The correlation decay (CD) tree and strong spatial mixing in multi-spin systems

This paper deals with the construction of a correlation decay tree (hypertree) for interacting systems modeled using graphs (hypergraphs) that can be used to compute the marginal probability of any vertex of interest. Local message passing equations have been used for some time to approximate the marginal probabilities in graphs but it is known that these equations are incorrect for graphs with loops. In this paper we construct, for any finite graph and a fixed vertex, a finite tree with appropriately defined boundary conditions so that the marginal probability on the tree at the vertex matches that on the graph. For several interacting systems, we show using our approach that if there is very strong spatial mixing on an infinite regular tree, then one has strong spatial mixing for any given graph with maximum degree bounded by that of the regular tree. Thus we identify the regular tree as the worst case graph, in a weak sense, for the notion of strong spatial mixing.