10570218
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.
117292360
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.
8906228
Nonuniversal critical dynamics in Monte Carlo simulations.
A new approach to Monte Carlo simulations is presented, giving a highly efficient method of simulation for large systems near criticality. The algorithm violates dynamic universality at second-order phase transitions, producing unusually small values of the dynamical critical exponent.
21262870
Generalization of the Fortuin-Kasteleyn-Swendsen-Wang representation and Monte Carlo algorithm.
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.
17289447
Discontinuity of the magnetization in one-dimensional 1/¦x−y¦2 Ising and Potts models
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.
15140012
Critical Percolation on any Nonamenable Group has no Infinite Clusters
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.
121342458
Exact sampling with coupled Markov chains and applications to statistical mechanics
For many applications it is useful to sample from a finite set of objects in accordance with some particular distribution. One approach is to run an ergodic (i.e., irreducible aperiodic) Markov chain whose stationary distribution is the desired distribution on this set; after the Markov chain has run for M steps, with M sufficiently large, the distribution governing the state of the chain approximates the desired distribution. Unfortunately, it can be difficult to determine how large M needs to be. We describe a simple variant of this method that determines on its own when to stop and that outputs samples in exact accordance with the desired distribution. The method uses couplings which have also played a role in other sampling schemes; however, rather than running the coupled chains from the present into the future, one runs from a distant point in the past up until the present, where the distance into the past that one needs to go is determined during the running of the algorithm itself. If the state space has a partial order that is preserved under the moves of the Markov chain, then the coupling is often particularly efficient. Using our approach, one can sample from the Gibbs distributions associated with various statistical mechanics models (including Ising, random-cluster, ice, and dimer) or choose uniformly at random from the elements of a finite distributive lattice. © 1996 John Wiley & Sons, Inc.
117822524
Amenability and Phase Transition in the Ising Model
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<∞.
18890502
The random cluster model on a general graph and a phase transition characterization of nonamenability
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.
119137791
The random geometry of equilibrium phases
Publisher Summary This chapter discusses the random geometry of equilibrium phases. Percolation will come into play here on various levels. Its concepts like clusters, open paths, connectedness etc. will be useful for describing certain geometric features of equilibrium phases, allowing characterizations of phases in percolation terms. Examples are presented where the (thermal) phase transition goes hand in hand with a phase transition in an associated percolation process. Percolation techniques can be used to obtain specific information about the phase diagram of the system. For example, equilibrium correlation functions are sometimes dominated by connectivity functions in an associated percolation problem which is easier to investigate. Further, representations in terms of percolation models yield explicit relations between certain observables in equilibrium models and some corresponding percolation quantities.
189837514
Simultaneous uniqueness of infinite clusters in stationary random labeled graphs
In processes such as invasion percolation and certain models of continuum percolation, in which a possibly random labelf(b) is attached to each bondb of a possibly random graph, percolation models for various values of a parameterr are naturally coupled: one can define a bondb to be occupied at levelr iff(b)≦r. If the labeled graph is stationary, then under the mild additional assumption of positive finite energy, a result of Gandolfi, Keane, and Newman ensures that, in lattice models, for each fixedr at which percolation occurs, the infinite cluster is unique a.s. Analogous results exist for certain continuum models. A unifying framework is given for such fixed-r results, and it is shown that if the site density is finite and the labeled graph has positive finite energy, then with probability one, uniqueness holds simultaneously for all values ofr. An example is given to show that when the site density is infinite, positive finite energy does not ensure uniqueness, even for fixedr. In addition, with finite site density but without positive finite energy, one can have fixed-r uniqueness a.s. for eachr, yet not have simultaneous uniqueness.
258599
The covariance matrix of the Potts model: A random cluster analysis
We consider the covariance matrix,Gmm=q2, of thed-dimensionalq-states Potts model, rewriting it in the random cluster representation of Fortuin and Kasteleyn. In any of theq ordered phases, we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances, thereby attributing algebraic significance to these stochastic geometric quantities. We also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers. For dimensiond=2, we show that this correlation length and the correlation length of the two-point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two. For systems with first-order transitions, this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self-dual point βo. For systems with second order transitions, this relation implies the equality of the correlation length exponents from above and below threshold, as well as an amplitude ratio of two. In the course of proving the above results, we establish several properties of independent interest, including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions, and left continuity of the two-dimensional free boundary condition percolation probability at βo. We also introduce DLR equations for the random cluster model and use them to establish ergodicity of the free measure. In order to prove these results, we introduce a new class of events which we call decoupling events and two inequalities for these events. The first is similar to the FKG inequality, but holds for events which are neither increasing nor decreasing; the second is similar to the van den Berg-Kesten inequality in standard percolation. Both inequalities hold for an arbitrary FKG measure.
23472024
Density and uniqueness in percolation
Two results on site percolation on thed-dimensional lattice,d≧1 arbitrary, are presented. In the first theorem, we show that for stationary underlying probability measures, each infinite cluster has a well-defined density with probability one. The second theorem states that if in addition, the probability measure satisfies the finite energy condition of Newman and Schulman, then there can be at most one infinite cluster with probability one. The simple arguments extend to a broad class of finite-dimensional models, including bond percolation and regular lattices.
10350129
Phase transitions on nonamenable graphs
We survey known results about phase transitions in various models of statistical physics when the underlying space is a nonamenable graph. Most attention is devoted to transitive graphs and trees.
122591119
Infinite clusters in dependent automorphism invariant percolation on trees
We study dependent bond percolation on the homogeneous tree T n of order n ≥ 2 under the assumption of automorphism invariance. Excluding a trivial case, we find that the number of infinite clusters a.s. is either 0 or ∞. Furthermore, each infinite cluster a.s. has either 1, 2 or infinitely many topological ends, and infinite clusters with infinitely many topological ends have a.s. a branching number greater than 1. We also show that if the marginal probability that a single edge is open is at least 2/(n + 1), then the existence of infinite clusters has to have positive probability. Several concrete examples are considered.
15104841
Indistinguishability of Percolation Clusters
We show that when percolation produces infinitely many infinite clusters on a Cayley graph, one cannot distinguish the clusters from each other by any invariantly defined property. This implies that uniqueness of the infinite cluster is equivalent to nondecay of connectivity (a.k.a. long-range order). We then derive applications concerning uniqueness in Kazhdan groups and in wreath products and inequalities for p u .
8686726
The Ising model on diluted graphs and strong amenability
Say that a graph has persistent transition if the Ising model on the graph can exhibit a phase transition (nonuniqueness of Gibbs measures) in the presence of a nonzero external field. We show that for nonamenable graphs, for Bernoulli percolation with p close to 1, all the infinite clusters have persistent transition. On the other hand, we show that for transitive amenable graphs, the infinite clusters for any stationary percolation do not have persistent transition. This extends a result of Georgii for the cubic lattice. A geometric consequence of this latter fact is that the infinite clusters are strongly amenable (i.e., their anchored Cheeger constant is 0). Finally we show that the critical temperature for the Ising model with no external field on the infinite clusters of Bernoulli percolation with parameter p, on an arbitrary bounded degree graph, is a continuous function of p.
7061578
Strict inequality for critical values of Potts models and random-cluster processes
We prove that the critical value βc 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.
12378666
The Stochastic Random-Cluster Process and the Uniqueness of Random-Cluster Measures
The random-cluster model is a generalization of percolation and ferromagnetic Potts models, due to Fortuin and Kasteleyn. 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. Second, 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.
13932663
Iterated Random Functions
Iterated random functions are used to draw pictures or simulate large Ising models, among other applications. They offer a method for studying the steady state distribution of a Markov chain, and give useful bounds on rates of convergence in a variety of examples. The present paper surveys the field and presents some new examples. There is a simple unifying idea: the iterates of random Lipschitz functions converge if the functions are contracting on the average.
52873689
Propp–Wilson Algorithms and Finitary Codings for High Noise Markov Random Fields
In this paper, we combine two previous works, the first being by the first author and K. Nelander, and the second by J. van den Berg and the second author, to show (1) that one can carry out a Propp–Wilson exact simulation for all Markov random fields on ℤd satisfying a certain high noise assumption, and (2) that all such random fields are a finitary image of a finite state i.i.d. process. (2) is a strengthening of the previously known fact that such random fields are so-called Bernoulli shifts.
109927566
Level set representation for the Gibbs states of the ferromagnetic ising model
SummaryWe prove the existence of a real valued random field with parameters in thed-dimensional cubic lattice, such that the distribution of the level set of this random field is a Gibbs state for the nearest neighbour ferromagnetic Ising model. Using this, we prove the continuity of the percolation probability with respect to the parameter (β,h) in the uniqueness region except on the critical curve Γc={(β,hc(β))}, wherehc(β) is the critical level of the external field above which percolation takes place.
118237497
On the random-cluster model: III. The simple random-cluster model
Abstract The possibility of the occurence of a phase transition in an infinite simple random-cluster model (0≤ p ≤1, k ≥1), which includes the percolation model and the ferromagnetic Ising and Ashkin-Teller-Potts model, is studied by means of several criteria for large-range connectivity. It is shown that graphs which contain the square lattice exhibit a phase transition in this sense. The large-range connectivities in the simple random-cluster model turn out to have the same properties as those in the percolation model. Furthermore, it is shown that in graphs with a lattice structure the generalized spontaneous magnetization is strongly related to global large-range connectivity.
120560563
Monotonicity of uniqueness for percolation on Cayley graphs: all infinite clusters are born simultaneously
Abstract. Consider site or bond percolation with retention parameter p on an infinite Cayley graph. In response to questions raised by Grimmett and Newman (1990) and Benjamini and Schramm (1996), we show that the property of having (almost surely) a unique infinite open cluster is increasing in p. Moreover, in the standard coupling of the percolation models for all parameters, a.s. for all p2>p1>pc, each infinite p2-cluster contains an infinite p1-cluster; this yields an extension of Alexander's (1995) “simultaneous uniqueness” theorem. As a corollary, we obtain that the probability θv(p) that a given vertex v belongs to an infinite cluster is depends continuously on p throughout the supercritical phase p>pc. All our results extend to quasi-transitive infinite graphs with a unimodular automorphism group.
121501717
Stability of infinite clusters in supercritical percolation
Abstract. A recent theorem by Häggström and Peres concerning independent percolation is extended to all the quasi-transitive graphs. This theorem states that if 0<p1<p2≤1 and percolation occurs at level p1, then every infinite cluster at level p2 contains some infinite cluster at level p1. Consequences are the continuity of the percolation probability above the percolation threshold and the monotonicity of the uniqueness of the infinite cluster, i.e., if at level p1 there is a unique infinite cluster then the same holds at level p2. These results are further generalized to graphs with a “uniform percolation” property. The threshold for uniqueness of the infinite cluster is characterized in terms of connectivities between large balls.
122292502
Følner Independence and the amenable Ising model
We define a criterion called Folner Independence for a stationary process over an amenable group. Intuitively, a process satisfies the criterion if, for sufficiently invariant Folner sets, the process in the Folner set is nearly independent of the process outside. We show that Folner Independence implies Finitely Determined. As an application, we show that, in its extreme Gibbs states, the amenable attractive Ising model is Folner Independent (hence Finitely Determined, hence Bernoulli).
122389862
On the norms of group-invariant transition operators on graphs
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.
27943325
Infinite clusters in percolation models
The qualitative nature of infinite clusters in percolation models is investigated. The results, which apply to both independent and correlated percolation in any dimension, concern the number and density of infinite clusters, the size of their external surface, the value of their (total) surface-to-volume ratio, and the fluctuations in their density. In particular it is shown thatN0, the number of distinct infinite clusters, is either 0, 1, or ∞ and the caseN0=∞ (which might occur in sufficiently high dimension) is analyzed.
14666744
Percolation Beyond $Z^d$, Many Questions And a Few Answers
A comprehensive study of percolation in a more general context than the usual $Z^d$ setting is proposed, with particular focus on Cayley graphs, almost transitive graphs, and planar graphs. Results concerning uniqueness of infinite clusters and inequalities for the critical value $p_c$ are given, and a simple planar example exhibiting uniqueness and non-uniqueness for different $p>p_c$ is analyzed. Numerous varied conjectures and problems are proposed, with the hope of setting goals for future research in percolation theory.
18183748
Percolation on Transitive Graphs as a Coalescent Process : Relentless Merging Followed by Simultaneous
Consider i.i.d. percolation with retention parameter p on an infinite graph G. There is a well known critical parameter pc ∈ [0, 1] for the existence of infinite open clusters. Recently, it has been shown that when G is quasi-transitive, there is another critical value pu ∈ [pc, 1] such that the number of infinite clusters is a.s. ∞ for p ∈ (pc, pu), and a.s. one for p > pu. We prove a simultaneous version of this result in the canonical coupling of the percolation processes for all p ∈ [0, 1]. Simultaneously for all p ∈ (pc, pu), we also prove that each infinite cluster has uncountably many ends. For p > pc we prove that all infinite clusters are indistinguishable by robust properties. Under the additional assumption that G is unimodular, we prove that a.s. for all p1 < p2 in (pc, pu), every infinite cluster at level p2 contains infinitely many infinite clusters at level p1. We also show that any Cartesian product G of d infinite connected graphs of bounded degree satisfies pu(G) ≤ pc(Z ). Research supported by a grant from the Swedish Natural Science Research Council Research partially supported by NSF grant # DMS-9803597 Research partially supported by NSF grant # DMS-9703814
15917659
A mean field spin glass with short-range interactions
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.
122952364
The random-cluster model on a homogeneous tree
SummaryThe random-cluster model on a homogeneous tree is defined and studied. It is shown that for 1≦q≦2, the percolation probability in the maximal random-cluster measure is continuous inp, while forq>2 it has a discontinuity at the critical valuep=pc(q). It is also shown that forq>2, there is nonuniqueness of random-cluster measures for an entire interval of values ofp. The latter result is in sharp contrast to what happens on the integer lattice Zd.
60742687
Gibbs Measures and Phase Transitions
From a review of the first edition: "This book [...] covers in depth a broad range of topics in the mathematical theory of phase transition in statistical mechanics. [...] It is in fact one of the author's stated aims that this comprehensive monograph should serve both as an introductory text and as a reference for the expert." (F. Papangelou, "Zentralblatt MATH"). The second edition has been extended by a new section on large deviations and some comments on the more recent developments in the area.
1834425
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.
17979380
On the purity of the limiting gibbs state for the Ising model on the Bethe lattice
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.
1482684
Broadcasting on trees and the Ising model
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122104944
Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses
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.
55129428
Growth functions on Fuchsian groups and the Euler characteristic
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 [12] 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 ([5] and [6]) 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
16928810
1997 ON PROBLEMS RELATED TO GROWTH , ENTROPY AND SPECTRUM IN GROUP
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.
122620379
Exponential decay of connectivities in the two-dimensional ising model
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.
17441562
Robust Phase Transitions for Heisenberg and Other Modelson General
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 …
14156357
A Remark on the Notion of Robust Phase Transitions
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.