Fractal Failure of Quasilocality for a Majority Rule Transformation on a Tree

We provide a transformation of the Ising model on a Cayley tree leading to non-Gibbsianness at any temperature, i.e. even within the uniqueness regime. We also introduce a new type of pathologies of renormalized Gibbs measures, called the fractal failure of quasilocality, and exhibit a concrete example.     

Variational Principle and Almost Quasilocality for Renormalized Measures

We study the variational principle for some non-Gibbsian measures. We give a necessary and sufficient condition for the validity of the implication zero relative entropy density implies common version of conditional probabilities (so-called second part of the variational principle). Applying this to noisy decimations of the low-temperature phases of the Ising model, we obtain almost sure quasilocality for these measures and the second part of the variational principle. For the projection of low temperature Ising phases on a one-dimensional layer, we also obtain the second part of the variational principle.     

Random-cluster representation of the ashkin-teller model

We show that a class of spin models, containing the Ashkin-Teller model, admits a generalized random-cluster (GRC) representation. Moreover, we show that basic properties of the usual representation, such as FKG inequalities and comparison inequalities, still hold for this generalized random-cluster model. Some elementary consequences are given. We also consider the duality transformations in the spin representation and in the GRC model and show that they commute.     

Reflection Positivity of the Random-Cluster Measure Invalidated for Noninteger q

We consider the random-cluster Potts measure on a lattice torus that weights each connected component by a positive number q. We show, by constructing a counterexample, that this measure is not reflection-positive unless q is integer.     

Weakly Gibbsian Measures and Quasilocality: A Long-Range Pair-Interaction Counterexample

We exhibit an example of a measure on a discrete and finite spin system whose conditional probabilities are given in terms of an almost everywhere absolutely summable potential but are discontinuous almost everywhere.     

Random-Cluster Representation of the Blume–Capel Model

The so-called diluted-random-cluster model may be viewed as a random-cluster representation of the Blume–Capel model. It has three parameters, a vertex parameter a, an edge parameter p, and a cluster weighting factor q. Stochastic comparisons of measures are developed for the ‘vertex marginal’ when q ∊ [1,2], and the ‘edge marginal’ when q ∊ [1,∞). Taken in conjunction with arguments used earlier for the random-cluster model, these permit a rigorous study of part of the phase diagram of the Blume–Capel model. Mathematics Subject Classification (2000): 82B20, 60K35.     

Entanglement in the Quantum Ising Model

We study the asymptotic scaling of the entanglement of a block of spins for the ground state of the one-dimensional quantum Ising model with transverse field. When the field is sufficiently strong, the entanglement grows at most logarithmically in the number of spins. The proof utilises a transformation to a model of classical probability called the continuum random-cluster model, and is based on a property of the latter model termed ratio weak-mixing. In an intermediate result, we establish an exponentially decaying bound on the operator norm of differences of the reduced density operator. Of special interest is the mathematical rigour of this work, and the fact that the proof applies equally to a large class of disordered interactions.     

Rigidity of the Interface in Percolation and Random-Cluster Models

We study conditioned random-cluster measures with edge-parameter p and cluster-weighting factor q satisfying q1. The conditioning corresponds to mixed boundary conditions for a spin model. Interfaces may be defined in the sense of Dobrushin, and these are proved to be rigid in the thermodynamic limit, in three dimensions and for sufficiently large values of p. This implies the existence of non-translation-invariant (conditioned) random-cluster measures in three dimensions. The results are valid in the special case q=1, thus indicating a property of three-dimensional percolation not previously noted.     

Random-Cluster Representation for the Blume–Capel Model

We present in this paper a way to perform the mapping of the spin-1 Blume–Capel model into a random-cluster model, and analyze thermodynamic properties of the former model in terms of geometric properties of clusters generated in the random-cluster representation. It is shown that there are two different relevant types of cluster, and that one of them is the exact analogue of the type of cluster generated in the Ising model. We use this result to derive expressions for thermodynamical properties on the second-order transition line which are equivalent to the ones found in the Ising model. The other type of cluster is responsible for the first-order transitions, and we may see the tricritical point as a point where both types of cluster compete on the same footing.     

Random-Cluster Analysis of a Class of Binary Lattice Gases

We introduce a class of binary lattice gases which can be viewed as a lattice analogue of the continuum Widom–Rowlinson model, and which also is related to the beach model of Burton and Steif. This new model is shown to exhibit phase transition for large particle intensities. Stochastic monotonicity results of varying strength are derived in various parts of the parameter space. The main tool is a random-cluster representation of the model, analogous to the Fortuin–Kasteleyn representation of the Potts model.     

Comparison and disjoint-occurrence inequalities for random-cluster models

A principal technique for studying percolation, (ferromagnetic) Ising, Potts, and random-cluster models is the FKG inequality, which implies certain stochastic comparison inequalities for the associated probability measures. The first result of this paper is a new comparison inequality, proved using an argument developed elsewhere in order to obtain strict inequalities for critical values. As an application of this inequality, we prove that the critical pointp _c (q) of the random-cluster model with cluster-weighting factorq (1) is strictly monotone inq. Our second result is a BK inequality for the disjoint occurrence of increasing events, in a weaker form than that available in percolation theory.     

One-Sided Continuity Properties for the Schonmann Projection

We consider the plus-phase of the two-dimensional Ising model below the critical temperature. In 1989 Schonmann proved that the projection of this measure onto a one-dimensional line is not a Gibbs measure. After many years of continued research which have revealed further properties of this measure, the question whether or not it is a Gibbs measure in an almost sure sense remains open. In this paper we study the same measure by interpreting it as a temporal process. One of our main results is that the Schonmann projection is almost surely a regular g-measure. That is, it does possess the corresponding one-sided notion of almost Gibbsianness. We further deduce strong one-sided mixing properties which are of independent interest. Our proofs make use of classical coupling techniques and some monotonicity properties which are known to hold for one-sided, but not two-sided conditioning for FKG measures.     

Potts models and random-cluster processes with many-body interactions

Known differential inequalities for certain ferromagnetic Potts models with pair interactions may be extended to Potts models with many-body interactions. As a major application of such differential inequalities, we obtain necessary and sufficient conditions on the set of interactions of such a Potts model in order that its critical point be astrictly monotonic function of the strengths of interactions. The method yields some ancillary information concerning the equality of certain critical exponents for Potts models; this amounts to a small amount of rigorous universality. These results are achieved in the context of a Fortuin-Kasteleyn representation of Potts models with many-body interactions. For such a Potts model, the corresponding random-cluster process is a (random) hypergraph.     

Almost sure quasilocality in the random cluster model

We investigate the Gibbsianness of the random cluster measures ^{q, p} and , obtained as the infinite-volume limit of finite-volume measures with free and wired boundary conditions. Forq>1, the measures are not Gibbs measures, but it turns out that the conditional distribution on one edge, given the configuration outside that edge, is almost surely quasilocal.     

Existence of Delaunay Pairwise Gibbs Point Process with Superstable Component

The present stuffy deals with the existence of Delaunay pairwise Gibbs point process with superstable component by using the well-known Preston theorem. In particular, we prove the stability, the lower regularity, and the quasilocality properties of the Delaunay model.     

Opinion evolution analysis for short-range and long-range Deffuant–Weisbuch models

In this paper, we propose and then analyze two generalized Deffuant–Weisbuch (DW) models. The generalized models extend the conventional DW model by taking multiple choices in two different ways. First, we demonstrate the almost sure convergence of the agent opinions for the short-range multi-choice DW dynamics when only the opinions within confidence regions may be count in. Then we analyze dynamical behavior about the long-range multi-choice DW model when some opinions out of the confidence ranges are considered with a weighted combination. Moreover, both theoretical and simulation results show that the dynamical behaviors of the two models are totally different.     

Decay of Correlations in Random-Cluster Models

We prove exponential decay for the tail of the radius R of the cluster at the origin, for subcritical random-cluster models, under an assumption slightly weaker than that (here, d is the number of dimensions). Specifically, if throughout the subcritical phase, then for some α > 0. This implies the exponential decay of the two-point correlation function of subcritical Potts models, subject to a hypothesis of (at least) polynomial decay of this function. Similar results are known already for percolation and Ising models, and for Potts models when the number q of available states is sufficiently large; indeed the hypothesis of polynomial decay has been proved rigorously for these cases. In two dimensions, the hypothesis that is weaker than requiring that the susceptibility be finite, i.e., that the two-point function be summable. The principal new technique is a form of Russo's formula for random-cluster models reported by Bezuidenhout, Grimmett, and Kesten. For the current application, this leads to an analysis of a first-passage problem for random-cluster models, and a proof that the associated time constant is strictly positive if and only if the tail of R decays exponentially. Received: 25 September 1996 / Accepted: 21 February 1997     

Strict inequality for critical values of Potts models and random-cluster processes

We prove that the critical value _c of a ferromagnetic Potts model is astrictly decreasing function of the strengths of interaction of the process. This is achieved in the (more) general context of the random-cluster representation of Fortuin and Kasteleyn, by deriving and utilizing a formula which generalizes the technique known in percolation theory as Russo's formula. As a byproduct of the method, we present a general argument for showing that, at any given point on the critical surface of a multiparameter process, the values of a certain critical exponent do not depend on the direction of approach of that point. Our results apply to all random-cluster processes satisfying the FKG inequality.G.R.G. acknowledges support from Cornell University, and also partial support by the U.S. Army Research Office through the Mathematical Sciences Institute of Cornell University. H.K. was supported in part by the N.S.F. through a grant to Cornell University     

Critical behavior of the Chayes-Machta-Swendsen-Wang dynamics

We study the dynamic critical behavior of the Chayes-Machta dynamics for the Fortuin-Kasteleyn random-cluster model, which generalizes the Swendsen-Wang dynamics for the q-state Potts model to noninteger q, in two and three spatial dimensions, by Monte Carlo simulation. We show that the Li-Sokal bound z >or= alpha/nu is close to but probably not sharp in d = 2 and is far from sharp in d = 3, for all q. The conjecture z >or= beta/nu is false (for some values of q) in both d = 2 and d = 3.