complex temperature singularities in potts models on the square lattice we report some results on the complex temperature ct singularities of q state potts models on the square lattice we concentrate on the problematic region mathrm re a l0 where a e k in which ct zeros of the partition function are sensitive to finite lattice artifacts from analyses of low temperature series expansions for 3l ql 8 we establish the existence in this region of complex conjugate ct singularities at which the magnetization and susceptibility diverge from calculations of zeros of the partition function we obtain evidence consistent with the inference that these singularities occur at endpoints a e a e of arcs protruding into the complex temperature extension of the ferromagnetic phase exponents for these singularities are determined e g for q 3 we find ensuremath beta e ensuremath 0 125 1 consistent with ensuremath beta e ensuremath frac 1 8 by duality these results also imply associated arcs extending into the ct extension of the symmetric paramagnetic phase analytic expressions are suggested for the positions of some of these singularities e g for q 5 our finding is consistent with the exact value a e a e 2 ensuremath 1 ensuremath mp i further discussions of complex temperature phase diagrams are given

zeroes of the triangular potts model partition function a conjectured distribution abstract we consider the q state potts model on the triangular lattice with two and three site interactions in alternate triangular faces and determine zeroes of the partition function numerically in the case of pure three site interactions on the basis of a rigorous reciprocal symmetry and results on the zeroes for finite lattices we conjecture that zeroes of the partition function of the triangular potts model with pure three site interactions in alternate triangular faces lie on a circle and a segment of the negative real axis it is shown that the conjecture holds for q 2 and that it reproduces the known critical point for general q including the q 1 site percolation

critical frontier of the potts and percolation models on triangular type and kagome type lattices i closed form expressions we consider the potts model and the related bond site and mixed site bond percolation problems on triangular type and kagome type lattices and derive closed form expressions for the critical frontier for triangular type lattices the critical frontier is known usually derived from a duality consideration in conjunction with the assumption of a unique transition our analysis however is rigorous and based on an established result without the need of a uniqueness assumption thus firmly establishing all derived results for kagome type lattices the exact critical frontier is not known we derive a closed form expression for the potts critical frontier by making use of a homogeneity assumption the closed form expression is unique and we apply it to a host of problems including site bond and mixed site bond percolations on various lattices it yields exact thresholds for site percolation on kagome martini and other lattices and is highly accurate numerically in other applications when compared to numerical determination

a rigorous theory of finite size scaling at first order phase transitions a large class of classical lattice models describing the coexistence of a finite number of stable states at low temperatures is considered the dependence of the finite volume magnetizationm per h l in cubes of sizel d under periodic boundary conditions on the external fieldh is analyzed for the case where two phases coexist at the infinite volume transition pointh t we prove that independent of the details of the model the finite volume magnetization per lattice site behaves likem per h t m tanh ml d h ht withm per h denoting the infinite volume magnetization and m 1 2 m per h t 0 m per h t 0 introducing the finite size transition pointh m l as the point where the finite volume susceptibility attains the maximum we show that in the case of asymmetric field driven transitions its shift ish t h m l o l 2d in contrast to claims in the literature starting from the obvious observation that the number of stable phases has a local maximum at the transition point we propose a new way of determining the pointh t from finite size data with a shift that is exponentially small inl finally the finite size effects are discussed also in the case where more than two phases coexist

complex temperature phase diagrams for the q state potts model on self dual families of graphs and the nature of the q limit exact calculations of the potts model partition function z g q v have been presented for arbitrary q and temperature like variable v on self dual strip graphs g of the square lattice with fixed width l y and arbitrarily great length lx with two types of boundary conditions letting lx infinity the resultant free energy and complex temperature phase diagram have been computed including the locus b where the free energy is nonanalytic results are analyzed for widths l y 1 2 3 these results have been used to study the approach to the large q limit of b

partition function zeros of a restricted potts model on lattice strips and effects of boundary conditions we calculate exactly the partition function z g q v of the q state potts model with temperature like boltzmann variable v for strip graphs g of the square and triangular lattices of various widths ly and arbitrarily great lengths lx with a variety of boundary conditions and with q and v restricted to satisfy conditions corresponding to the ferromagnetic phase transition on the associated two dimensional lattices from these calculations in the limit lx we determine the continuous accumulation loci of the partition function zeros in the v and q planes strips of the honeycomb lattice are also considered we discuss some general features of these loci

about the critical condition of the q state potts model on the anisotropic hypercubic lattice a critical condition is obtained for the ferromagnetic model on the hypercubic lattice in d dimensions with different potts interactions along the different lattice axes it is done by extending a simple procedure which is shown to be exact for the square lattice in d 2 dimensions

zeros of jones polynomials for families of knots and links abstract we calculate jones polynomials v l t for several families of alternating knots and links by computing the tutte polynomials t g x y for the associated graphs g and then obtaining v l t as a special case of the tutte polynomial for each of these families we determine the zeros of the jones polynomial including the accumulation set in the limit of infinitely many crossings a discussion is also given of the calculation of jones polynomials for non alternating links

yang lee zeros of the potts model the yang lee zeros of the three component ferromagnetic potts model in one dimension in the complex plane of an applied field are determined the phase diagram consists of a triple point where three phases coexist emerging from the triple point are three lines on which two phases coexist and which terminate at critical points yang lee edge singularity the zeros do not all lie on the imaginary axis but along the three two phase lines the model can be generalized to give rise to a tricritical point which is a new type of yang lee edge singularity gibbs phase rule is generalized to apply to coexisting phases in the complex plane

flow polynomials and their asymptotic limits for lattice strip graphs we present exact calculations of flow polynomials f g q for lattice strips of various fixed widths l y 4 and arbitrarily great lengths l x with several different boundary conditions square honeycomb and triangular lattice strips are considered we introduce the notion of flows per face fl in the infinite length limit we study the zeros of f g q in the complex q plane and determine exactly the asymptotic accumulation sets of these zeros in the infinite length limit for the various families of strips the function fl is nonanalytic on this locus the loci are found to be noncompact for many strip graphs with periodic or twisted periodic longitudinal boundary conditions and compact for strips with free longitudinal boundary conditions we also find the interesting feature that aside from the trivial case l y 1 the maximal point q cf where crosses the real axis is universal on cyclic and mobius strips of the square lattice for all widths for which we have calculated it and is equal to the asymptotic value q cf 3 for the infinite square lattice

a mapping relating complex and physical temperatures in the 2d q state potts model and some applications we show an exact equivalence of the free energy of the q state potts antiferromagnet on a lattice for the full temperature interval and the free energy of the q state potts model on the dual lattice for a semi infinite interval of complex temperatures cts this implies the existence of two quite different types of ct singularities the generic kind which does not obey universality or various scaling relations and a special kind which does obey such properties and encodes information of direct physical relevance we apply this observation to characterize ct properties of the potts model on several lattices to rule out two existing conjectures and to determine the critical value of q above which the potts antiferromagnet on the diced lattice has no phase transition

s4 symmetry on the checkerboard potts model the large q expansion of the partition function z of the checkerboard q state potts model with a magnetic field is given up to sixth order in q 12 up to this order z is shown to exhibit an unexpected s4 symmetry using various arguments exact results expansion etc the authors suggest the general validity of this new symmetry property of the considered model an exact expression for the magnetisation jump is also proposed and checked on the large q expansion

the partition function zeros for a potts model of helix coil transition with three site interactions abstract the partition function fisher zeros of the q state potts model with three site interactions on a zigzag ladder are studied by a transfer matrix approach this q state model is proposed as a description of helix coil transition in homopolymers the fisher zeros are exactly derived for arbitrary value of the system parameter q it is shown that there is only a quasi phase transition for all temperatures the dependence of the helix coil transition on solvent is analyzed using fisher zeros of the model the densities of the fisher zeros are singular at the edge singularity points and the critical exponent 1 2

exact potts model partition function on strips of the triangular lattice abstract in this paper we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex strip graphs of width ly 2 and arbitrary length of the triangular lattice with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of the graphs and the thermodynamics is discussed considering the full generalization to arbitrary complex q and temperature we determine the singular locus b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros comparisons are made with our previous exact calculation of potts model partition functions for the corresponding strips of the square lattice our present calculations yield as special cases several quantities of graph theoretic interest

spanning forests and the q state potts model in the limit q 0 we study the q state potts model with nearest neighbor coupling v e j 1 in the limit q v 0 with the ratio w v q held fixed combinatorially this limit gives rise to the generating polynomial of spanning forests physically it provides information about the potts model phase diagram in the neighborhood of q v 0 0 we have studied this model on the square and triangular lattices using a transfer matrix approach at both real and complex values of w for both lattices we have computed the symbolic transfer matrices for cylindrical strips of widths 2 l 10 as well as the limiting curves b of partition function zeros in the complex w plane for real w we find two distinct phases separated by a transition point w w0 where w0 1 4 resp w0 0 1753 0 0002 for the square resp triangular lattice for w w0 we find a non critical disordered phase that is compatible with the predicted asymptotic freedom as w for w w0 our results are compatible with a massless berker kadanoff phase with central charge c 2 and leading thermal scaling dimension xt 1 2 marginally irrelevant operator at w w0 we find a first order critical point the first derivative of the free energy is discontinuous at w0 while the correlation length diverges as w w0 and is infinite at w w0 the critical ehavior at w w0 seems to be the same for both lattices and it differs from that of the berker kadanoff phase our results suggest that the central charge is c 1 the leading thermal scaling dimension is xt 1 0 and the critical exponents are 1 d 1 2 and 1

the ferro antiferromagnetic q state potts model the critical properties of the mixed ferro antiferromagnetic q state potts model on the square lattice are investigated using the numerical transfer matrix technique the transition temperature is found to be substantially lower than previously found for q 3 it is conjectured that there is no transition for q 3 in contradiction with previous results

multi domain simulations of the time dependent navier stokes equations benchmark error analysis of some nesting procedures this paper presents a benchmark error analysis of various approaches for treating multiple domain calculations within an anelastic finite difference model one way and two way interactive nesting errors with and without temporal refinement are evaluated the two way interactive nesting approach is one where solutions between fine and coarse grid domains are matched through the simple post insertion of data on the other hand the equations can be matched by using the pressure defect correction approach it is shown that for the present model the two way interactive nesting method gives identical results to multi domain solutions using the pressure defect correction approach the present results indicate that in this type of analastic framework a priori matching of the equations is equivalent to the a posteriori matching of the solutions this result is attributed to the inflexible nature of the neumann boundary conditions on the fine mesh pressure which need to be specified from the coarse mesh since a large number of meterological models employ the hydrostatic assumption it is of interest to know of nesting errors attributable to this approximation the results presented indicate essentially equivalent error levels for both the hydrostatic and nonhydrostatic systems of equations for the present case of airflow over an isolated mountain it is shown how nesting technology can be used in a virtual sense to reduce the central memory requirements for large array sized numerical simulations nesting can be used in this sense to decompose the maximum memory working space required without affecting the results

generalized chebyshev polynomials associated with affine weyl groups we begin with a compact figure that can be folded into smaller replicas of itself such as the interval or equilateral triangle such figures are in one to one correspondence with affine weyl groups for each such figure in n dimensional euclidean space we construct a sequence of polynomials pk rn rn so that the mapping pk is conjugate to stretching the figure by a factor k and folding it back onto itself if n 1 and the figure is the interval this construction yields the chebyshev polynomials up to conjugation the polynomials pk are orthogonal with respect to a suitable measure and can be extended in a natural way to a complete set of orthogonal polynomials

topology on words we investigate properties of topologies on sets of finite and infinite words over a finite alphabet the guiding example is the topology generated by the prefix relation on the set of finite words considered as a partial order this partial order extends naturally to the set of infinite words hence it generates a topology on the union of the sets of finite and infinite words we consider several partial orders which have similar properties and identify general principles according to which the transition from finite to infinite words is natural we provide a uniform topological framework for the set of finite and infinite words to handle limits in a general fashion

finite size scaling for potts models recently borgs and koteck developed a rigorous theory of finite size effects near first order phase transitions here we apply this theory to the ferromagneticq state potts model which forq large andd 2 undergoes a first order phase transition as the inverse temperature is varied we prove a formula for the internal energy in a periodic cube of side lengthl which describes the rounding of the infinite volume jump e in terms of a hyperbolic tangent and show that the position of the maximum of the specific heat is shifted by m l inq e l d o l 2d with respect to the infinite volume transition point t we also propose an alternative definition of the finite volume transition temperature t l which might be useful for numerical calculations because it differs only by exponentially small corrections from t

hyperbolic coxeter groups for triangular potts models the symmetry groups generated by the inversion relations of lattice models of statistical mechanics on triangular lattices are analysed for vertex models and for the standard scalar potts model with two and three site interactions these groups are generated by three inversion relations and are seen to be generically very large ones hyperbolic groups two situations for which the representations of these groups degenerate into smaller ones hopefully compatible with integrability are considered the first reduction for the vertex triangular model corresponds to the situation where the vertex of the triangular model coincides with the left or right hand side of a yang baxter relation in this case the representation of the group is isomorphic up to a semi direct product by a finite group to z z the second reduction for q state ports models occurs for particular values of q the so called tutte beraha numbers for this model algebraic varieties including the known ferromagnetic critical variety happen to be invariant under such large groups of symmetries as a byproduct this analysis provides nice birational representations of hyperbolic coxeter groups

exact results for a checkerboard ising model with crossing and four spin interactions a seven parameter ising model with crossing and four spin interactions on a checkerboard lattice is mapped by a duality transformation onto a general eight vertex model which has only five independent parameters in a six dimensional subspace of the ising model for which the corresponding eight vertex model satisfies the free fermion condition an exact solution is found

scaling and density of lee yang zeros in the four dimensional ising model the scaling behaviour of the edge of the lee yang zeroes in the four dimensional ising model is analyzed this model is believed to belong to the same universality class as the phi 4 4 model which plays a central role in relativistic quantum field theory while in the thermodynamic limit the scaling of the yang lee edge is not modified by multiplicative logarithmic corrections such corrections are manifest in the corresponding finite size formulae the asymptotic form for the density of zeroes which recovers the scaling behaviour of the susceptibility and the specific heat in the thermodynamic limit is found to exhibit logarithmic corrections too the density of zeroes for a finite size system is examined both analytically and numerically

fa yueh wu s contributions in physics abstract we sketch a subset of professor f y wu s contributions in lattice statistical mechanics solid state physics graph theory enumerative combinatorics and other domains of physics and mathematics we will recall some of f y wu s most important and well known classic results and we will also sketch some of his most recent researches dedicated to the connections of lattice statistical mechanical models with deep problems in pure mathematics

phase diagram of a six state chiral potts model abstract we study the phase diagram of an isotropic six state chiral potts model on a square lattice by means of both exact and numerical methods the phase diagram of this model presents many similarities with the phase diagrams of the ashkin teller model or the models studied by zamolodchikov and monarstirskii a remarkable line globally invariant under a transformation generalizing the kramers wannier duality seems to correspond to a first order transition line up to a bifurcation point where this line splits into two second order lines all the numerical calculations are compared with exact results which can be performed using a canonical elliptic parametrization of this model the bifurcation point is found to correspond to the intersection of a generalized self dual line with an algebraic curve this curve corresponds to the set of points of the phase diagram for which a non trivial infinite symmetry group of the model degenerates into a finite group of order six the agreement between numerical and analytical results is very good

thermal transmissivity in discrete spin systems formulation and applications thermal transmissivity a notion which arises empirically and leads to considerable simplification of analyses of discrete spin systems is of fundamental interest in graphical and real space renormalization group approaches the authors present a formulation of the theory of thermal transmissivity they show that the situation is different depending on whether the interaction matrices commute in the commuting case the thermal transmissivity is given by the eigenvalues of the interaction matrix while in the non commuting case it is given by the eigenvalues as well as matrices which block diagonalize the interaction matrix the meaning of the thermal transmissivity in diagrammatic analyses of spin models is also elucidated and their results are illustrated by examples finally they present a general formulation of disorder solutions for spin models in terms of thermal transmissivities

inversion relations and disorder solutions on potts models the inversion symmetry and the automorphy group generated by the latter when combined with the spatial symmetries are studied for the anisotropic potts models on the triangular and checkerboard lattices the exact expression of the partition function which is known on some particular disorder subvarieties of the triangular model is checked and a generalisation is proposed for the checkerboard lattice the automorphy group is then used to extend the disorder solutions to the infinity of transformed subvarieties

on the density of partition function temperature zeros the density of partition function temperature zeros for the two dimensional spatially anisotropic ising model in the absence of a magnetic field satisfies a linear homogeneous partial differential equation a generalised form of this equation for a non zero specific heat exponent can be derived by applying non linear scaling to the expression for the specific heat as an integral over the density of zeros

valuations of languages with applications to fractal geometry abstract valuations morphisms from to 0 1 are a simple generalization of bernoulli morphisms distributions measures as introduced by berstel and perrin 1988 berstel and reutenauer 1988 blanchard and hansel 1984 eilenberg 1974 and hansel and perrin 1983 1989 this paper shows that valuations are useful not only within the theory of codes but also when dealing with ambiguity especially in context free grammars or for defining outer measures on the space of words which are of some importance to the theory of fractals these connections yield new formulae to determine the hausdorff dimension of fractal sets especially in euclidean spaces defined via formal languages the class of fractals describable with context free languages strictly includes that of mrfs fractals introduced by culik and dube 1990 and mauldin and williams 1988 some of the results of this paper also appear as part of the author s ph d thesis and other works fernau 1992a b 1993

landau singularities and singularities of holonomic integrals of the ising class we consider families of multiple and simple integrals of the ising class and the linear ordinary differential equations with polynomial coefficients they are solutions of we compare the full set of singularities given by the roots of the head polynomial of these linear odes and the subset of singularities occurring in the integrals with the singularities obtained from the landau conditions for these ising class integrals we show that the landau conditions can be worked out either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral the singular behaviour of these integrals is obtained in the self dual variable w s 2 1 s2 with s sinh 2k where k j kt is the usual ising model coupling constant switching to the variable s we show that the singularities of the analytic continuation of series expansions of these integrals actually break the kramers wannier duality we revisit the singular behaviour zenine et al 2005 j phys a math gen 38 9439 74 of the third contribution to the magnetic susceptibility of ising model 3 at the points 1 3w 4w2 0 and show that 3 s is not singular at the corresponding points inside the unit circle s 1 while its analytical continuation in the variable s is actually singular at the corresponding points 2 s s2 0 outside the unit circle s 1

the hyperdeterminant and triangulations of the 4 cube peter huggins bernd sturmfels josephine yu and debbie s yusterabstract the hyperdeterminant of format 2 2 2 2 is a polynomialof degree 24 in 16 unknowns which has 2894276 terms we compute thenewton polytope of this polynomial and the secondary polytope of the 4 cube the 87959448 regular triangulations of the 4 cube are classi ed into25448 d equivalence classes one for each vertex of the newton polytope the 4 cube has 80876 coarsest regular subdivisions one for each facet of thesecondary polytope but only 268 of them come from the hyperdeterminant

zeros of the partition function for the two dimensional ising model the zeros of the partition function for the zero field two dimensional ising model in the complex temperature plane are represented in terms of the eigenvalues of the transfer matrix the boundaries of the limiting distribution have a particular simple representation in terms of the correlation range along the two principal axes of the quadratic lattice examples of the distribution in the complex plane of z exp 2k are given for the anisotropic model when the horizontal and vertical interactions are in integer ratio

duality and fisher zeros in the two dimensional potts model on a square lattice a phenomenological approach to the ferromagnetic two dimensional 2d potts model on square lattice is proposed our goal is to present a simple functional form that obeys the known properties possessed by the free energy of the q state potts model the duality symmetry of the 2d potts model together with the known results on its critical exponent ensuremath alpha allows us to fix consistently the details of the proposed expression for the free energy the agreement of the analytic ansatz with numerical data in the q 3 case is very good at high and low temperatures as well as at the critical point it is shown that the qg4 cases naturally fit into the same scheme and that one should also expect a good agreement with numerical data the limiting q 4 case is shortly discussed

a fully multidimensional positive definite advection transport algorithm with small implicit diffusion in numerical modeling of physical phenomena it is often necessary to solve the advective transport equation for positive definite scalar functions numerical schemes of second or higher order accuracy can produce negative values in the solution due to the dispersive ripples lower order schemes such as the donor cell or lax friedrichs or higher order schemes with zeroth order diffusion added produce no ripples but suffer from excessive implicit diffusion in the last ten years a possible resolution of this dilemma has been developed in the form of hybrid schemes in which the advective fluxes are given as a weighted average of a first order positive definite scheme s fluxes and a higher order scheme s fluxes the difference in determination of the weights in the calculation of the average advective fluxes has led to different hybrid schemes two main hybrid type schemes have been developed one the so called flux corrected transport fct method was originated by boris and book i 31 and generalized by zalesak 141 the other was developed by harten and zwas 6 9 in the form of the self adjusting hybrid schemes sahs method both methods were constructed to deal effectively with shocks and contact discontinuities solutions of the advection transport equation obtained by using fct or sahs maintain positive definiteness of the initial condition and as can be seen from presented tests zalesak 141 harten 6 be very accurate unfortunately application of these methods to the modeling of complex multidimensional hydrodynamical systems like atmospheric phenomena is rather limited due to the excessive computer time required furthermore in many hydrodynamical systems

boundary conditions and the free energy for statistical mechanical models the author shows that the choice of boundary conditions can significantly affect the calculation of the free energy for discrete statistical mechanical models on finite or semi infinite lattices when the exponentiated coupling is extended into the complex plane he shows however that the boundary conditions most consistent with boundary configuration independence in the direction in which a lattice is infinite give a unique analytic function for the free energy this unique function is the free energy most appropriate for extrapolation to the full thermodynamic limit the author shows that such boundary conditions are orthogonal to all but one degenerate ferromagnetic ground states at zero temperature

zeroes of the jones polynomial abstract we study the distribution of zeroes of the jones polynomial v k t for a knot k we have computed numerically the roots of the jones polynomial for all prime knots with n 10 crossings and found the zeroes scattered about the unit circle t 1 with the average distance to the circle approaching a nonzero value as n increases for torus knots of the type m n we show that all zeroes lie on the unit circle with a uniform density in the limit of either m or n a fact confirmed by our numerical findings we have also elucidated the relation connecting the jones polynomial with the potts model and used this relation to derive the jones polynomial for a repeating chain knot with 3 n crossings for general n it is found that zeroes of its jones polynomial lie on three closed curves centered about the points 1 i and i in addition there are two isolated zeroes located one each near the points t e 2 i 3 at a distance of the order of 3 n 2 2 closed form expressions are deduced for the closed curves in the limit of n

three state chiral potts models in two dimensions integrability and symmetry abstract symmetry and integrability relations for the three state chiral potts model on two dimensional lattices are reviewed our detailed and systematic analysis leads to new integrability varieties for specific chiral potts models including new integrable points for the standard antiferromagnetic potts model we also describe a three state vertex model representation of the chiral potts model and write down the associated fundamental invariants for three coordinated lattices

holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

some analytical consequences of the inverse relation for the potts model the authors present some analytical results for the potts model using the symmetry group acting on its parameters generated by the inverse relation and other symmetries of this model in particular they find the critical manifolds and study the relationship between this group and the lee yang singularities in the complex plane for those cases where the symmetry group is finite they look at the possible consequences for some colouring problems in graph theory and more specifically for chromatic polynomials

free fermion checkerboard and z invariant lattice models in statistical mechanics it is shown that the two dimensional free fermion model is equivalent to a checkerboard ising model which is a special case of the general z invariant ising model expressions are given for the partition function and local correlations in terms of those of the regular square lattice ising model corresponding results are given for the self dual potts model and the application of the methods to the three dimensional zamolodchikov model is discussed the paper ends with a discussion of the critical and disorder surfaces of the checkerboard potts model

location of zeros in the complex temperature plane absence of lee yang theorem in the yang lee theory 1952 the occurrence of a phase transition at real physical values of the magnetic field is related to the behaviour of the density of zeros of the partition function in the complex magnetic field plane as shown by lee and yang these zeros fall on lines for a wide class of ferromagnetic models in extensions of the yang lee theory to the complex temperature plane it has often been assumed that the zeros of the partition function would again fall on lines rather than in areas though this happens to be true for the isotropic ising model the authors point out that there is in general no basis for this assumption they discuss the location of the zeros of the partition function of the anisotropic ising model in the complex temperature plane and show that they indeed always fall in areas

fisher zeros and singular behaviour of the two dimensional potts model in the thermodynamic limit the duality transformation is applied to the fisher zeros near the ferromagnetic critical point in the q 4 state two dimensional potts model a requirement that the locus of the duals of the zeros be identical to the dual of the locus of zeros in the thermodynamic limit i recovers the ratio of specific heat to internal energy discontinuity at criticality and the relationships between the discontinuities of higher cumulants and ii identifies duality with complex conjugation for the zeros near the ferromagnetic critical point the conjecture that all zeros governing ferromagnetic singular behaviour satisfy the latter requirement gives the full locus of such fisher zeros to be a circle this locus together with the density of zeros is then shown to be sufficient to recover the singular part of the thermodynamic functions in the thermodynamic limit their regular parts coming from separate loci of zeros not crossing the positive real temperature axis

complex temperature singularities in the d 2 ising model triangular and honeycomb lattices we study complex temperature singularities of the ising model on the triangular and honeycomb lattices we first discuss the complex t phases and their boundaries from exact results we determine the complex t singularities in the specific heat and magnetization for the triangular lattice we discuss the implications of the divergence of the magnetization at the point u d 1 where u d z 2 d e 4k and extend a previous study by guttmann of the susceptibility at this point with the use of differential approximants for the honeycomb lattice from an analysis of low temperature series expansions we have found evidence that the uniform and staggered susceptibilities n and n a both have divergent singularities at z d 1 z and our numerical values for the exponents are consistent with the hypothesis that the exact values are 0 d 0 a d 5 2 the critical amplitudes at this singularity were calculated using our exact results for 0 and together with numerical values for 0 from series analyses we find that the exponent relation 0 c 2 c 0 d 2 is violated at z d 1 on the honeycomb lattice the right hand side is consistent with being equal to 4 rather than 2 the connections of the critical exponents at these two singularities on the triangular and honeycomb lattice are discussed

the failure distribution in percolation models of breakdown the probability of finding a largest defect cluster of size n in a percolation network is calculated analytically using a new distribution function scaling equation from this result and the stress voltage enhancement at the tip of the most critical defect in the network the probability of failure in percolation models of breakdown is calculated for defect fractions less than the percolation point this distribution is found to be of the form exponential of an exponential numerical simulations on the two dimensional random fuse network confirm the new distribution function and convincingly distinguish between it and the weibull 1951 form most often used in the fitting of breakdown data

order and disorder lines in systems with competing interactions iii exact results from stochastic crystal growth the methods presented in the first two articles of this series are simplified and generalized by growing stationary stochastic crystals from a given ansatz layer on the disorder trajectory the free energy correlation functions and multicritical points are calculated explicitly for a large class of models with competing interactions including the staggered eight vertex model the general sixteen vertex model theq state potts model on a triangular lattice a generalz q model and restricted spin glass models in two dimensions

group velocity in finite difference schemes the relevance of group velocity to the behavior of finite difference models of time dependent partial differential equations is surveyed and illustrated applications involve the propagation of wave packets in one and two dimensions numerical dispersion the behavior of parasitic waves and the stability analysis of initial boundary value problems

inversion relations and symmetry groups for potts models on the triangular lattice inversion relations are obtained for the standard scalar q state triangular potts model with two and three spin interactions generalizing previously known results for two spin interaction models it is shown that these inversion relations generate a group of symmetries of the model which is naturally represented in terms of birational transformations in a four dimensional parameter space this group of birational transformations is generically a very large one namely a hyperbolic coxeter group in this framework of very large groups of symmetries a remarkable situation pops out the one for which q corresponds to tutte beraha numbers

algebraic approximations to the locus of partition function zeros applications of the mathematical formalism put forward in the previous paper see ibid p 3471 1987 are made to a number of well known lattice models in statistical mechanics sequences of algebraic curves are obtained from the branches of an algebraic function lambda 1 where lambda 1 at real temperatures is the branch of lambda 1 which is the partition function per site of an m infinity lattice section these sequences are viewed as approximations to parts of the limiting locus cinfinity of partition function zeros approximations to critical points can be obtained in a natural extension of these curves beyond the branch points of lambda 1

self consistent scaling theory for logarithmic correction exponents multiplicative logarithmic corrections frequently characterize critical behavior in statistical physics here a recently proposed theory relating the exponents of such terms is extended to account for circumstances which often occur when the leading specific heat critical exponent vanishes also the theory is widened to encompass the correlation function the new relations are then confronted with results from the literature and some new predictions for logarithmic corrections in certain models are made

generalized partition function zeros of 1d spin models and their critical behavior at edge singularities here we study the partition function zeros of the one dimensional blume emery griffiths model close to their edge singularities the model contains four couplings h j k including the magnetic field h and the ising coupling j we assume that only one of the three couplings j k is complex and the magnetic field is real the generalized zeros zi tend to form continuous curves on the complex z plane in the thermodynamic limit the linear density at the edges ze diverges usually with z z ze and 1 2 however as in the case of complex magnetic fields yang lee edge singularity if we have a triple degeneracy of the transfer matrix eigenvalues a new critical behavior with 2 3 can appear as we prove here explicitly for the cases where either or k is complex our proof applies for a general three state spin model with short range interactions the fisher zeros complex j are more involved in practice we have not been able to find an explicit example with 2 3 as far as the other couplings h k are kept as real numbers our results are supported by numerical computations of zeros we show that it is absolutely necessary to have a non vanishing magnetic field for a new critical behavior the appearance of 2 3 at the edge closest to the positive real axis indicates its possible relevance for tricritical phenomena in higher dimensional spin models

selected non holonomic functions in lattice statistical mechanics and enumerative combinatorics we recall that the full susceptibility series of the ising model modulo powers of the prime reduce to algebraic functions we also recall the nonlinear polynomial differential equation obtained by tutte for the generating function of the q coloured rooted triangulations by vertices which is known to have algebraic solutions for all the numbers of the form the holonomic status of being unclear we focus on the analysis of the case showing that the corresponding series is quite certainly non holonomic along the line of a previous work on the susceptibility of the ising model we consider this series modulo the first eight primes and show that this probably non holonomic function reduces modulo these primes to algebraic functions we conjecture that this probably non holonomic function reduces to algebraic functions modulo almost every prime or power of prime numbers this raises the question of whether such remarkable non holonomic functions can be seen as a ratio of diagonals of rational functions or even algebraic functions of diagonals of rational functions

infinite word languages and continuous mappings abstract languages with infinite words have often been studied as subsets of a topological space the topology used for this purpose was always the natural topology of sequences this paper suggests that another topology may be more useful it is obtained by taking lim w as the derived set of w the resulting topological space is shown to be completely regular strongly zero dimensional and not normal the languages appearing in the theory of automata are shown to be closely related to functionally closed sets continuous inverse images of the set 0 as a consequence a number of known results can be expressed and proved in terms of continuous mappings

some remarks on non algebraic adherences we present some properties of non algebraic adherences of languages which can be consistently called context sensitive adherences because it is proved that in the chomsky hierarchy of adherences context sensitive and recursive enumerable adherences coincide concerning the center mapping the traversing from algebraic to non algebraic takes simple recursive to not necessarily recursive enumerable except for the last equality the chomsky hierarchy of adherences is proper radh cfadh csadh readh and moreover it can be refined in the non algebraic case by considering the mcnaughton nivat adherences regarding the programming context adherences occur when we consider the set of divergent computations shapes of a program or a program scheme in this respect the interpretation is powerful enough to translate radh to csadh

random cluster models on the triangular lattice we study percolation and the random cluster model on the triangular lattice with 3 body interactions starting with percolation we generalize the star triangle transformation we introduce a new parameter the 3 body term and identify configurations on the triangles solely by their connectivity in this new setup necessary and sufficient conditions are found for positive correlations and this is used to establish regions of percolation and non percolation next we apply this set of ideas to the q 1 random cluster model we derive duality relations for the suitable random cluster measures prove necessary and sufficient conditions for them to have positive correlations and finally prove some rigorous theorems concerning phase transitions

discrete symmetry groups of vertex models in statistical mechanics we analyze discrete symmetry groups of vertex models in lattice statistical mechanics represented as groups of birational transformations they can be seen as generated by involutions corresponding respectively to two kinds of transformations onq q matrices the inversion of theq q matrix and an involutive permutation of the entries of the matrix we show that the analysis of the factorizations of the iterations of these transformations is a precious tool in the study of lattice models in statistical mechanics this approach enables one to analyze two dimensionalq4 state vertex models as simply as three dimensional vertex models or higher dimensional vertex models various examples of birational symmetries of vertex models are analyzed a particular emphasis is devoted to a three dimensional vertex model the 64 state cubic vertex model which exhibits a polynomial growth of the complexity of the calculations a subcase of this general model is seen to yield integrable recursion relations we also concentrate on a specific two dimensional vertex model to see how the generic exponential growth of the calculations reduces to a polynomial growth when the model becomes yang baxter integrable it is also underlined that a polynomial growth of the complexity of these iterations can occur even for transformations yielding algebraic surfaces or higher dimensional algebraic varieties

study of the potts model on the honeycomb and triangular lattices low temperature series and partition function zeros we present and analyse low temperature series and complex temperature partition function zeros for the q state potts model with q 4 on the honeycomb lattice and q 3 4 on the triangular lattice a discussion is given on how the locations of the singularities obtained from the series analysis correlate with the complex temperature phase boundary extending our earlier work we include a similar discussion for the potts model with q 3 on the honeycomb lattice and with q 3 4 on the kagom lattice

the multidimensional positive definite advection transport algorithm further development and applications abstract this paper presents further generalizations of the advection algorithm described in j comput phys 54 1984 325 time dependent velocity fields and a generalized form of the continuity equation are considered applicability of the algorithm to the diffusion equation and to the transport of nonpositive scalars is also discussed theoretical considerations are illustrated through numerical tests and applications to some particular geophysical fluid dynamics problems

the ising model from elliptic curves to modular forms and calabi yau equations we show that almost all the linear differential operators factors obtained in the analysis of the n particle contributions s of the susceptibility of the ising model for n 6 are linear differential operators associated with elliptic curves beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral e the second and third order differential operators z2 f2 f3 can actually be interpreted as modular forms of the elliptic curve of the ising model a last order 4 globally nilpotent linear differential operator is not reducible to this elliptic curve modular form scheme this operator is shown to actually correspond to a natural generalization of this elliptic curve modular form scheme with the emergence of a calabi yau equation corresponding to a selected 4f3 hypergeometric function this hypergeometric function can also be seen as a hadamard product of the complete elliptic integral k with a remarkably simple algebraic pull back square root extension the corresponding calabi yau fourth order differential operator having a symplectic differential galois group the mirror maps and higher order schwarzian odes associated with this calabi yau ode present all the nice physical and mathematical ingredients we had with elliptic curves and modular forms in particular an exact isogenies representation of the generators of the renormalization group extending the modular group to a symmetry group

disorder solutions for ising and potts models with a field the exact expressions of the disorder varieties and the corresponding values taken by the partition functions of the checkerboard ising and potts models with a magnetic field are obtained different specialisations and extensions of these exact results are examined

numerical results for the three state critical potts model on finite rectangular lattices partition functions for the three state critical potts model on finite square lattices and for a variety of boundary conditions are presented the distribution of their zeros in the complex plane of the spectral variable is examined and is compared to the expected infinite lattice result the partition functions are then used to test the finite size scaling predictions of conformal and modular invariance

the inversion relation method for some two dimensional exactly solved models in lattice statistics the partition functions per site of several two dimensional models notably the eight vertex self dual potts and hard hexagon models can be easily obtained by using an inversion relation for local transfer matrices together with symmetry and analyticity properties this technique is discussed the analyticity properties compared and some equivalences and nonequivalences pointed out in particular the critical hard hexagon model is found to have the same as the self dualq state potts model withq 3 5 2 2 618 the temperley lieb equivalence between the potts and six vertex models is found to fail in certain nonphysical antiferromagnetic cases

random matrix theory and higher genus integrability the quantum chiral potts model we perform a random matrix theory rmt analysis of the quantum fourstate chiral potts chain for different sizes of the chain up to size l 8 ou ra nalysis gives clear evidence of a gaussian orthogonal ensemble goe statistics suggesting the existence of a generalized time reversal invariance furthermore a change from the generic goe distribution to a poisson distribution occurs when the integrability conditions are met the chiral potts model is known to correspond to a star triangle integrability associated with curves of genus higher than zero or one therefore the rmt analysis can also be seen as a detector of higher genus integrability

large q expansion for the anisotropic potts model a large q expansion of the partition function of the anisotropic q state potts model in two dimensions is given up to the sixth order in q 12 at the critical temperature it is in agreement with the exact solutions for the partition function and for the latent heat the latter does not depend on the anisotropy it does not show any simple factorisation in the parameters of the model

triangular lattice potts models in this paper we study the 3 state potts model on the triangular lattice which has two and three site interactions using a peierls argument we obtain a rigorous bound on the transition temperature thereby disproving a conjecture on the location of its critical point low temperature series are generated and analyzed for three particular choices of the coupling constants a phase diagram is then drawn on the basis of these considerations our analysis indicates that the antiferromagnetic transition and the transition along the coexistence line are of first order implying the existence of a multicritical point in the ferromagnetic region relation of the triangularq state potts model with other lattice statistical problems is also discussed in particular an ashkin teller model and the hard hexagon lattice gas solved by baxter emerge as special cases in appropriate limits

critical exponents of two dimensional potts and bond percolation models critical exponents of the two dimensional q state potts model are calculated by means of finite size scaling and transfer matrix techniques for continuous q results for the temperature exponent agree accurately with the conjecture of den nijs 1979 the magnetic exponent is found to behave in accordance with the conjecture of nienhuis et al 1980

analytic properties of the partition function for statistical mechanical models the author explains with examples the relationship between zeros of the partition function analyticity and degeneracy of absolute magnitude of eigenvalues of the transfer matrix for statistical mechanical models it is shown how to write down a polynomial representation of the partition function for any model with a two sheet largest eigenvalue the staggered ice model representation of the q state potts model on a sequence of semi infinite strips is solved and it is shown that for 0 q 4 the potts model result is only projected out by special boundary conditions the non potts part of the result is obtained for a 4 infinity strip and gives baxter s antiferromagnetic critical curve 1982

exact potts model partition functions on ladder graphs abstract we present exact calculations of the partition function z of the q state potts model and its generalization to real q for arbitrary temperature on n vertex ladder graphs i e strips of the square lattice with width ly 2 and arbitrary length lx with free cyclic and mobius longitudinal boundary conditions these partition functions are equivalent to tutte whitney polynomials for these graphs the free energy is calculated exactly for the infinite length limit of these ladder graphs and the thermodynamics is discussed by comparison with strip graphs of other widths we analyze how the singularities at the zero temperature critical point of the ferromagnet on infinite length finite width strips depend on the width we point out and study the following noncommutativity at certain special values q s lim n lim q q s z 1 n lim q q s lim n z 1 n it is shown that the potts antiferromagnet on both the infinite length line and ladder graphs with cyclic or mobius boundary conditions exhibits a phase transition at finite temperature if 0 b in the corresponding c 2 space arising as the accumulation set of partition function zeros as n in particular we study the connection with the t 0 limit of the potts antiferromagnet where b reduces to the accumulation set of chromatic zeros certain properties of the complex temperature phase diagrams are shown to exhibit close connections with those of the model on the square lattice showing that exact solutions on infinite length strips provide a way of gaining insight into these complex temperature phase diagrams

zeros of the partition function for the triangular lattice three state potts model the authors obtain the zeros of the partition function for the isotropic triangular lattice three state potts model in a finite lattice the distribution exhibits new points on the real axis which are well fitted by an algebraic equation deduced from the inversion relation and the symmetries of the anisotropic model they identify a symmetry approximately relating all six transition points

partition function zeros for the two dimensional ising model iii abstract the density of the complex temperature zeros of the partition function of the two dimensional ising model on completely anisotropic triangular lattices is investigated near real and complex critical points the non uniform behaviour of the density of zeros at interior and boundary critical points is studied analytically and numerically for a lattice with interactions in the ratios 3 2 1 the limiting behaviour of the density at complex critical points depends on the direction of approach in the complex plane in the neighbourhood of interior critical points one finds only a single layer of zeros but generally there are two layers or superposed sets of zeros with different distributions and different limiting densities at boundary critical points on anisotropic quadratic and on partially anisotropic triangular lattices the two layers become identical by symmetry of the partition function the divergence of the two dimensional density of zeros at real critical points is discussed briefly in relation to scaling theory

q state potts model on the checkerboard lattice a large q expansion of the partition function of the checkerboard q state potts model is given up to sixth order in q 12 this expansion is used to discuss the following two points first on the critical manifold the large q expansion agrees up to this order with the minimal solution suggested by the group structure associated with this model secondly the latent heat and some correlation functions are discussed from the point of view of their functional dependence in the parameters

three state chiral potts models on the triangular lattice a monte carlo study abstract the phase transition of the three state chiral potts model is investigated by monte carlo simulations performed on two types of isotropic triangular lattices of various linear sizes up to 128 these two types of model are shown to have different phase diagrams in addition the critical antifferomagnetic point is analyzed in detail

on infinite words obtained by selective substitution grammars abstract we present a method of generating infinite words from selective substitution grammars introduced by rozenberg 1977 comparison is made with some of the well known limiting processes and limit language families and certain closure properties are examined a technique is given for obtaining infinite non repetitive words several decidability results are established

a heuristic principle for the phase boundaries of pure and quenched bond diluted spin models results from high temperature series for a class of discrete classical spin models on the square lattice suggest that the phase boundary is at least in part given by a simple heuristic principle it is shown how this principle can be extended to triangular and hexagonal lattices by making use of duality and the star triangle transformation for the potts model the exact phase transition is recovered not only for integer number of states but also in the percolation limit for more complicated models a number of phase diagrams are derived which are presumably exact except in regions where partially broken symmetry phases or massless phases intervene the principle can also be applied to quenched bond diluted spin models the results for the potts model are obtained for all three lattices and shown to be exact in certain limiting cases the m to 1 limit of the wreath product model s m s m which corresponds to double percolation is studied to provide an example for which the principle is completely incorrect

rational mappings arborescent iterations and the symmetries of integrability we describe a class of nonlinear birational representations of groups generated by a finite number of involutions these groups are symmetries of the yang baxter equations and their higher dimensional generalizations they provide discrete dynamical systems with a variety of behaviors from chaotic to integrable according to the number of invariants of the representation

critical point of a triangular potts model with two and three site interactions the q state potts model on the triangular lattice with nearest neighbour interactions and three site interactions in half of the triangular faces is considered the exact duality relation is re examined from the point of view of determining its critical point using the continuity and uniqueness arguments the authors determine the exact critical point in the ferromagnetic model it is argued that a transition exists in an antiferromagnetic model only for q 3 a conjecture is then made on the phase diagram for the q 3 isotropic model these results are used to determine the exact criticality of a dilute potts model on the honeycomb lattice

an exact critical frontier for the potts model on the 3 12 lattice an exact critical frontier for the potts model on the 3 12 lattice which includes the kagome lattice as a special case is determined in a special parameter space of the two and three site interactions the determination is made possible by the use of a star triangle transformation which converts the lattice into one whose exact critical point is known

inverse functional relation on the potts model the two dimensional anisotropic potts model at all temperatures satisfies a local inverse relation leading to an inverse functional equation on the transfer matrix the pertinence of the related functional equation for the partition function is discussed and verified perturbatively by introducing a diagrammatic expansion corresponding to a partial resummation on one parameter of the model possible exploitations of the inverse relation in order to determine the partition functions are sketched this indicates that the inverse relation can be used without referring to a star triangle relation

automorphisms of algebraic varieties and yang baxter equations it is shown that commuting transfer matrix models in statistical mechanics are parametrized by algebraic varieties having a set of automorphisms deduced from the so called inversion relation in general this set of automorphisms is infinite this shows that for algebraic varieties of dimension 1 the models are parametrized by algebraic curves of genus 0 or 1

massless phases for the zp model on the union jack lattice the authors introduce and analyse zp symmetric models on the union jack lattice they show that these models have the same self dual structure already known for other zp systems by performing renormalisation group calculations as well as monte carlo simulations they analyse their phase diagrams showing in particular that for p or 5 they exhibit a disordered massless phase

the potts model on kagome and honeycomb lattices low temperature series have been derived for theq state potts model on the kagom e lattice qd 3 and 4 and for the 3 state honeycomb lattice the series are derived by the finite lattice method with many additional terms obtained by noting the structure of the correction terms accurate estimates of the critical points are found which confirm exact but not rigorous results in the case of the honeycomb lattice and refute previous conjectures in the case of the kagom e lattice in this paper we report on the derivation and analysis of low temperature series for the 3 state honeycomb and 3 and 4 state kagomlattice potts model from the series we obtain estimates for the critical points which for the honeycomb model confirms the validity of the exact but not rigorous result 7 and for the kagomlattice provides accurate results against which conjectures can be tested and found not to hold we comment on properties that the solution must have in an unsuccessful attempt to find an alternative conjecture in addition we obtain estimates for the location of various non physical singularities the remainder of the paper is organized as follows in section 2 we briefly describe the finite lattice method of series expansions give details regarding the specific implementations for the kagomand honeycomb lattices and show how the finite lattice method can be supplemented by an extension procedure allowing us to derive more series terms details regarding the analysis of the series are given in section 3 and the implications for possible conjectures for the critical point on the kagom