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.

13 pages, 3 figures