Coarsening dynamics on $\mathbb{Z}^d$ with frozen vertices
  We study Markov processes in which $\pm 1$-valued random variables $\sigma_x(t), x\in \mathbb{Z}^d$, 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 $\rho^+$ (resp., $\rho^-$), we study the prevalence of vertices that are (eventually) fixed plus or fixed minus or flippers (changing forever). Our main results are that, for $\rho^+ >0$ and $\rho^- =0$, all sites are fixed plus, while for $\rho^+ >0$ and $\rho^-$ very small (compared to $\rho^+$), the fixed minus and flippers together do not percolate. We also obtain some results for deterministic placement of frozen vertices. 
Michael Damron, Sinziana M. Eckner, Hana Kogan, Charles M. Newman,   Vladas Sidoravicius
math.PR