ_x(t), x\in \mathbb{Z}^d$, update by taking the value of a majority of their nearest neighbors or else tossing a fair coin in case of a tie. In the presence of a random environment of frozen plus (resp., minus) vertices with density $ρ^+$ (resp., $ρ^-$), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for $ρ^+ >0$ and $ρ^- =0$, all sites are fixed plus, while for $ρ^+ >0$ and $ρ^-$ very small (compared to $ρ^+$), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices.

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