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 fails for the random-cluster model on a tree

We study the random-cluster model on a homogeneous tree, and show that the following three conditions are equivalent for a random-cluster measure: quasilocality, almost sure quasilocality, and the almost sure nonexistence of infinite clusters. As a consequence of this, we find that the plus measure for the Ising model on a tree at sufficiently low temperatures can be mapped, via a local stochastic transformation, into a measure which fails to be almost surely quasilocal.     

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.     

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.     

On two correlation inequalities for Potts models

The Fortuin-Kasteleyn random cluster representation ofq-state Potts models is used to extend to everyq two correlation inequalities proven previously only for even values ofq.     

More correlation inequalities for a class of even ferromagnets

Rigorous correlation inequalities are presented for a class of even ferromagnets, which includes the spin-1/2 Ising model and scalar ⁴ models. One of them leads to an extension of the Glimm and Jaffe uniform upper bound on the ⁴ renormalized coupling constant into the nonsymmetric regime.     

Correlation inequalities for a class of even ferromagnets

We present rigorous correlation inequalities for connectedn-point functions in a class of even ferromagnets. The class includes spin-1/2 Ising models and scalar field models with potential functionV which is even and continuously differentiable withV convex on [0, ). These inequalities are obtained by pushing ahead with the method of Ellis, Monroe, and Newman at its maximum.     

Critical Value of the Quantum Ising Model on Star-Like Graphs

We present a rigorous determination of the critical value of the ground-state quantum Ising model in a transverse field, on a class of planar graphs which we call star-like. These include the junction of several copies of ℤ at a single point. Our approach is to use the graphical, or fk-, representation of the model, and the probabilistic and geometric tools associated with it. This research was carried out during the author’s Ph.D. studentship at the University of Cambridge, UK, and the Royal Institute of Technology (KTH), Sweden. The author gratefully acknowledges funding from KTH during this period. The author would also like to thank Riddarhuset, Stockholm, for generous support during his studies.     

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.     

The Phase Transition of the Quantum Ising Model is Sharp

An analysis is presented of the phase transition of the quantum Ising model with transverse field on the d-dimensional hypercubic lattice. It is shown that there is a unique sharp transition. The value of the critical point is calculated rigorously in one dimension. The first step is to express the quantum Ising model in terms of a (continuous) classical Ising model in d+1 dimensions. A so-called ‘random-parity’ representation is developed for the latter model, similar to the random-current representation for the classical Ising model on a discrete lattice. Certain differential inequalities are proved. Integration of these inequalities yields the sharpness of the phase transition, and also a number of other facts concerning the critical and near-critical behaviour of the model under study.     

A study of perfect wetting for Potts and Blume-Capel models with correlation inequalities

The so-called perfect wetting phenomenon is studied for theq-state,d2 Potts model. Using a new correlation inequality, a general inequality is established for the surface tension between ordered phases ( ^a,b ) and the surface tension between an ordered and the disordered phases ( ^a,f ) for any even value ofq. This result implies in particular at the transition point _t where the previous phases coexist forq large. This inequality is connected to perfect wetting at the transition point using thermodynamic considerations. The same kinds of results are derived for the Blume-Capel model.     

Onsager's reaction field for the Potts model from the path integral

A new path integral formulation for theq-state Potts model is proposed. This formulation reproduces known results for the Ising model (q=2) and naturally extends these results for arbitraryq. The mean field results for both the Ising and the Potts models are obtained as a leading saddle point contribution to the corresponding functional integrals, while the systematic computation of corrections to the saddle point contribution produces the Onsager reaction field terms, which forq=2 coincide with results already known for the Ising model.     

Droplets in two-dimensional Ising and Potts models

Droplets on a wall and droplets around a nucleus in the center of the lattice are studied in the two-dimensional Ising and three-state Potts models using Monte Carlo techniques. Finite-size effects are discussed by applying a scaling argument and by relating the shape of a droplet to a random walk.     

The distribution of clusters for the Ising model

We rigorously prove that the probabilityP _n for the origin to belong to a cluster of exactlyn positive spins in thev-dimensional Ising model behaves as exp(–n^{(v – 1)/v}) in various regions, including in particular the low-temperature positive and negative phases in zero magnetic field.     

Some correlation inequalities for Ising antiferromagnets

We prove some inequalities for two-point correlations of Ising antiferromagnets and derive inequalities relating correlations of ferromagnets to correlations of antiferromagnets whose interactions and field strengths have equal magnitudes. The proofs are based on the method of duplicate spin variables introduced by J. Percus and used by several authors to derive correlation inequalities for Ising ferromagnets.     

Ising models on hyperbolic graphs

We consider Ising models on a hyperbolic graph which, loosely speaking, is a discretization of the hyperbolic planeH ² in the same sense asZ ^d is a discretization ofR ^d . We prove that the models exhibit multiple phase transitions. Analogous results for Potts models can be obtained in the same way.     

Correlation inequalities for the truncated two-point function of an Ising ferromagnet

We establish the following new correlation inequalities for the truncated twopoint function of an Ising ferromagnet in a positive external field: _j ; _l ^T _j ; _k ^T _k ; _l ^T , and _j ; _l ^T _{k K j} ; _k ^T _{k l} , whereK is any set of sites which separatesj froml. The inequalities are also valid for the pure phases with zero magnetic field at all temperatures. Above the critical temperature they reduce to known inequalities of Griffiths and Simon, respectively.NSERC Postgraduate Fellow, 1978–1981. Research supported in part by NSF Grant No. PHY-78-25390-A02.     

Logarithmic Sobolev inequalities and stochastic Ising models

We use logarithmic Sobolev inequalities to study the ergodic properties of stochastic Ising models both in terms of large deviations and in terms of convergence in distribution.     

Percolation and the Potts model

The Kasteleyn-Fortuin formulation of bond percolation as a lattice statistical model is rederived using an alternate approach. It is shown that the quantities of interest arising in the percolation problem, including the critical exponents, can be obtained from the solution of the Potts model. We also establish the Griffith inequality for critical exponents for the bond percolation problem.Work supported in part by NSF Grant No. D MR 76-20643.     

Inequalities for continuous-spin Ising ferromagnets

We investigate correlation inequalities for Ising ferromagnets with continuous spins, giving a simple unified derivation of inequalities of Griffiths, Ginibre, Percus, Lebowitz, and Ellis and Monroe. The single-spin measure and Hamiltonian for which an inequality may be proved become more restricted as the inequality becomes more complex. However, all results hold for a model with ferromagnetic pair interactions, positive (nonuniform) external field, and single-spin measure eitherv() = [1/(l + 1)] x f=0/l (–l +2j +) (spinl/2) ordv() = exp [–P()]d, whereP is an even polynomial all of whose coefficients must be positive except the quadratic, which is arbitrary. The Lebowitz correlation inequality is a corollary of the Ellis-Monroe inequality. As an application, we generalize the method of van Beijeren to establish a sharp phase interface at low temperature in nearest neighbor ferromagnets of at least three dimensions with arbitrary (symmetric) single-spin measure.Supported in part by the National Science Foundation under Grants MPS 73-05037 and MPS 75-20638. Much of this research was performed while the author was a student at the Massachusetts Institute of Technology and Harvard University, Cambridge, Masachusetts.