PR ] 2 4 A ug 2 00 0 Explicit isoperimetric constants , phase transitions in the random-cluster and Potts models , and Bernoullicity

The random-cluster model is a dependent percolation model that has applications in the study of Ising and Potts models. In this paper, several new results for the random-cluster model with cluster parameter q ≥ 1 are obtained. These include an explicit pointwise dynamical construction of random-cluster measures for arbitrary graphs, and for unimodular transitive graphs, lack of percolation for the free random-cluster measure at the lower critical value on nonamenable graphs, and a number of inequalities for the critical values. Some of these inequalities lead to considerations of isoperimetric constants in certain hyperbolic graphs, and the first nontrivial explicit calculations of such constants are obtained. Applications to the Potts model include Bernoullicity in the Z case at all temperatures, and non-robust phase transition in the case of nonamenable regular graphs.

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title: On the random-cluster model: I. Introduction and relation to other models
abstract: The random-cluster model is defined as a model for phase transitions and other phenomena in lattice systems, or more generally in systems with a graph structure. The model is characterized by a (probability) measure on a graph and a real parameter κ. By specifying the value of κ to 1, 2, 3, 4, … is shown that the model covers the percolation model, the Ising model, the Ashkin-Teller-Potts model with 3, 4, … states per atom, respectively, and thereby, contains information on graph-colouring problems; in the limit κ ↓ 0 it describes linear resistance networks. It is shown that the function which for the random-cluster model plays the role of a partition function, is a generalization of the dichromatic polynomial earlier introduced by Tutte, and related polynomials.

title: Nonuniversal critical dynamics in Monte Carlo simulations.
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title: Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm.
abstract: We give a simple explanation of the Swendsen-Wang algorithm for Potts models in terms of a joint model of Potts spin variables interacting with bond occupation variables. We then show how to generalize this representation, as well as the corresponding Monte Carlo algorithm, to arbitrary models. We give initial results of tests of the new algorithm on the two-dimensional XY model.

title: Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models
abstract: Results from percolation theory are used to study phase transitions in one-dimensional Ising andq-state Potts models with couplings of the asymptotic formJx,y≈ const/¦x−y¦2. For translation-invariant systems with well-defined limx→∞x2Jx=J+ (possibly 0 or ∞) we establish: (1) There is no long-range order at inverse temperaturesβ withβJ+⩽1. (2) IfβJ+>q, then by sufficiently increasingJ1 the spontaneous magnetizationM is made positive. (3) In models with 0<J+<∞ the magnetization is discontinuous at the transition point (as originally predicted by Thouless), and obeysM(βc)⩾1/(βcJ+)1/2. (4) For Ising (q=2) models withJ+<∞, it is noted that the correlation function decays as 〈σxσy〉(β)≈c(β)/|x−y|2 wheneverβ<βc. Points 1–3 are deduced from previous percolation results by utilizing the Fortuin-Kasteleyn representation, which also yields other results of independent interest relating Potts models with different values ofq.

title: Critical Percolation on any Nonamenable Group has no Infinite Clusters
abstract: We show that independent percolation on any Cayley graph of a nonamenable group has no infinite components at the critical parameter. This result was obtained by the present authors earlier as a corollary of a general study of group-invariant percolation. The goal here is to present a simpler self-contained proof that easily extends to quasi-transitive graphs with a unimodular automorphism group. The key tool is a “mass-transport” method, which is a technique of averaging in nonamenable settings.

title: Exact sampling with coupled Markov chains and applications to statistical mechanics
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title: Amenability and Phase Transition in the Ising Model
abstract: We consider the Ising model with external field h and coupling constant J on an infinite connected graph G with uniformly bounded degree. We prove that if G is nonamenable, then the Ising model exhibits phase transition for some h≠0 and some J<∞. On the other hand, if G is amenable and quasi-transitive, then phase transition cannot occur for h≠0. In particular, a group is nonamenable if and only if the Ising model on one (all) of its Cayley graphs exhibits a phase transition for some h≠0 and some J<∞.

title: The random cluster model on a general graph and a phase transition characterization of nonamenability
abstract: The random cluster model on a general infinite graph with bounded degree wired at infinity is considered and a "ghost vertex" method is introduced in order to explicitly construct random cluster measures satisfying the Dobrushin-Lanford-Ruelle condition for q[greater-or-equal, slanted]1. It is proved that on a regular nonamenable graph there is a q0 such that for q[greater-or-equal, slanted]q0 there is a phase transition for an entire interval of values of p, whereas on a quasi-transitive amenable graph there is a phase transition for at most a countable number of values of p. In particular, a transitive graph is nonamenable if and only if there is a phase transition for an entire interval of p-values for large enough q. It is also observed that these results have a Potts model interpretation. In particular, a transitive graph is nonamenable if and only if the q-state Potts model on that graph has the property that for q large enough there is an entire interval of temperatures for which the free Gibbs state is not a convex combination of the q Gibbs states obtained from one-spin boundary conditions. It is also proved that on the regular tree, , with q[greater-or-equal, slanted]1 and p close enough to 1, there is unique random cluster measure despite the presence of more than one infinite cluster. This partly proves Conjecture 1.9 of H.

title: The random geometry of equilibrium phases
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title: Simultaneous uniqueness of infinite clusters in stationary random labeled graphs
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title: The covariance matrix of the Potts model: A random cluster analysis
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title: The Ising model on diluted graphs and strong amenability
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title: Strict inequality for critical values of Potts models and random-cluster processes
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title: The Stochastic Random-Cluster Process and the Uniqueness of Random-Cluster Measures
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title: Iterated Random Functions
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title: Propp–Wilson Algorithms and Finitary Codings for High Noise Markov Random Fields
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title: Level set representation for the Gibbs states of the ferromagnetic ising model
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title: Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously
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title: Stability of infinite clusters in supercritical percolation
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title: On the norms of group-invariant transition operators on graphs
abstract: In this paper we consider reversible random walks on an infinite grapin, invariant under the action of a closed subgroup of automorphisms which acts with a finite number of orbits on the vertex-set. Thel2-norm (spectral radius) of the simple random walk is equal to one if and only if the group is both amenable and unimodular, and this also holds for arbitrary random walks with bounded invariant measure. In general, the norm is bounded above by the Perron-Frobenius eigenvalue of a finite matrix, and this bound is attained if and only if the group is both amenable and unimodular.

title: Infinite clusters in percolation models
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title: Percolation Beyond $Z^d$, Many Questions And a Few Answers
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title: A mean field spin glass with short-range interactions
abstract: We formulate and study a spin glass model on the Bethe lattice. Appropriate boundary fields replace the traditional self-consistent methods; they give our model well-defined thermodynamic properties. We establish that there is a spin glass transition temperature above which the single-site magnetizations vanish, and below which the Edwards-Anderson order parameter is strictly positive. In a neighborhood below the transition temperature, we use bifurcation theory to establish the existence of a nontrivial distribution of single-site magnetizations. Two properties of this distribution are studied: the leading perturbative correction to the Gaussian scaling form at the transition, and the (nonperturbative) behavior of the tails.

title: The random-cluster model on a homogeneous tree
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title: Gibbs Measures and Phase Transitions
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title: Multiplicity of Phase Transitions and Mean-Field Criticality on Highly Non-Amenable Graphs
abstract: We consider independent percolation, Ising and Potts models, and the contact process, on infinite, locally finite, connected graphs.It is shown that on graphs with edge-isoperimetric Cheeger constant sufficiently large, in terms of the degrees of the vertices of the graph, each of the models exhibits more than one critical point, separating qualitatively distinct regimes. For unimodular transitive graphs of this type, the critical behaviour in independent percolation, the Ising model and the contact process are shown to be mean-field type.For Potts models on unimodular transitive graphs, we prove the monotonicity in the temperature of the property that the free Gibbs measure is extremal in the set of automorphism invariant Gibbs measures, and show that the corresponding critical temperature is positive if and only if the threshold for uniqueness of the infinite cluster in independent bond percolation on the graph is less than 1.We establish conditions which imply the finite-island property for independent percolation at large densities, and use those to show that for a large class of graphs the q-state Potts model has a low temperature regime in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases.In the case of non-amenable transitive planar graphs with one end, we show that the q-state Potts model has a critical point separating a regime of high temperatures in which the free Gibbs measure is extremal in the set of automorphism-invariant Gibbs measures from a regime of low temperatures in which the free Gibbs measure decomposes as the uniform mixture of the q ordered phases.

title: On the purity of the limiting gibbs state for the Ising model on the Bethe lattice
abstract: We give a proof that for the Ising model on the Bethe lattice, the limiting Gibbs state with zero effective field (disordered state) persists to be pure for temperature below the ferromagnetic critical temperatureTcF until the critical temperatureTcSG of the corresponding spin-glass model. This new proof revises the one proposed earlier.

title: Broadcasting on trees and the Ising model
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title: Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses
abstract: SummaryWe extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on ℤd or ℤd × ℕ with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on ℤd or on a slab ℤd × {0,...K} and prove both that percolation occurs and that the infinite component is unique forV=ℤ2×{0,1} or larger.

title: Growth functions on Fuchsian groups and the Euler characteristic
abstract: Let G be a finitely generated group, and 2; a finite generating set for G. Then 2; determines a norm on G, called the word norm, defined by letting Ig[ be the minimal length ofg as a word in 27. Following Milnor [,15], we define the growth series g(z) = 1 + a2z +. . . + a,z"..., where a, is the number of elements of G of word norm exactly n. Much of the early work (e.g., [-1, 16, 17, 23]) on growth series was concerned with asymptotic properties of the coefficients of g(z). This work culminated in Gromov 's theorem  that G has polynomial growth if and only if it is virtually nilpotent and in Grigorchuk's theorem [11] that there are finitely generated groups whose growth is neither polynomial nor exponential. If G is a Coxeter group and 2; is the standard generating set for G, Bourbaki [3] showed that the growth series g(z) is the series of a rational function f (z) and Serre [-20] showed that f(1) = I/z(G), where z(G) is the rational Euler characteristic of G. Note that if G is finite then g(1)=order(G)=l/z(G), but g(z) is not defined at one if G is infinite. If G is a compact hyperbolic or irreducible Euclidean Coxeter group and 27 is the standard generating set for G, Serre showed that f (1 /z )= +_ f(z), and so the poles of f are algebraic units. In contrast to the work on the asymptotic properties of the coefficients of g(z), these results raised the possibility of a beautiful theory of the exact structure of growth functions on groups. However, a decade went by before there was much further work on this subject. In two papers ( and ) Cannon studied growth functions for cocompact hyperbolic groups. If G is a cocompact hyperbolic group and X is any finite generating set for G, he showed that the coefficients a, of the growth series g(z) satisfy a linear recursion, and hence g(z) is the power series of a rational function f(z). He also computed some examples when G is a closed surface group or a compact hyperbolic Coxeter group. If G is a closed surface group and 2; is a geometric generating set (see below for a definition) for G coming from a one relator presentation, then f is reciprocal ( f ( z )=f (1 /z ) ) , f (1)=l /x (G) , the

title: 1997 ON PROBLEMS RELATED TO GROWTH , ENTROPY AND SPECTRUM IN GROUP
abstract: We review some known results and open problems related to the growth of groups. For a nitely generated group ?; given whenever necessary together with a nite generating set, we discuss the notions of (A) uniformly exponential growth, (B) growth tightness, (C) regularity of growth series, (D) classical growth series (with respect to word lengths), (E) growth series with respect to weights, (F) complete growth series, (G) spectral radius of simple random walks on Cayley graphs. From the modern point of view a dynamical system is a pair (G; X) where G is a group (or a semi-group) and X is a set on which G acts; introducing diierent structures on G and X we get diierent directions for the Theory of Dynamical Systems. Many of the dynamical properties of the pair (G; X) depend on appropriate properties of the group G: In this paper we discuss such notions for a group G as growth, entropy, amenability and spectral radius of random walks, which are closely related to the notions of growth of manifolds and foliations, entropy of geodesic ows and of symbolic systems, spectral theory of Laplace operators and the theory of harmonic functions on Riemannian manifolds. It is a great pleasure for the rst author to mention that one of the results that is discussed in this paper, namely the cogrowth criterium for amenability, was rstly reported at the seminar of Prof.

title: Exponential decay of connectivities in the two-dimensional ising model
abstract: We prove some results concerning the decay of connectivities in the low-temperature phase of the two-dimensional Ising model. These provide the bounds necessary to establish, nonperturbatively, large-deviation properties for block magnetizations in these systems. We also obtain estimates on the rate at which the finite-volume, plus-boundary-condition expectation of the spin at the origin converges to the spontaneous magnetization.

title: Robust Phase Transitions for Heisenberg and Other Modelson General
abstract: We study several statistical mechanical models on a general tree. Particular attention is devoted to the classical Heisenberg models, where the state space is the d{dimensional unit sphere and the interactions are proportional to the cosines of the angles between neighboring spins. The phenomenon of interest here is the classiica-tion of phase transition (non-uniqueness of the Gibbs state) according to whether it is robust. In many cases, including all of the Heisenberg and Potts models, occurrence of robust phase transition is determined by the geometry (branching number) of the tree in a way that parallels the situation with independent percolation and usual phase transition for the Ising model. The critical values for robust phase transition for the Heisenberg and Potts models are also calculated exactly. In some cases, such as the q 3 Potts model, robust phase transition and usual phase transition do not coincide , while in other cases, such as the Heisenberg models, we conjecture that robust phase transition and usual phase transition are equivalent. In addition, we show that symmetry breaking is equivalent to the existence of a phase transition, a fact believed but not known for the rotor model on Z Z 2. 1 Deenition of the model and main results Particle systems on trees have produced the rst and most tractable examples of certain qualitative phenomena. For example, the contact process on a tree has multiple phase transitions, ((19, 12, 22]) and the critical temperature for the Ising model on a tree is determined by its branching number or Hausdorr dimension ((13, 8, 20]), which makes the Ising model intimately related to independent percolation whose critical value is also determined by the branching number (see 14]). In this paper we study several models on general innnite trees, including the classical Heisenberg and Potts models. Our aim is to exhibit a distinction between two kinds of phase transitions, robust and non-robust, as well as to investigate conditions under which robust phase transitions occur. In many cases, including the Heisenberg and Potts models, the existence of a robust phase transition is determined by the branching number. However, in some cases (including the q > 2 Potts model), the critical temperature for the existence of usual phase transition is not determined by the branching number. Thus robust phase transition behaves in a more universal manner than non-robust phase transition, being a function of the branching number alone, as …

title: A Remark on the Notion of Robust Phase Transitions
abstract: We point out that the high-qPotts model on a regular lattice at its transition temperature provides an example of a nonrobust—in the sense recently proposed by Pemantle and Steif—phase transition.