The random cluster model on a general graph and a phase transition characterization of nonamenability
The paper describes the connection between the amenability of a graph and the properties (namely the phase transitions) of random cluster models. A graph is set to be amenable if the ``boundary to volume ratio'' tends to zero for some sequence of subsets of the graph. Otherwise, i.e. if the ratios are bounded from below by a positive constant, it is called nonamenable. Let \(G=(V,E)\) be an infinite graph. The random field \((X(s))_{s\in E}\) is said to be an (infinity-wired) random cluster model with parameters \(p\) and \(q\) if \[ P(X(S) = (\eta \mid X(E\setminus S) = \xi) = {1\over U_{S,\xi}^{p,q}} \left (\prod _{e\in S} p^{\eta (e)}(1-p)^{1-\eta (e)} \right)q^{\ell (\eta ,\xi)} \] for all finite \(S\subset E,\;\eta \in \{0,1\}^S\) and a.e. \(\xi \in \{0,1\}^{E\setminus S}\), where \(U_{S,\xi}^{p,q}\) is the proper normalizing constant and \(\ell (\eta ,\xi)\) is the number of finite connected components in the configuration given by \((\eta ,\xi)\) on \(E\) that intersect \(V(S) = \{v\in V;\exists e\in S \) such that \(e\) is incident to \(v\}\). Thus the definition agrees with the general concept of Gibbs random fields and the phenomenon of phase transitions occurs if there are more measures on \(\{0,1\}^E\) with the above defined finite conditional marginals. The main result f the paper states that (under some regularity conditions) the nonamenability off the graph is equivalent to the existence of phase transitions for \(q\) large enough and \(p\) belonging to a whole closed interval.
scientific article; zbMATH DE number 1554289