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

Abstract

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.