Gibbs States

Brook's theorem states that a positive probability measure on a finite product may be decomposed into factors indexed by the cliques of its dependency graph. Closely related to this is the well known fact that a positive measure is a spatial Markov field on a graph G if and only if it is a Gibbs state. The Ising and Potts models are introduced, and the n-vector model is mentioned. ⊥ is thus symmetric, and it gives rise to a graph G with vertex set V and edge-set E = {i, j : i ⊥ j }, called the dependency graph of X (or of its law). We shall see that the law of X may be expressed as a product over terms corresponding to complete subgraphs of G. A complete subgraph of G is called a clique, and we write K for the set of all cliques of G. For notational simplicity later, we designate the empty subset of V to be a clique, and thus ∅ ∈ K. A clique is maximal if no strict superset is a clique, and we write M for the set of maximal cliques of G. We assume for simplicity that the X i take values in some countable subset S of the reals R. The law of X gives rise to a probability mass function π on S n given by It is easily seen by the definition of independence that i ⊥ j if and only if π may be factorized in the form π(x)