on the layering transition of an sos surface interacting with a wall i equilibrium results we consider the model of a 2d surface above a fixed wall and attracted toward it by means of a positive magnetic field h in the solid on solid sos approximation when the inverse temperature fl is very large and the external field h is exponentially small in ft we improve considerably previous results by dinaburg and mazel on the competition between the external field and the entropic repulsion with the wall leading in this case to the phenomenon of layering phase transitions in particular we show using the pirogov sinai scheme as given by zahradnik that there exists a unique critical value h fl in the interval 88 4 k 4e 4 k such that for all h h i h and fl large enough there exists a unique infinite volume gibbs state the typical configurations are small perturbations of the ground state represented by a surface at height k 1 above the wall moreover for the same choice of the thermodynamic parameters the influence of the boundary conditions of the gibbs measure in a finite cube decays exponentially fast with the distance from the boundary when h h fl we prove instead the convergence of the cluster expansion for both k and k 1 boundary conditions this fact signals the presence of a phase transition in the second paper of this series we will consider a glauber dynamics for the above model and we will study the rate of approach to equilibrium in a large finite cube with arbitrary boundary conditions as a function of the external field h using the results proven in this paper we will show that there is a dramatic slowing down in the approach to equilibrium when the magnetic field takes one of the critical values and the boundary conditions are free absent

exact large deviation bounds up to t c for the ising model in two dimensions we prove an upper large deviation bound for the block spin magnetization in the 2d ising model in the phase coexistence region the precise rate given by the wulff construction is shown to hold true for all c combined with the lower bounds derived in i those results yield an exact second order large deviation theory up to the critical temperature

the logarithmic sobolev inequality for discrete spin systems on a lattice for finite range lattice gases with a finite spin space it is shown that the dobrushin shlosman mixing condition is equivalent to the existence of a logarithmic sobolev inequality for the associated unique gibbs state in addition implications of these considerations for the ergodic properties of the corresponding glauber dynamics are examined

harmonic crystal on the wall a microscopic approach a three dimensional winterbottom type construction in the regime of partial wetting is derived in a scaling limit of a gas of microscopic gaussian sos droplets under the fixed volume constraint the proof is based on a coarse graining of the random microscopic region wetted by the crystal random walk representations of various quantities related to free massless fields and a stability analysis of the torsional rigidity problem

renormalization group in the uniqueness region weak gibbsianity and convergence we analyze the block averaging transformation applied to lattice gas models with short range interaction in the uniqueness region below the critical temperature we prove weak gibbsianity of the renormalized measure and convergence of the renormalized potential in a weak sense since we are arbitrarily close to the coexistence region we have a diverging characteristic length of the system the correlation length or the critical length for metastability or both thus to perturbatively treat the problem we have to use a scale adapted expansion moreover such a model below the critical temperature resembles a disordered system in the presence of griffiths singularity then the cluster expansion that we use must be graded with its minimal scale length diverging when the coexistence line is approached

slab percolation for the ising model for the fk representation of the ising model we prove that the slab percolation threshold coincides with the critical temperature in any dimension d 3

large deviations for the 2d ising model a lower bound without cluster expansions we show that a lower large deviation bound for the block spin magnetization in the 2d ising model can be pushed all the way forward toward its correct wulff value for all c

letter to the editor on the conditions for the coincidence of two cubic bezier curves in a recent article wang et al 2 derive a necessary and sufficient condition for the coincidence of two cubic bezier curves with non collinear control points the condition reads that their control points must be either coincident or in reverse order we point out that this uniqueness of the control points for polynomial cubics is a straightforward consequence of a previous and more general result of barry and patterson namely the uniqueness of the control points for rational bezier curves moreover this uniqueness applies to properly parameterized polynomial curves of arbitrary degree

a differential operator realisation approach for constructing casimir operators of non semisimple lie algebras we introduce a search algorithm that utilises differential operator realisations to find polynomial casimir operators of lie algebras to demonstrate the algorithm we look at two classes of examples 1 the model filiform lie algebras and 2 the schrodinger lie algebras we find that an abstract form of dimensional analysis assists us in our algorithm and greatly reduces the complexity of the problem

large deviations and continuum limit in the 2d ising model we study the 2d ising model in a rectangular box l of linear size o l we determine the exact asymptotic behaviour of the large deviations of the magnetization p t2 l ae t when l 1 for values of the parameters of the model corresponding to the phase coexistence region where the order parameter m is strictly positive we study in particular boundary eects due to an arbitrary real valued boundary magnetic field using the self duality of the model a large part of the analysis consists in deriving properties of the covariance function hae 0 ae t i as t 1 at dual values of the parameters of the model to do this analysis we establish new results about the high temperature representation of the model these results are valid for dimensions d 2 and up to the critical temperature they give a complete non perturbative exposition of the high temperature representation we then study the gibbs measure conditioned by p t2 l ae t im l l l ic with 0 c 1 4 and im m m we construct the continuum limit of the model and describe the limit by the solutions of a variational problem of isoperi metric type

approach to equilibrium of glauber dynamics in the one phase region i the attractive case various finite volume mixing conditions in classical statistical mechanics are reviewed and critically analyzed in particular somefinite size conditions are discussed together with their implications for the gibbs measures and for the approach to equilibrium of glauber dynamics inarbitrarily large volumes it is shown that dobrushin shlosman s theory ofcomplete analyticity and its dynamical counterpart due to stroock and zegarlinski cannot be applied in general to the whole one phase region since it requires mixing properties for regions ofarbitrary shape an alternative approach based on previous ideas of oliveri and picco is developed which allows to establish results on rapid approach to equilibrium deeply inside the one phase region in particular in the ferromagnetic case we considerably improve some previous results by holley and aizenman and holley our results are optimal in the sene that for example they show for the first time fast convergence of the dynamicsfor any temperature above the critical one for thed dimensional ising model with or without an external field in part ii we extensively consider the general case not necessarily attractive and we develop a new method based on renormalizations group ideas and on an assumption of strong mixing in a finite cube to prove hypercontractivity of the markov semigroup of the glauber dynamics

equivalences of the large deviation principle for gibbs measures and critical balance in the ising model in this paper we obtain the equivalence of the large deviation principle for gibbs measures with and without an external field for the ising model the equivalence allows us to study the result of competing influences of a positive external fieldh and a negative boundary condition in the cube b h ash 0 for variousb we find a critical balance at a valueb0 ofb

large deviations for the empirical field of a gibbs measure on applique le principe de contraction pour obtenir un principe de grande deviation pour la distribution des distributions empiriques on donne une description detaillee de la fonction de taux resultante

the conditions for the coincidence or overlapping of two bezier curves in a recent article chen et al 2013 derive the condition for the coincidence of two quartic bezier curves different from straight lines the condition reads that either their control polygons coincide or the curves are degenerate representations of the same quadratic curve we point out that this uniqueness of the control points for properly parameterized polynomial curves is already well known not only for quartics but for curves of arbitrary degree this is a direct consequence of a previous more general result by berry and patterson namely the uniqueness of the control points for rational bezier curves in addition two conjectures posed in chen et al 2013 also derive as straightforward corollaries finally the condition for overlapping of two properly parameterized curves is also derived unsurprisingly it amounts to the coincidence of the bezier polygons for the overlapping segments of the given curves corresponding to the overlapping portions of their domains

random walks in space time mixing environments we prove that random walks in random environments that are exponentially mixing in space and time are almost surely diffusive in the sense that their scaling limit is given by the wiener measure

constrained variational problem with applications to the ising model we continue our study of the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh initiated in our earlier work we strengthen further a result previously proven by martirosyan at low enough temperature which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the system if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the system in our earlier work this result was extended to every subcritical value of the temperature here for every subcritical value of the temperature we show the existence of a critical valueb0 t which separates the two regimes specified above we also find the asymptotic shape of the region occupied by the phase in the second regime which turns out to be a squeezed wulff shape the main step in our study is the solution of the variational problem of finding the curve minimizing the wulff functional which curve is constrained to the unit square other tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field extended to all temperatures below the critical one

for 2 d lattice spin systems weak mixing implies strong mixing we prove that for finite range discrete spin systems on the two dimensional latticez2 the weak mixing condition which follows for instance from the dobrushin shlosman uniqueness condition for the gibbs state implies a stronger mixing property of the gibbs state similar to the dobrushin shlosman complete analyticity condition but restricted to all squares in the lattice or more generally to all sets multiple of a large enough square the key observation leading to the proof is that a change in the boundary conditions cannot propagate either in the bulk because of the weak mixing condition or along the boundary because it is one dimensional as a consequence we obtain for ferromagnetic ising type systems proofs that several nice properties hold arbitrarily close to the critical temperature these properties include the existence of a convergent cluster expansion and uniform boundedness of the logarithmic sobolev constant and rapid convergence to equilibrium of the associated glauber dynamics on nice subsets ofz2 including the full lattice

on the layering transition of an sos surface interacting with a wall ii the glauber dynamics we continue our study of the statistical mechanics of a 2d surface above a fixed wall and attracted towards it by means of a very weak positive magnetic fieldh in the solid on solid sos approximation when the inverse temperature is very large in particular we consider a glauber dynamics for the above model and study the rate of approach to equilibrium in a large cube with arbitrary boundary conditions using the results proved in the first paper of this series we show that for allh h k 1 h k h k being the critical values of the magnetic field found in the previous paper the gap in the spectrum of the generator of the dynamics is bounded away from zero uniformly in the size of the box and in the boundary conditions on the contrary forh h k and free boundary conditions we show that the gap in a cube of sidel is bounded from above and from below by a negative exponential ofl our results provide a strong indication that contrary to what happens in two dimensions for the three dimensional dynamical ising model in a finite cube at low temperature and very small positive external field with boundary conditions that are opposite to the field on one face of the cube and are absent free on the remaining faces the rate of exponential convergence to equilibrium which is positive in infinite volume may go to zero exponentially fast in the side of the cube

on weak mixing in lattice models for lattice models on d weak mixing is the property that the influence of the boundary condition on a finite decays exponentially with distance from that region for a wide class of models on 2 including all finite range models we show that weak mixing is a consequence of gibbs uniqueness exponential decay of an appropriate form of connectivity and a natural coupling property in particular on 2 the fortuin kasteleyn random cluster model is weak mixing whenever uniqueness holds and the connectivity decays exponentially and the q state potts model above the critical temperature is weak mixing whenever correlations decay exponentially a hypothesis satisfied if q is sufficiently large ratio weak mixing is the property that uniformly over events a and b occurring on subsets and respectively of the lattice p a b p a p b 1 decreases exponentially in the distance between and we show that under mild hypotheses for example finite range weak mixing implies ratio weak mixing

coupling surfaces and weak bernoulli in one and higher dimensions abstract we propose a notion of weak bernoulli in all dimensions which generalizes the usual definition in dimension 1 the key idea is the concept of a coupling surface we relate this notion to previously studied properties and discuss a number of possible variants in dimension 1 we also show that the ising model at low temperature is weak bernoulli with an explicit description of the coupling surface

a new look at dimensional analysis from a group theoretical viewpoint the dimensionalisation hypothesis given in a previous paper 1981 is modified and established in a way which is as independent of arbitrary choices as possible the process of group contraction is also analysed as well as the relation between dimensions in the original and contracted group the theory is illustrated with the example of cayley klein geometries and an ultrarelativistic limit of the poincare group

kinematical superalgebras and lie algebras of order 3 we study and classify kinematical algebras which appear in the framework of lie superalgebras or lie algebras of order 3 all these algebras are related through generalized inonu wigner contractions from either the orthosymplectic superalgebra or the de sitter lie algebra of order 3

large deviations for gibbs random fields a large deviation principle for gibbs random fields on zd is proven and a corresponding large deviations proof of the gibbs variational formula is given a generalization of the lanford theory of large deviations is also obtained

almost gibbsianness and parsimonious description of the decimated 2d ising model in this paper we complete and provide details for the existing characterizations of the decimation of the ising model on mathbb z 2 in the generalized gibbs context we first recall a few features of the dobrushin program of restoration of gibbsianness and present the construction of global specifications consistent with the extremal decimated measures we use them to prove that these renormalized measures are almost gibbsian at any temperature and to analyse in detail its convex set of dlr measures we also recall the weakly gibbsian description and complete it using a potential that admits a quenched correlation decay i e a well defined configuration dependent length beyond which this potential decays exponentially we use these results to incorporate these decimated measures in the new framework of parsimonious random fields that has been recently developed to investigate probability aspects related to neurosciences

spectral gap and logarithmic sobolev inequality for kawasaki and glauber dynamics we prove that the spectral gap of the kawasaki dynamics shrink at the rate of 1 l2 for cubes of sizel provided that some mixing conditions are satisfied we also prove that the logarithmic sobolev inequality for the glauber dynamics in standard cubes holds uniformly in the size of the cube if the dobrushin shlosman mixing condition holds for standard cubes

complete analyticity of the 2d potts model above the critical temperature we investigate the complete analyticity ca of the two dimensional q state potts model for large values of q we are able to prove it for every temperature provided we restrict ourselves to nice subsets their niceness depending on the temperature t contrary to this restricted complete analyticity rca the full ca is known to fail for some values of the temperature above

rigorous probabilistic analysis of equilibrium crystal shapes the rigorous microscopic theory of equilibrium crystal shapes has made enormous progress during the last decade we review here the main results that have been obtained both in two and higher dimensions in particular we describe how the phenomenological wulff and winterbottom constructions can be derived from the microscopic description provided by the equilibrium statistical mechanics of lattice gases we focus on the main conceptual issues and describe the central ideas of the existing approaches

consistent estimation of joint distributions for sufficiently mixing random fields the joint distribution of a d dimensional random field restricted to a box of size k can be estimated by looking at a realization in a box of size n k and computing the empirical distribution this is done by sliding a box of size k around in the box of size n and computing frequencies we show that when k k n grows as a function of n then the total variation distance between this empirical distribution and the true distribution goes to 0 a s as n provided k n d log n d h e where h is the entropy of the random field and providing the random field satisfies a condition called quite weak bernoulli with exponential rate this class of processes studied previously includes the plus state for the ising model at a variety of parameter values and certain measures of maximal entropy for certain subshifts of finite type marton and shields have proved such results in one dimension and this paper is an attempt to extend their results to some extent to higher dimensions

finite volume glauber dynamics in a small magnetic field we consider glauber dynamics on a finite cube in d dimensional lattice d 2 which is associated with basic ising model at temperature t 1 1 under a magnetic field h 0 we prove that if the effective magnetic field is positive then the relaxation of the glauber dynamics in the uniform norm is exponentially fast uniformly over the size of underlying cube the result covers the case of the free boundary condition with arbitrarily small positive magnetic field this paper is a continuation of an attempt initiated earlier by schonmann and yoshida to shed more light on the relaxation of the finite volume glauber dynamics when the thermodynamic parameter h is so near the phase transition line h c h 0 that the dobrushin shlosman mixing condition is no longer available

on the ergodic properties of glauber dynamics we show that if there is an infinite volume gibbs measure which satisfies a logarithmic sobolev inequality with local coefficients of moderate growth then the corresponding stochastic dynamics decays to equilibrium exponentially fast in the uniform norm

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

ill defined block spin transformations at arbitrarily high temperatures examples are presented of block spin transformations which map the gibbs measures of the ising model in two or more dimensions at temperature intervals extending to arbitrarily high temperatures onto non gibbsian measures in this way we provide the first example of this kind of pathology for very high temperatures and as a corollary also the first example of such a pathology happening at a critical point

slow droplet driven relaxation of stochastic ising models in the vicinity of the phase coexistence region we consider the stochastic ising models glauber dynamics corresponding to the infinite volume basic ising model in arbitrary dimensiond 2 with nearest neighbor interaction and under a positive external magnetic fieldh under minimal assumptions on the rates of flip so that all the common choices are included we obtain results which state that when the system is at low temperaturet the relaxation time when the evolution is started with all the spins down blows up whenh 0 as exp t hd 1 the precise results are lower and upper bounds of this form moreover after a time which does not scale withh and before a time which also grows as an exponential of a multiple of 1 hd 1 ash 0 the law of the state of the process stays whenh is small close to the minus phase of the same ising model without an external field these results may be considered as a partial vindication of a conjecture raised by aizenman and lebowitz in connection to the metastable behavior of these stochastic ising models

relaxed criteria of the dobrushin shlosman mixing condition an interacting particle system glauber dynamics which evolves on a finite subset in thed dimensional integer lattice is considered it is known that a mixing property of the gibbs state in the sense of dobrushin and shlosman is equivalent to several very strong estimates in terms of the glauber dynamics we show that similar but seemingly much milder estimates are again equivalent to the dobrushin shlosman mixing condition hence to the original ones found by stroock and zegarlinski this may be understood as the absence of intermediate speed of convergence to equilibrium

graded cluster expansion for lattice systems in this paper we develop a general theory which provides a unified treatment of two apparently different problems the weak gibbs property of measures arising from the application of renormalization group maps and the mixing properties of disordered lattice systems in the griffiths phase we suppose that the system satisfies a mixing condition in a subset of the lattice whose complement is sparse enough namely large regions are widely separated we then show how it is possible to construct a convergent multi scale cluster expansion

absence of renormalization group pathologies near the critical temperature two examples we consider real space renormalization group transformations for ising type systems which are formally defined by exp left h sigma right sum limits sigma t sigma sigma exp left h sigma right wheret is a probability kernel i e t 1 for every configuration for each choice of the block spin configuration let let be the measure on spin configurations which is formally given by taking the probability of to be proportional tot exp h we give a condition which is sufficient to imply that the renormalized hamiltonianh is defined roughly speaking the condition is that the collection of measures is in the high temperature phase uniformly in the block spin configuration the proof of this result uses methods of olivieri and picco we use our theorem to prove that the first iteration of the renormalization group transformation is defined in the following two examples decimation with spacingb 2 on the square lattice with 1 36 c and the kadanoff transformation with parameterp on the trian gular lattice in a subset of the p plane that includes values of greater than c

complete analyticity for 2d ising completed we study the behavior of the two dimensional nearest neighbor ferromagnetic ising model under an external magnetic fieldh we extend to every subcritical value of the temperature a result previously proven by martirosyan at low enough temperature and which roughly states that for finite systems with boundary conditions under a positive external field the boundary effect dominates in the bulk if the linear size of the system is of orderb h withb small enough while ifb is large enough then the external field dominates in the bulk as a consequence we are able to complete the proof that complete analyticity for nice sets holds for every value of the temperature and external field in the interior of the uniqueness region in the phase diagram of the model the main tools used are the results and techniques developed to study large deviations for the block magnetization in the absence of the magnetic field and recently extended to all temperatures below the critical one by ioffe

non semisimple lie algebras with levi factor fraktur so 3 fraktur sfraktur l 2 bbb r and their invariants we analyse the number of functionally independent generalized casimir invariants for non semisimple lie algebras r with levi factors isomorphic to 3 and 2 in dependence of the pair r formed by a representation r of and a solvable lie algebra we show that for any dimension n 6 there exist lie algebras r with non trivial levi decomposition such that r 0

wulff droplets and the metastable relaxation of kinetic ising models we consider the kinetic ising models glauber dynamics corresponding to the infinite volume ising model in dimension 2 with nearest neighbor ferromagnetic interaction and under a positive external magnetic field h minimal conditions on the flip rates are assumed so that all the common choices are being considered we study the relaxation towards equilibrium when the system is at an arbitrary subcritical temperature t and the evolution is started from a distribution which is stochastically lower than the phase we show that as h 0 the relaxation time blows up as exp c t h with lgr c t w t 2 12 t m t here m t is the spontaneous magnetization and w t is the integrated surface tension of the wulff body of unit volume moreover for 0 c the state of the process at time exp h is shown to be close when h is small to the phase the difference between this state and the phase can be described in terms of an asymptotic expansion in powers of the external field this expansion can be interpreted as describing a set of continuations in h of the family of gibbs distributions with the negative magnetic fields into the region of positive fields

large deviations and concentration properties for ϕ interface models we consider the massless field with zero boundary conditions outside d n d d n n d a suitable subset of d i e the continuous spin gibbs measure n on d n with hamiltonian given by h x y x y 1 v x y and x 0 for x d n c the interaction v is taken to be strictly convex and with bounded second derivative this is a standard effective model for a d 1 dimensional interface represents the height of the interface over the base d n due to the choice of scaling of the base we scale the height with the same factor by setting n n we study various concentration and relaxation properties of the family of random surfaces n and of the induced family of gradient fields n n as the discretization step 1 n tends to zero n in particular we prove a large deviation principle for n and show that the corresponding rate function is given by d u x dx where is the surface tension of the model this is a multidimensional version of the sample path large deviation principle we use this result to study the concentration properties of n under the volume constraint i e the constraint that 1 n d x dn n x stays in a neighborhood of a fixed volume v 0 and the hard wall constraint i e n x 0 for all x this is therefore a model for a droplet of volume v lying above a hard wall we prove that under these constraints the field n of rescaled heights concentrates around the solution of a variational problem involving the surface tension as it would be predicted by the phenomenological theory of phase boundaries our principal result however asserts local relaxation properties of the gradient field n n to the corresponding extremal gibbs states thus our approach has little in common with traditional large deviation techniques and is closer in spirit to hydrodynamic limit type of arguments the proofs have both probabilistic and analytic aspects essential analytic tools are p estimates for elliptic equations and the theory of young measures on the side of probability tools a central role is played by the helffer sjostrand 31 pde representation for continuous spin systems which we rewrite in terms of random walk in random environment and by recent results of t funaki and h spohn 25 on the structure of gradient fields

a combinatorial proof of tree decay of semi invariants we consider finite range gibbs fields and provide a purely combinatorial proof of the exponential tree decay of semi invariants supposing that the logarithm of the partition function can be expressed as a sum of suitable local functions of the boundary conditions this hypothesis holds for completely analytical gibbs fields in this context the tree decay of semi invariants has been proven via analyticity arguments however the combinatorial proof given here can be applied also to the more complicated case of disordered systems in the so called griffiths phase when analyticity arguments fail

efficient detection of symmetries of polynomially parametrized curves we present efficient algorithms for detecting central and mirror symmetry for the case of algebraic curves defined by means of polynomial parametrizations the algorithms are based on the existence of a linear relationship between two proper polynomial parametrizations of the curve which leads to a triangular polynomial system with complex unknowns that can be solved in a very fast way in particular curves parametrized by polynomials of serious degrees can be analyzed in a few seconds in our analysis we provide a good number of theoretical results on symmetries of polynomial curves algorithms for detecting rotation and mirror symmetry and closed formulas to determine the symmetry center and the symmetry axis when they exist a complexity analysis of the algorithms is also given

on the speed of convergence for two dimensional first passage ising percolation consider first passage ising percolation on z2 let denote the reciprocal temperature and let h denote an external magnetic field denote by 8 the critical temperature and for j3 t let h f h sup h 0 3 h 0 where 0 8 h is the probability that the origin is connected by an infinite cluster with these definitions let us consider first passage ising percolation on z2 let ao 7 denote the first passage time from 0 0 to n 0 it follows from a subadditive argument that

coexistence of infinite clusters ii ising percolation in two dimensions we show a strong type of conditionally mixing property for the gibbs states ofd dimensional ising model when the temperature is above the critical one by using this property we show that there is always coexistence of infinite and clusters when is smaller than c andh 0 in two dimensions it is also possible to show that this coexistence region extends to some non zero external field case i e for every c there exists someh c 0 such that h h c implies the coexistence of infinite clusters with respect to the gibbs state for h

the asymmetric random cluster model and comparison of ising and potts models we introduce the asymmetric random cluster or arc model which is a graphical representation of the potts lattice gas and establish its basic properties the arc model allows a rich variety of comparisons in the fkg sense between models with different parameter values we give for example values h for which the 0 s configuration in the potts lattice gas is dominated by the configuration of the h ising model the potts model with possibly an external field applied to one of the spins is a special case of the potts lattice gas which allows our comparisons to yield rigorous bounds on the critical temperatures of potts models for example we obtain 0 571 1 exp c 0 600 for the 9 state potts model on the hexagonal lattice another comparison bounds the movement of the critical line when a small potts interaction is added to a lattice gas which otherwise has only interparticle attraction arc models can also be compared to related models such as the partial fk model obtained by deleting a fraction of the nonsingleton clusters from a realization of the fortuin kasteleyn random cluster model this comparison leads to bounds on the effects of small annealed site dilution on the critical temperature of the potts model

kinematic groups and dimensional analysis it is shown that group theory may be useful in relation to dimensional analysis the group theoretical support of dimensional analysis in universes described by kinematic groups is analysed its close relation to the structure of the corresponding kinematic group is also displayed by means of a simple dimensionalisation hypothesis the scheme of contractions relating the kinematic groups enables one to discuss the dimensionalisation method the same problem is also looked at in a different way the possibility of adding dilatation like transformations is studied finally the role of mass in both relativistic and non relativistic theories is examined

approach to equilibrium of glauber dynamics in the one phase region ii the general case we develop a new method based on renormalization group ideas block decimation procedure to prove under an assumption of strong mixing in a finite cube 0 a logarithmic sobolev inequality for the gibbs state of a discrete spin system as a consequence we derive the hypercontractivity of the markov semigroup of the associated glauber dynamics and the exponential convergence to equilibrium in the uniform norm in all volumes multiples of the cube 0

exponential relaxation of glauber dynamics with some special boundary conditions we consider attractive finite range glauber dynamics and show that if a certain mixing condition is satisfied then the system evolving on arbitrary subsets of the lattice with appropriate boundary conditions converges to equilibrium exponentially fast in the uniform sense uniformly over the subsets of the lattice this result applies for instance to the ferromagnetic nearest neighbor ising model in the so called basuev region where complete analyticity is expected to fail

on the two dimensional stochastic ising model in the phase coexistence region near the critical point we consider the two dimensional stochastic ising model in finite square with free boundary conditions at inverse temperature 0 and zero external field using duality and recent results of ioffe on the wulff construction close to the critical temperature we extend some of the results obtained by martinelli in the low temperature regime to any temperature below the critical one in particular we show that the gap in the spectrum of the generator of the dynamics goes to zero in the thermodynamic limit as an exponential of the side length of with a rate constant determined by the surface tension along one of the coordinate axes we also extend to the same range of temperatures the result due to shlosman on the equilibrium large deviations of the magnetization with free boundary conditions

exponential convergence under mixing we show that for a mixing stationary sequence of bounded random variables the average of the firstn variables converges exponentially fast withn to the mean value of these random variables

conditions for coincidence of two cubic bezier curves abstract this paper presents a necessary and sufficient condition for judging whether two cubic bezier curves are coincident two cubic bezier curves whose control points are not collinear are coincident if and only if their corresponding control points are coincident or one curve is the reversal of the other curve however this is not true for degree higher than 3 this paper provides a set of counterexamples of degree 4

conditions for the coincidence of two quartic bezier curves this paper presents efficient and necessary coincidence conditions for two quartic bezier curves i e either their two control polygons are coincident or the two curves can be reparameterized or degenerated into the same quadratic bezier curve

renormalization group transformations under strong mixing conditions gibbsianness and convergence of renormalized interactions in this paper we study a renormalization group map the block averaging transformation applied to gibbs measures relative to a class of finite range lattice gases when suitable strong mixing conditions are satisfied using a block decimation procedure cluster expansion and detailed comparison between statistical ensembles we are able to prove gibbsianness and convergence to a trivial i e gaussian and product fixed point our results apply to the 2d standard ising model at any temperature above the critical one and arbitrary magnetic field