random-cluster model is a generalisation of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn (see [29]). Not only is the random-cluster model a worthwhile topic for study in its own right, but also it provides much information about phase transitions in the associated physical models. This paper serves two functions. First, we introduce and survey random-cluster measures from the probabilist’s point of view, giving clear statements of some of the many open problems. Secondly, we present new results for such measures, as follows. We discuss the relationship between weak limits of random-cluster measures and measures satisfying a suitable DLR condition. Using an argument based on the convexity of pressure, we prove the uniqueness of random-cluster measures for all but (at most) countably many values of the parameter p. Related results concerning phase transition in two or more dimensions are included, together with various stimulating conjectures. The uniqueness of the infinite cluster is employed in an intrinsic way, in part of these arguments. In the second part of this paper is constructed a Markov process whose level-sets are reversible Markov processes with random-cluster measures as unique equilibrium measures. This construction enables a coupling of random-cluster measures for all values of p. Furthermore it leads to a proof of the semicontinuity of the percolation probability, and provides a heuristic probabilistic justification for the widely held belief that there is a first-order phase transition if and only if the cluster-weighting factor q is sufficiently large. 1.

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Abstract. The class of random-cluster models is a unification of a variety of stochastic processes of significance for probability and statistical physics, including percolation, Ising, and Potts models; in addition, their study has impact on the theory of certain random combinatorial structures, and of electrical networks. Much (but not all) of the physical theory of Ising/Potts models is best implemented in the context of the random-cluster representation. This systematic summary of random-cluster models includes accounts of the fundamental methods and inequalities, the uniqueness and specification of infinite-volume measures, the existence and nature of the phase transition, and the structure of the subcritical and supercritical phases. The theory for two-dimensional lattices is better developed than for three and more dimensions. There is a rich collection of open problems, including some of substantial significance for the general area of disordered systems, and these are highlighted when encountered. Amongst the major open questions, there is the problem of ascertaining the exact nature of the phase transition for general values of the cluster-weighting factor q, and the problem of proving that the critical random-cluster model in two

Abstract. We consider the covariance matrix G mn (x − y) = 〈qδ(σx, m) qδ(σy, n) 〉 − 〈qδ(σx, m) 〉 〈qδ(σy, n)〉 wir of the d-dimensional q-states Potts model, rewriting it in terms of the connectivity, the finite-cluster connectivity and the infinite-cluster covariance in the random cluster representation of Fortuin and Kasteleyn. In any of the q ordered phases, we show that – in addition to the trivial eigenvalue 0 – the matrix Gmn (x − y) has one simple eigenvalue G (1) (x − y) and one (q − 2)-fold degenerate eigenvalue G(2) (x − y). Furthermore, we identify the eigenvalues both in terms of representations of the unbroken symmetry group of the model, and in terms of connectivities and cluster covariances, thereby attributing algebraic significance to these stochastic geometric quantities. In addition to establishing the existence of the correlation lengths ξ (1) wir and ξ(2)

. Consider a translation-invariant bond percolation model on the integer lattice which has exponential decay of connectivities, that is, the probability of a connection 0 $ x by a path of open bonds decreases like e \Gammam(`)jxj for some positive constant m(`) which may depend on the direction ` = x=jxj. In two and three dimensions, it is shown that if the model has an appropriate mixing property and satisfies a special case of the FKG property, then there is at most a power-law correction to the exponential decay---there exist A and C such that e \Gammam(`)jxj P (0 $ x) Ajxj \GammaC e \Gammam(`)jxj for all nonzero x. In four or more dimensions, a similar bound holds with jxj \GammaC replaced by e \GammaC(log jxj) 2 . In particular the power-law lower bound holds for the Fortuin-Kasteleyn random cluster model in two dimensions whenever the connectivity decays exponentially, since the mixing property is known to hold in that case. Consequently a similar bound holds for ...

An infinite-volume mixing or exponential-decay property in a spin system or percolation model reflects the inability of the influence of the conguration in one region to propagate to distant regions, but in some circumstances where such properties hold, propagation can nonetheless occur in finite volumes endowed with boundary conditions. We establish the absense of such propagation, particularly in two dimensions in finite volumes which are simply connected, under a variety of conditions, mainly for the Potts model and the Fortuin-Kasteleyn (FK) random cluster model, allowing external fields. For example, for the FK model in two dimensions we show that exponential decay of connectivity in infinite volume implies exponential decay in simply connected finite volumes, uniformly over all such volumes and all boundary conditions, and implies a strong mixing property for such volumes with certain types of boundary conditions. For the Potts model in two dimensions we show that exponential decay of correlations in infinite volume implies a strong mixing property in simply connected finite volumes, which includes exponential decay of correlations in simply connected finite volumes, uniformly over all such volumes and all boundary conditions.

Abstract. For a family of bond percolation models on Z 2 that includes the Fortuin-Kasteleyn random cluster model, we consider properties of the “droplet ” that results, in the percolating regime, from conditioning on the existence of an open dual circuit surrounding the origin and enclosing at least (or exactly) a given large area A. This droplet is a close surrogate for the one obtained by Dobrushin, Koteck´y and Shlosman by conditioning the Ising model; it approximates an area-A Wulff shape. The local part of the deviation from the Wulff shape (the “local roughness”) is the inward deviation of the droplet boundary from the boundary of its own convex hull; the remaining part of the deviation, that of the convex hull of the droplet from the Wulff shape, is inherently long-range. We show that the local roughness is described by at most the exponent 1/3 predicted by nonrigorous theory; this same prediction has been made for a wide class of interfaces in two dimensions. Specifically, the average of the local roughness over the droplet surface is shown to be O(l 1/3 (log l) 2/3) in probability, where l = √ A is the linear scale of the droplet. We also bound the maximum of the local roughness over the droplet surface and bound the long-range part of the deviation from a Wulff shape, and we establish the absense of “bottlenecks,” which are a form of self-approach by the droplet boundary, down to scale log l. Finally, if we condition instead on the event that the total area of all large droplets inside a finite box exceeds A, we show that with probability near 1 for large A, only a single large droplet is present. 1.

. We introduce the asymmetric random cluster (or ARC) model, which is a graphical representation of the Potts lattice gas, and establish its basic properties. The ARC model allows a rich variety of comparisons (in the FKG sense) between models with different parameter values; we give, for example, values (fi; h) for which the 0's configuration in the Potts lattice gas is dominated by the "+" configuration of the (fi; h) Ising model. The Potts model, with possibly an external field applied to one of the spins, is a special case of the Potts lattice gas, which allows our comparisons to yield rigorous bounds on the critical temperatures of Potts models. For example, we obtain :571 1 \Gamma exp(\Gammafi c ) :600 for the 9-state Potts model on the hexagonal lattice. Another comparison bounds the movement of the critical line when a small Potts interaction is added to a lattice gas which otherwise has only interparticle attraction. ARC models can also be compared to related models such as ...

The fuzzy Potts model is obtained by looking at the Potts model with a pair of glasses that prevents distinguishing between some of the spin values. We show that the fuzzy Potts model on Z (d 2) is Gibbsian at high temeratures and non-Gibbsian at low temperatures.

We sketch elementary results and open problems in the theory of percolation and random-cluster models. The presentation is rather selective, and is intended to stimulate interest rather than to survey the established theory. In the case of the random-cluster model, we include sketch proofs of basic material such as the FKG inequality and the comparison inequalities.