A simple proof of the stability criterion of Gray and Griffeath

Gray and Griffeath studied attractive nearest neighbor spin systems on the integers having "all 0's" and "all 1's" as traps. Using the contour method, they established a necessary and sufficient condition for the stability of the "all 1's" equilibrium under small perturbations. In this paper we use a renormalized site construction to give a much simpler proof. Our new approach can be used in many situations as a substitute for the contour method.