Generalized criteria for uniqueness of Gibbs measures

We study the uniqueness of Gibbs measures constructed on infinite graphs, in which the vertexes admit certain markov properties related to the connectivity properties of the graphs. Inspired by Weitz and Winklerpsilas work, we generalize Dobrushinpsilas influence of site to site via sites to a more general form of block to block via blocks by path coupling technique. Our condition for uniqueness of Gibbs measure, is a mathematical statement of the spatial correlation decay conjecture.