Coarsening with a frozen vertex

Abstract

In the standard nearest-neighbor coarsening model with state space $\{-1,+1\}^{\mathbb{Z}^2}$ and initial state chosen from symmetric product measure, it is known (see~\cite{NNS}) that almost surely, every vertex flips infinitely often. In this paper, we study the modified model in which a single vertex is frozen to $+1$ for all time, and show that every other site still flips infinitely often. The proof combines stochastic domination (attractivity) and influence propagation arguments.