Coarsening Dynamics on Z^d with Frozen Vertices

We study Markov processes in which ±1-valued random variables σ_x(t),x∈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.