theta function parametrization and fusion for 3d integrable boltzmann weights we report progress in constructing boltzmann weights for integrable threedimensional lattice spin models we show that a large class of vertex solutions to the modified tetrahedron equation mte can be conveniently parametrized in terms of nth roots of theta functions on the jacobian of a compact algebraic curve fay s identity guarantees the fermat relations and the classical equations of motion for the parameters determining the boltzmann weights our parametrization allows us to write a simple formula for fused boltzmann weights r which describe the partition function of an arbitrary open box and which also obey the modified tetrahedron equation imposing periodic boundary conditions we observe that the r satisfy the normal tetrahedron equation the scheme described contains the zamolodchikov baxter bazhanov model and the chessboard model as special cases

uniqueness of gradient gibbs measures with disorder we consider in a uniformly strictly convex potential regime two versions of random gradient models with disorder in model a the interface feels a bulk term of random fields while in model b the disorder enters through the potential acting on the gradients we assume a general distribution on the disorder with uniformly bounded finite second moments it is well known that for gradient models without disorder there are no gibbs measures in infinite volume in dimension d 2 while there are shift invariant gradient gibbs measures describing an infinite volume distribution for the gradients of the field as was shown by funaki and spohn commun math phys 185 1 36 1997 van enter and kulske proved in ann appl probab 18 1 109 119 2008 that adding a disorder term as in model a prohibits the existence of such gradient gibbs measures for general interaction potentials in d 2 in cotar and kulske ann appl probab 22 5 1650 1692 2012 we proved the existence of shift covariant random gradient gibbs measures for model a when d ge 3 the disorder is i i d and has mean zero and for model b when d ge 1 and the disorder has a stationary distribution in the present paper we prove existence and uniqueness of shift covariant random gradient gibbs measures with a given expected tilt u in mathbb r d and with the corresponding annealed measure being ergodic for model a when d ge 3 and the disordered random fields are i i d and symmetrically distributed and for model b when d ge 1 and for any stationary disorder dependence structure we also compute for both models for any gradient gibbs measure constructed as in cotar and kulske ann appl probab 22 5 1650 1692 2012 when the disorder is i i d and its distribution satisfies a poincare inequality assumption the optimal decay of covariances with respect to the averaged over the disorder gradient gibbs measure

delocalization of two dimensional random surfaces with hard core constraints we study the fluctuations of random surfaces on a two dimensional discrete torus the random surfaces we consider are defined via a nearest neighbor pair potential which we require to be twice continuously differentiable on a possibly infinite interval and infinity outside of this interval no convexity assumption is made and we include the case of the so called hammock potential when the random surface is uniformly chosen from the set of all surfaces satisfying a lipschitz constraint our main result is that these surfaces delocalize having fluctuations whose variance is at least of order log n where n is the side length of the torus we also show that the expected maximum of such surfaces is of order at least log n the main tool in our analysis is an adaptation to the lattice setting of an algorithm of richthammer who developed a variant of a mermin wagner type argument applicable to hard core constraints we rely also on the reflection positivity of the random surface model the result answers a question mentioned by brascamp et al on the hammock potential and a question of velenik

symmetry relations in exactly soluble models symmetry relations such as the star triangle or the inverse relation are very useful in determining the partition function of two dimensional exactly soluble models a common construction of the three dimensional equivalents of these symmetry relations is presented they are used to derive in a geometric way applying simultaneously to different kinds of spin models the consequent global properties i e the commutativity of the transfer matrices and the inverse functional equations on the transfer matrix and the partition function the usefulness of the inverse relation is illustrated by an application to the three dimensional ising model

quantization of three wave equations the subject of this paper is the revision of discretization and quantization of two similar classical integrable hamiltonian systems in three dimensional spacetime the standard three wave equations and the modified three wave equations the darboux manakov zakharov system in two distinguished gauges the quantized systems in discrete spacetime may be understood as the regularized integrable quantum field theories integrability of the theories and in particular the quantum tetrahedron equations for vertex operators follow from the quantum auxiliary linear problems the principal object of the lattice field theories is the heisenberg discrete time evolution operator constructed with the help of vertex operators

the large n limits of the chiral potts model abstract in this paper we study the large n limits of the integrable n state chiral potts model three chiral solutions of the star triangle equations are derived with states taken from all integers or from a finite or infinite real interval these solutions are expected to be chiral field lattice deformations of parafermionic conformal field theories a new two sided hypergeometric identity is derived as a corollary

the eight tetrahedron equations in this paper we derive from arguments of string scattering a set of eight tetrahedron equations with different index orderings it is argued that this system of equations is the proper system that represents integrable structures in three dimensions generalizing the yang baxter equation under additional restrictions this system reduces to the usual tetrahedron equation in the vertex form most known solutions fall under this class but it is by no means necessary comparison is made with the work on braided monoidal 2 categories also leading to eight tetrahedron equations

on the absence of breakdown of symmetry for the plane rotator model with long range random interaction we study the plane rotator model with hamiltonian frac 1 2 sum limits x ne y frac jxy cos mathcal o x mathcal o y left x y right 3 mathcal e where jxy for different pair x y are independent symmetric unbounded random variables it is proved that for almost all j all gibbs states p j are invariant by rotation

clustering in low dimensional so n invariant statistical models with long range interactions we obtain upper bounds for the two point correlation functions in statistical models in one or two dimensions which haveso n symmetry this clarifies upper bounds for long range interactions for which there exists clustering

energy entropy inequalities for classical lattice systems a new characterization of equilibrium states for classical lattice systems is given in terms of correlation inequalities their physical meaning is found to express thermodynamic stability we demonstrate the applicability of the inequalities in specific models

on zamolodchikov s solution of the tetrahedron equations the tetrahedron equations arise in field theory as the condition for thes matrix in 2 1 dimensions to be factorizable and in statistical mechanics as the condition that the transfer matrices of three dimensional models commute zamolodchikov has proposed what appear from numerical evidence and special cases to be non trivial particular solutions of these equations but has not fully verified them here it is proved that they are indeed solutions

zn n 1 generalization of the chiral potts model we show that ther matrix which intertwines twon by nn 1 state cyclicl operators related with a generalization ofuq sl n algebra can be considered as a boltzmann weight of four spin box for a lattice model with two spin interaction just as ther matrix of the checkerboard chiral potts model the rapidity variables lie on the algebraic curve of the genusg n2 n 1 n 1 n n 1 defined by 2n 3 independent moduli this curve is a natural generalization of the curve which appeared in the chiral potts model factorization properties of thel operator and its connection to the sos models are also discussed

the vertex formulation of the bazhanov baxter model in this paper we formulate an integrable model on the simple cubic lattice then valued spin variables of the model belong to edges of the lattice the boltzmann weights of the model obey the vertex type tetrahedron equation in the thermodynamic limit our model is equivalent to the bazhanov baxter model in the case whenn 2 we reproduce korepanov s and hietarinta s solutions of the tetrahedron equation as special cases

an integrable 3d lattice model with positive boltzmann weights in this paper we construct a three dimensional 3d solvable lattice model with non negative boltzmann weights the spin variables in the model are assigned to edges of the 3d cubic lattice and run over an infinite number of discrete states the boltzmann weights satisfy the tetrahedron equation which is a 3d generalization of the yang baxter equation the weights depend on a free parameter 0 q 1 and three continuous field variables the layer to layer transfer matrices of the model form a two parameter commutative family this is the first example of a non trivial solvable 3d lattice model with non negative boltzmann weights

strict convexity of the free energy for a class of non convex gradient models we consider a gradient interface model on the lattice with interaction potential which is a non convex perturbation of a convex potential we show using a one step multiple scale analysis the strict convexity of the surface tension at high temperature this is an extension of funaki and spohn s result 8 where the strict convexity of potential was crucial in their proof

permutation type solutions to the yang baxter and other n simplex equations we study permutation type solutions to n simplex equations that is solutions whose matrix form can be written as with some matrix a and vector b both over with this ansatz the equations of the n simplex equation reduce to a matrix equation over we have completely analysed the 2 3 and 4 simplex equations in the generic d case the solutions show interesting patterns that seem to continue to still higher simplex equations

quantum 2 1 evolution model a quantum evolution model in 2 1 discrete spacetime connected with a 3d fundamental map is investigated map is derived as a map providing a zero curvature of a 2d linear lattice system called the current system in a special case of the local weyl algebra for dynamical variables the map appears to be canonical and it corresponds to the known operator valued matrix the current system is a type of the linear problem for the 2 1 evolution model a generating function for the integrals of motion for the evolution is derived with the help of the current system thus the complete integrability in 3d is proved directly

on equivalence of spin and field pictures of lattice systems we investigate the spin and field systems on a lattice connected by the kac siegert transform it is shown that the structures of corresponding theories are equivalent in the sense of isomorphy of space of gibbs states and order parameters using the idea of equivalence of spin and field pictures we exhibit a class of lattice systems possessing infinitely uncountably many ground states the systems of this type with infinite range slow decaying interactions are expected to have a spin glass phase transition

star triangle relation for a three dimensional model the solvablesl n chiral potts model can be interpreted as a three dimensional lattice model with local interactions to within a minor modification of the boundary conditions it is an ising type model on the body centered cubic lattice with two and three spin interactions the corresponding local boltzmann weights obey a number of simple relations including a restricted star triangle relation which is a modified version of the well known star triangle relation appearing in two dimensional models we show that these relations lead to remarkable symmetry properties of the boltzmann weight function of an elementary cube of the lattice related to the spatial symmetry group of the cubic lattice these symmetry properties allow one to prove the commutativity of the row to row transfer matrices bypassing the tetrahedron relation the partition function per site for the infinite lattice is calculated exactly

spectral gap and decay of correlations in u 1 symmetric lattice systems in dimensions d 2 we consider many body systems with a global u 1 symmetry on a class of lattices with the fractal dimension d 2 and their zero temperature correlations whose observables behave as a vector under the u 1 rotation for a wide class of the models we prove that if there exists a spectral gap above the ground state then the correlation functions have a stretched exponentially decaying upper bound this is an extension of the mcbryan spencer method at finite temperatures to zero temperature the class includes quantum spin and electron models on the lattices and our method also allows finite or infinite quasi degeneracy of the ground state the resulting bounds rule out the possibility of the corresponding magnetic and electric long range orders

large deviation principle for one dimensional vector spin models with kac potentials we consider the one dimensional planar rotator and classical heisenberg models with a ferromagnetic kac potential j r j yr j with compact support below the lebowitz penrose critical temperature the limit mean field theory exhibits a phase transition with a continuum of equilibrium states indexed by the magnetization vectors m s s any unit vector and m the curie weiss spontaneous magnetization we prove a large deviation principle for the associated gibbs measures then we study the system in the limit 0 below the above critical temperature we prove that the norm of the empirical spin average in blocks of order 1 converges to m uniformly in intervals of order p for any p 1 we also give a lower bound to the scale on which the change of phase occurs by showing that the empirical spin average is approximately constant on intervals having length of order 1 with 0 1 small enough

complex of three dimensional solvable models an overview is given of integrable models of quantum classical and statistical mechanics defined as evolution models in wholly discrete 2 1 dimensional space time and based on a special type of auxiliary linear problem

thermodynamic limit for a spin lattice an integrable spin lattice is a higher dimensional generalization of integrable spin chains in this paper we consider a special spin lattice related to quantum mechanical interpretation of the three dimensional lattice model in statistical mechanics zamolodchikov and baxter the integrability means the existence of a set of mutually commuting operators expressed in the terms of local spin variables the significant difference between spin chain and spin lattice is that the commuting set for the latter is produced by a transfer matrix with two equitable spectral parameters there is a specific bilinear functional equation for the eigenvalues of this transfer matrix the spin lattice is investigated in this paper in the limit when both sizes of the lattice tend to infinity the limiting form of bilinear equation is derived it allows to analyze the distributions of eigenvalues of the whole commuting set the ground state distribution is obtained explicitly a structure of excited states is discussed

on the absence of spontaneous symmetry breaking and of crystalline ordering in two dimensional systems we develop a unified approach based on araki s relative entropy concept to proving absence of spontaneous breaking of continuous internal symmetries and translation invariance in two dimensional statistical mechanical systems more precisely we show that under rather general assumptions on the interactions all equilibrium states of a two dimensional system have all the symmetries compact internal and spatial of the dynamics except possibly rotation invariance rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance we also prove that two dimensional systems with a non compact internal symmetry group like anharmonic crystals typically do not have gibbs states

a new proof of ruelle s superstability bounds for a slightly stronger assumptions on the interaction we give a very transparent proof of ruelle s result in the language of poisson integral measure representation for the correlation functions on the configuration space using some kind of cluster expansion in the densities of configurations

u 1 su 2 gauge invariance of non relativistic quantum mechanics and generalized hall effects we show that the non relativistic quantum mechanics of particles with spin coupled to an electromagnetic field has a naturalu 1 su 2 gauge invariance ward identities reflecting this gauge invariance combined with an assumption of incompressibility of a system of such particles in an appropriate external field and for suitable values of the particle density permit us to determine the form of the effective action of the system as a functional of small fluctuations in the electromagnetic field in the large distance adiabatic limit in this limit the action is found to have a universal form we present explicit results for two dimensional incompressible electron fluids and apply them to derive the equations of linear response theory describing a variety of generalized hall effects sum rules for the hall conductivities magnetic susceptibilities and other quantities of physical interest are found

explicit free parametrization of the modified tetrahedron equation the modified tetrahedron equation mte with affine weyl quantum variables at the nth root of unity is solved by a rational mapping operator which is obtained from the solution of a linear problem we show that the solutions can be parametrized in terms of eight free parameters and 16 discrete phase choices thus providing a broad starting point for the construction of three dimensional integrable lattice models the fermat curve points parametrizing the representation of the mapping operator in terms of cyclic functions are expressed in terms of the independent parameters an explicit formula for the density factor of the mte is derived for the example n 2 we write the mte in full detail

on the symmetry of the gibbs states in two dimensional lattice systems under fairly general conditions if a two dimensional classical lattice system has an internal symmetry groupg which is a compact connected lie group then all gibbs states areg invariant

euclidean invariance in statistical mechanics of classical continuous system it is shown that equilibrium states of classical particles with short range interactions are euclidean invariant whenever their correlation functions have a clustering which is integrable the relation between invariance and clustering is analysed for spatial rotations and internal rotational degrees of freedom the analysis is then extended to the case of long range interactions including the coulomb force and jellium systems

labelling schemes for tetrahedron equations and dualities between them zamolodchikov s tetrahedron equations which were derived by considering the scattering of straight strings can be written in three different labelling schemes one can use as labels the states of the vacua between the strings the states of the string segments or the states of the particles at the intersections of the strings we give a detailed derivation of the three corresponding tetrahedron equations and also show how the frenkel moore equations fits in as a non local string labelling we then discuss how an analogue of the wu kadanoff duality can be defined between each pair of the above three labelling schemes it turns out that there are two cases for which one can simultaneously construct a duality between all three pairs of labellings

large deviations and variational principle for harmonic crystals we consider massless gaussian fields with covariance related to the green function of a long range random walk on d these are viewed as gibbs measures for a linear quadratic interaction we establish thermodynamic identities and prove a version of gibbs variational principle showing that translation invariant gibbs measures are characterized as minimizers of the relative entropy density we then study the large deviations of the empirical field of a gibbs measure we show that a weak large deviation principle holds at the volume order with rate given by the relative entropy density

tetrahedron equations and the relativistic s matrix of straight strings in 2 1 dimensions the quantums matrix theory of straight strings infinite one dimensional objects like straight domain walls in 2 1 dimensions is considered thes matrix is supposed to be purely elastic and factorized the tetrahedron equations which are the factorization conditions are investigated for the special two colour model the relativistic three strings matrix which apparently satisfies this tetrahedron equation is proposed

the bazhanov stroganov model from 3d approach we apply a three dimensional approach to describe a new parametrization of the l operators for the two dimensional bazhanov stroganov bs integrable spin model related to the chiral potts model this parametrization is based on the solution of the associated classical discrete integrable system using a three dimensional vertex satisfying a modified tetrahedron equation we construct an operator which generalizes the bs quantum intertwining matrix s this operator describes the isospectral deformations of the integrable bs model

the mermin wagner phenomenon and cluster properties of one and two dimensional systems we give optimal conditions concerning the range of interactions for the absence of spontaneous breakdown of continuous symmetries for one and two dimensional quantum and classical lattice and continuum systems for a class of models verifying infrared bounds our conditions are necessary and sufficient using the same techniques we obtain a priori bounds on clustering for systems with continuous symmetry improving results of jasnow and fisher

some properties of correlations of quantum lattice systems in thermal equilibrium simple proofs of uniqueness of the thermodynamic limit of kms states and of the decay of equilibrium correlations are presented for a large class of quantum lattice systems at high temperatures new quantum correlation inequalities for general heisenberg models are described finally a simplified derivation of a general result on power law decay of correlations in 2d quantum lattice systems with continuous symmetries is given extending results of mcbryan and spencer for the 2d classical xy model

tetrahedron equation and generalized quantum groups we construct families of solutions of the yang baxter equation from n products of three dimensional r and l operators satisfying the tetrahedron equation they are identified with the quantum r matrices for the hopf algebras known as generalized quantum groups depending on the number of r s and l s involved in the product the trace construction interpolates the symmetric tensor representations of and the antisymmetric tensor representations of whereas a boundary vector construction interpolates the q oscillator representation of and the spin representation of the intermediate cases are associated with an affinization of quantum superalgebras

absence of phase transitions in certain one dimensional long range random systems an ising chain is considered with a potential of the formj i j i j where thej i j are independent random variables with mean zero the chain contains both randomness and frustration and serves to model a spin glass a simple argument is provided to show that the system does not exhibit a phase transition at a positive temperature if 1 this is to be contrasted with a ferromagnetic interaction which requires 2 the basic idea is to prove that the surface free energy between two half lines is finite although the surface energy may be unbounded ford dimensional systems it is shown that the free energy does not depend on the specific boundary conditions if 1 2 d

zamolodchikov s tetrahedron equation and hidden structure of quantum groups the tetrahedron equation is a three dimensional generalization of the yang baxter equation its solutions define integrable three dimensional lattice models of statistical mechanics and quantum field theory their integrability is not related to the size of the lattice therefore the same solution of the tetrahedron equation defines different integrable models for different finite periodic cubic lattices obviously any such three dimensional model can be viewed as a two dimensional integrable model on a square lattice where the additional third dimension is treated as an internal degree of freedom therefore every solution of the tetrahedron equation provides an infinite sequence of integrable 2d models differing by the size of this hidden third dimension in this paper we construct a new solution of the tetrahedron equation which provides in this way the two dimensional solvable models related to finite dimensional highest weight representations for all quantum affine algebra where the rank n coincides with the size of the hidden dimension these models are related to an anisotropic deformation of the sl n invariant heisenberg magnets they were extensively studied for a long time but the hidden 3d structure was hitherto unknown our results lead to a remarkable exact rank size duality relation for the nested bethe ansatz solution for these models note also that the above solution of the tetrahedron equation arises in the quantization of the resonant three wave scattering model which is a well known integrable classical system in 2 1 dimensions

tetrahedral zamolodchikov algebras corresponding to baxter s l operators tetrahedral zamolodchikov algebras are structures that occupy an intermediate place between the solutions of the yang baxter equation and its generalization onto 3 dimensional mathematical physics the tetrahedron equation these algebras produce solutions to the tetrahedron equation and besides specific two layer solutions to the yang baxter equation here the tetrahedral zamolodchikov algebras are studied that arise froml operators of the free fermion case of baxter s eight vertex model

let s baxterise we recall the concept of baxterisation of an r matrix or of a monodromy matrix which corresponds to build from one point in the r matrix parameter space the algebraic variety where the spectral parameter s live we show that the baxterisation which amounts to studying the iteration of a birational transformation is a win win strategy it enables to discard efficiently the non integrable situations focusing directly on the two interesting cases where the algebraic varieties are of the so called general type finite order iteration or are abelian varieties infinite order iteration we emphasize the heuristic example of the sixteen vertex model and provide a complete description of the finite order iterations situations for the baxter model we show that the baxterisation procedure can be introduced in much larger frameworks where the existence of some underlying yang baxter structure is not used we baxterise l operators local quantum lax matrices and quantum hamiltonians

three dimensional vertex model in statistical mechanics from baxter bazhanov model we find that the boltzmann weight of the three dimensional baxter bazhanov model is dependent on four spin variables which are the linear combinations of the spins on the corner sites of the cube and the wu kadanoff wegner duality between the cube and vertex type tetrahedron equations is obtained explicitly for the baxter bazhanov model then a three dimensional vertex model is obtained by considering the symmetry property of the weight function which corresponds to the three dimensional baxter bazhanov model the vertex type weight function is parametrized as the dihedral angles between the rapidity planes connected with the cube we write down the symmetry relations of the weight functions under the actions of the symmetry groupg of the cube the six angles with a constraint condition appearing in the tetrahedron equation can be regarded as the six spectra connected with the six spaces in which the vertextype tetrahedron equation is defined

quantum energy entropy inequalities a new method for proving the absence of symmetry breaking for quantum systems we develop a new method based on a general energy entropy inequality to rule out spontaneous breaking of symmetries the main advantage of our scheme consists in its clear cut physical significance and its new areas of applicability in particular we can handle discrete symmetry groups as well as continuous ones finally a few illustrations are discussed

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

a mermin wagner theorem for gibbs states on lorentzian triangulations we establish a mermin wagner type theorem for gibbs states on infinite random lorentzian triangulations lt arising in models of quantum gravity such a triangulation is naturally related to the distribution p of a critical galton watson tree conditional upon non extinction at the vertices of the triangles we place classical spins taking values in a torus m of dimension d with a given group action of a torus g of dimension d d in the main body of the paper we assume that the spins interact via a two body nearest neighbor potential u x y invariant under the action of g we analyze quenched gibbs measures generated by u and prove that for p almost all lorentzian triangulations every such gibbs measure is g invariant which means the absence of spontaneous continuous symmetry breaking

on estimating the derivatives of symmetric diffusions in stationary random environment with applications to ϕ interface model we consider diffusions on d or random walks on d in a random environment which is stationary in space and in time and with symmetric and uniformly elliptic coefficients we show existence and holder continuity of second space derivatives and time derivatives for the annealed kernels of such diffusions and give estimates for these derivatives in the case of random walks these estimates are applied to the ginzburg landau interface model

new solvable lattice models in three dimensions in this paper we establish a remarkable connection between two seemingly unrelated topics in the area of solvable lattice models the first is the zamolodchikov model which is the only nontrivial model on a three dimen sional lattice so far solved the second is the chiral potts model on the square lattice and its generalization associated with theu q sl n algebra which is of current interest due to its connections with high genus algebraic curves and with representations of quantum groups at roots of unity we show that this last sl n generalized chiral potts model can be interpreted as a model on a threedimensional simple cubic lattice consisting ofn square lattice layers with ann valued n 2 spin at each site further in then 2 case this three dimen sional model reduces after a modification of the boundary conditions to the zamolodchikov model we mentioned above

phase structure of two dimensional spin models and percolation for a class of classical spin models in 2d satisfying a certain continuity constraint it is proven that some of their correlations do not decay exponentially the class contains discrete and continuous spin systems with abelian and non abelian symmetry groups for the discrete models our results imply that they show either long range order or are in a soft phase characterized by powerlike decay of correlations for the continuous models only the second possibility exists the continuous models include a version of the plane rotator o 2 model for this model we rederive modulo two conjectures the frohlich spencer result on the existence of the kosterlitz thouless phase in a very simple way the proof is based on percolation theoretic and topological arguments

the global markov property for equilibrium states which are determined by correlations in a strip consider an interaction round face potential on the lattice d which may include nearest neighbor and next nearest neighbor pair interactions as well as more general plaquette terms assuming some periodicity of the potential it is shown that under the condition that equilibrium states can be distinguished by expectation values of sufficiently local observables equilibrium states possess a global markov property the condition under which this markov property is shown to hold is met in particular in case the equilibrium state is unique or determined by the magnetization the proof is based on an application of the variational principle to states which are constructed like markov chains