duality relations and equilibrium crystal shapes of potts models on triangular and honeycomb lattices abstract we consider the q state potts models on triangular and honeycomb lattices for each lattice it is shown that the anisotropic interfacial tension is related to the anisotropic correlation length on the dual lattice using the star triangle transformation we also find that the anisotropic correlation length on the triangular lattice is the same as that of the honeycomb lattice for q 4 at the first order transition point the anisotropic correlation length is exactly calculated at the first order transition point by a method which introduces auxiliary edges into the models from the calculated anisotropic correlation length the equilibrium droplet shape of one ordered phase embedded inside another is derived via the wulff construction the equilibrium shape is written as a simple algebraic curve which corresponds to the dispersion relation of bound states in the terminology of spin chains

finite size corrections and numerical calculations for long spin 1 2 heisenberg chains in the critical region leading and next to leading order finite size corrections to the ground and first excited states are calculated for the spin 1 2 anisotropic heisenberg model in the critical region the analytic results are compared to numerical data obtained for chains up to a length of n 1024 it is found that near the isotropic point the asymptotic region where the results obtained for n to infinity are applicable sets in at very large n values and for obtaining good accuracy in fitting the numerical data one has to take into account several correction terms even at large n 100 chain lengths

the covariance matrix of the potts model a random cluster analysis we consider the covariance matrix g mm q2 of thed dimensionalq states potts model rewriting it in the random cluster representation of fortuin and kasteleyn in any of theq ordered phases we identify the eigenvalues of this matrix both in terms of representations of the unbroken symmetry group of the model and in terms of random cluster connectivities and covariances thereby attributing algebraic significance to these stochastic geometric quantities we also show that the correlation length corresponding to the decay rate of one of the eigenvalues is the same as the inverse decay rate of the diameter of finite clusers for dimensiond 2 we show that this correlation length and the correlation length of the two point function with free boundary conditions at the corresponding dual temperature are equal up to a factor of two for systems with first order transitions this relation helps to resolve certain inconsistencies between recent exact and numerical work on correlation lengths at the self dual point o for systems with second order transitions this relation implies the equality of the correlation length exponents from above and below threshold as well as an amplitude ratio of two in the course of proving the above results we establish several properties of independent interest including left continuity of the inverse correlation length with free boundary conditions and upper semicontinuity of the decay rate for finite clusters in all dimensions and left continuity of the two dimensional free boundary condition percolation probability at o we also introduce dlr equations for the random cluster model and use them to establish ergodicity of the free measure in order to prove these results we introduce a new class of events which we call decoupling events and two inequalities for these events the first is similar to the fkg inequality but holds for events which are neither increasing nor decreasing the second is similar to the van den berg kesten inequality in standard percolation both inequalities hold for an arbitrary fkg measure

partition function zeros of the one dimensional potts model the recursive method the yang lee fisher and potts zeros of the one dimensional q state potts model are studied using the theory of dynamical systems an exact recurrence relation for the partition function is derived it is shown that zeros of the partition function may be associated with neutral fixed points of the recurrence relation further a general equation for zeros of the partition function is found and a classification of the yang lee fisher and potts zeros is given it is shown that the fisher zeros in a nonzero magnetic field are located on several lines in the complex temperature plane and that the number of these lines depends on the value of the magnetic field analytical expressions for the densities of the yang lee fisher and potts zeros are derived it is shown that densities of all types of zeros of the partition function are singular at the edge singularity points with the same critical exponent

exact results for the zeros of the partition function of the potts model on finite lattices the yang lee zeros of the q state potts model are investigated in one two and three dimensions analytical results derived from the transfer matrix for the one dimensional model reveal a systematic behavior of the locus of zeros as a function of q for 1 2 they lie outside the unit circle for finite temperature in the special case q 2 the zeros lie exactly on the unit circle as proved by lee and yang in two and three dimensions the zeros are calculated numerically and behave in the same way results are also presented for the critical line of the potts model in an external field as determined from the zeros of the partition function in the complex temperature plane

on the phase diagram of the discrete z6 spin models we point out some problems with the previously proposed phase diagram of the z6 spin models consideration of the diagram near to the decoupling surface using both exact and approximate arguments suggests a modification which remedies these deficiencies with the aid of a new parametrization of the phase space we study the models numerically with results which support our conjectures

hard hexagon model calculation of anisotropic interfacial tension from asymptotic degeneracy of largest eigenvalues of row row transfer matrix to find the directional dependence of the interfacial tension of the hard hexagon model an inhomogeneous system is studied this system is defined on a square lattice with 1 v m columns so that the lhs of the m 1 th column is the hard hexagon model and the rhs of the m 1 th column works as the operator which shifts the particle configuration of a column downward a triplet of the largest eigenvalues of the row row transfer matrix are asymptotically degenerate asm under the conditions that 1 v m 0 mod 3 withv being fixed to be constant the interfacial tension of a tilted interface is calculated from the finite correction terms in this limit

critical behavior of a three dimensional random bond ising model using finite time scaling with extensive monte carlo renormalization group method we have investigated the critical behavior of a three dimensional random bond ising model for a series of the disorder strength by a finite time scaling combining with monte carlo renormalization group method in the presence of a linearly varying temperature the method enables us to estimate a lot of critical exponents of both static and dynamic nature independently as well as the critical temperatures the static exponents obtained agree well with most existing results verify both the hyperscaling and the rushbrooke scaling laws and their combined scaling law which in turn validate their asymptotic nature and corroborate the universality of the relevant random fixed point with respect to the forms of disorder the dynamic critical exponent z is estimated to 2 114 51 which is compatible with those obtained from experiments and renormalization group analyses the exponents at low and high disorder strengths do not satisfy all scaling laws and are argued to be crossover exponents that reflect crossover from the random fixed point to the pure and the percolation fixed point they also indicate that the exponents that were previously suggested to be a distinct universality class for strong disorder strength in the literature may be just crossover our results demonstrate the effectiveness of the finite time scaling method

a regularity theorem for minimizers of quasiconvex integrals we prove c1 partial regularity for minimizers of functionals with quasiconvex integrand f x u du depending on vector valued functions u the integrand is required to be twice continuously differentiable in du and no assumption on the growth of the derivatives of f is made a polynomial growth is required only on f itself

random geometric graphs we analyze graphs in which each vertex is assigned random coordinates in a geometric space of arbitrary dimensionality and only edges between adjacent points are present the critical connectivity is found numerically by examining the size of the largest cluster we derive an analytical expression for the cluster coefficient which shows that the graphs are distinctly different from standard random graphs even for infinite dimensionality insights relevant for graph bipartitioning are included

hard squares with diagonal attractions the authors complete the solution of the square lattice gas with nearest neighbour exclusion and attractive next nearest neighbour diagonal interactions on the special surface corresponding to regimes iii and iv of the generalised hard hexagon model the interfacial tension correlation length and sublattice density difference are calculated throughout these regimes by obtaining the eigenvalues of the row to row and corner transfer matrices the associated critical exponents are found to be mu v 5 4 beta 3 32 in regime iii and mu v 5 2 beta 1 4 in regime iv in particular their results confirm the recent proposal by huse that regime iii is the first order coexistence surface separating the disordered fluid phase from the square ordered solid phases and that the regime iii regime iv boundary is a line of tricritical points

short time dynamics of cluster growth in the potts model the two and three dimensional q state potts model is simulated by a cluster algorithm and the relaxation dynamics of average cluster size and the order parameter during initial thermalization are studied the dependence of these quantities on the lattice size and q in the early stages of monte carlo iterations reveals valuable information on the nature of the phase transition occurring in the system from the initial thermalization data of the 2d 7 state potts model surface tension is also calculated the value is in good agreement with the exact result

infinite clusters in percolation models the qualitative nature of infinite clusters in percolation models is investigated the results which apply to both independent and correlated percolation in any dimension concern the number and density of infinite clusters the size of their external surface the value of their total surface to volume ratio and the fluctuations in their density in particular it is shown thatn0 the number of distinct infinite clusters is either 0 1 or and the casen0 which might occur in sufficiently high dimension is analyzed

hard hexagon model anisotropy of correlation length and interfacial tension the directional dependence of the correlation length of the hard hexagon model is calculated by a new method which introduces the shift operator into the usual transfer matrix method this method is also applied to the calculation of the interfacial tension of the hard hexagon model which is anisotropic in addition the equilibrium droplet shape of one phase embedded inside another is obtained from the analysis of the interfacial tension by the use of wulff s construction

anisotropic correlation length in the eight vertex model we investigate the directional dependence or anisotropy of the correlation length of the eight vertex model by a method which introduces the shift operator into the usual transfer matrix argument for a given x 0 x 1 there are two cases with respect to a parameter q in the case 0 q x3 the anisotropic correlation length acl is independent of q noting that the eight vertex model factors into two ising lattices in the q x4 limit we can directly relate the acl to that of the square lattice ising model when x3 q x2 the acl depends on q using an algebraic curve we show that the acl in this case is connected with that of the ising model on a 4 8 or union jack lattice we suggest that a wide class of models have essentially the same acl as the eight vertex model has

the partition function zeros in the one dimensional q state potts model the zeros of the partition function in the one dimensional q state potts model with arbitrary and continuous q or 0 have been studied using a transfer matrix the location of zeros and the yang lee edge singularity have been analysed and two different regimes corresponding to q 1 and q 1 have been observed a duality relation has also been derived which relates the zeros in the complex field plane to those in the complex temperature plane

inhomogeneous six vertex model with domain wall boundary conditions and bethe ansatz in this note we consider the six vertex model with domain wall boundary conditions defined on an m m lattice in the inhomogeneous case where the partition function depends on 2m inhomogeneities j and k for a particular choice of the set of j we find a new determinant representation for the partition function which allows evaluation of the bulk free energy in the thermodynamic limit this provides a new connection between two types of determinant formulas we also show in a special case that spin correlations on the horizontal line going through the center coincide with the ones for periodic boundary conditions

thin fisher zeros various authors have suggested that the loci of partition function zeros can profitably be regarded as phase boundaries in the complex temperature or field planes we obtain the fisher zeros for ising and potts models on non planar thin regular random graphs using this approach and note that the locus of fisher zeros on a bethe lattice is identical to the corresponding random graph since the number of states q appears as a parameter in the potts solution the limiting locus of chromatic zeros is also accessible

partial regularity for non autonomous degenerate quasi convex functionals with general growth abstract we study partial c 1 regularity of minimizers of quasi convex variational integrals with non standard growth we assume in particular that the relevant integrands satisfy an orlicz s type growth condition i e a so called general growth condition moreover the functionals are supposed to be non autonomous and possibly degenerate

surface critical behaviour of an o n loop model related to two manhattan lattice walk problems we find and discuss the scaling dimensions of the branch 0 manifold of the nienhuis o n loop model on the square lattice concentrating on the surface dimensions the results are extracted from a bethe ansatz calculation of the finite size corrections to the eigenspectrum of the six vertex model with free boundary conditions these results are especially interesting for polymer physics at two values of the crossing parameter lambda interacting self avoiding walks on the manhattan lattice at the collapse temperature lambda pi 3 and hamiltonian walks on the manhattan lattice lambda pi 2 are discussed in detail our calculations illustrate the importance of examining both odd and even strip widths when performing finite size correction calculations to obtain scaling dimensions

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

boundary correlation functions of the six vertex model we consider the six vertex model on an n n square lattice with the domain wall boundary conditions boundary one point correlation functions of the model are expressed as determinants of n n matrices generalizing the known result for the partition function in the free fermion case explicit answers are obtained the introduced correlation functions are closely related to the problem of enumeration of alternating sign matrices and domino tilings

surface operator content of the al face models abstract a set of fixed boundary weights for both the critical dense and dilute a l face models is constructed from the known boundary weights of the related loop model the surface operator content and the conformal partition functions then follow from the results obtained via the bethe equations

the free energies of six vertex models and the n equivalence relation the free energies of six vertex models on a general domain d with various boundary conditions are investigated with the use of the n equivalence relation which help classify the thermodynamic limit properties it is derived that the free energy of the six vertex model on the rectangle is unique in the limit height width it is derived that the free energies of the model on the domain d are classified through the densities of left down arrows on the boundary specifically the free energy is identical to that obtained by lieb phys rev lett 18 1046 1967 19 108 1967 phys rev 162 162 1967 and sutherland phys rev lett 19 103 1967 with the cyclic boundary condition when the densities are both equal to 12 this fact explains several results already obtained through the transfer matrix calculation the relation to the domino tiling or dimer or matching problems is also noted

reply to comment on equilibrium crystal shape of the potts model at the first order transition point the eight vertex model is defined on the square lattice rotated through an arbitrary angle with respect to the coordinate axes we re examine the analysis of the anisotropic correlation length in a previous paper fujimoto m 1996 physica a 233 485 502 we point out that the asymptotic form of the correlation function is expressed by the use of differential forms on a riemann surface of genus 1 combined with the symmetry of the square lattice this fact explains that the anisotropic correlation length is represented in terms of simple algebraic curves the argument is applicable to a wide class of lattice models including unsolvable ones

magnetization at corners in two dimensional ising models we investigate the corner spin magnetization of two dimensional ferromagnetic ising models in various wedge geometries results are obtained for triangular and square lattices by numerical studies on finite wedges using the star triangle transformation as well as analytic calculations using corner transfer matrices and a fermionic representation of the row to row transfer matrix the corner magnetizations vanish at the bulk critical temperature tc with an exponent c differing from the bulk exponent for isotropic systems with free edges we find that c is given simply by c 2 where is the angle at the corner for apex magnetizations of conical lattices we obtain the strikingly similar result a 4 these formulas apply equally well to anisotropic lattices if the angle is interpreted as an effective angle eff determined by the relative strengths of the interactions

anisotropic interfacial tension and the equilibrium crystal shape of kagome lattice eight vertex model we exactly calculate the anisotropic interfacial tension of the kagome lattice eight vertex model with the help of two types of square lattice models each square lattice model is inhomogeneous but still possesses a one parameter family of commuting transfer matrices from the anisotropic interfacial tension the equilibrium shape of an inclusion of one ordered phase in an environment of another is obtained by applying the wulff construction calculations are expressed by the use of differential forms on a riemann surface of genus 1 and the equilibrium shape is written as an elliptic curve we suggest the possibility of interpreting critical phenomena in terms of the moduli space of riemann surfaces of genus 1

partial holder regularity for elliptic systems with non standard growth abstract we study partial holder regularity for elliptic systems with non standard growth we consider general systems with orlicz growth and discontinuous coefficient factor for which we prove that their weak solutions are partially holder continuous for any holder exponents in addition we also obtain a similar result for systems with double phase growth

partial regularity for minimizers of discontinuous quasi convex integrals with degeneracy abstract in this paper we are concerned with the regularity of minimizers u w 1 p r n of quasi convex integral functionals of the type f u f x u d u d x the crucial point here is that the integrand f admits very weak regularity properties with respect to the gradient variable it satisfies degenerate singular p growth conditions without necessarily possessing a quasi diagonal uhlenbeck type structure and with respect to the x variable the integrand might be even discontinuous it is only assumed that a certain vmo condition holds under those assumptions we prove partial holder continuity of minimizers i e holder continuity of u for any holder exponent 0 1 outside a set of measure zero under such weak assumptions regularity results for the gradient of minimizers is not expected to hold since even in the scalar case counterexamples to c 1 regularity are known

equilibrium crystal shape of the potts model at the first order transition point we investigate the q state two dimensional potts model the anisotropic interfacial tension is related by duality to the anisotropic correlation length for q 4 we calculate exactly the anisotropic correlation length at the first order transition point from the calculated anisotropic correlation length the equilibrium crystal shape ecs is derived via the wulff construction the ecs is expressed by means of a simple algebraic curve regarding q as a temperature scale we show that the potts model has the same ecs as the eight vertex model we discuss a connection between the ecs and the q deformed pseudo euclidian algebra

inversion identities for the self dual potts and ashkin teller models the commuting diagonal to diagonal transfer matrices of the self dual potts and ashkin teller models on the square lattice are shown to satisfy special functional equations called inversion identities these identities generalise the known local inversion or unitarity relations for interaction round a face or irf models satisfying yang baxter or star triangle equations

a monte carlo renormalisation group study of the two dimensional discrete cubic model a hint for superconformal invariance the discrete cubic model for six states of the lattice variables is investigated at one special point of the phase diagram where a number of contradicting results are known concerning the order of the phase transition and the critical exponents a new implementation of the mcrg method is used for determining the critical point with great accuracy and for calculating the critical exponents a continuous phase transition is found following the recent results for six state self dual quantum chains with cubic symmetry where for a special point superconformal invariance is discovered the critical exponents obtained are explained in the framework of this theory indicating a superconformal point with a conformal anomaly of c 1 25

a numerical method to compute exactly the partition function with application to z n theories in two dimensions i present a new method to exactly compute the partition function of a class of discrete models in arbitrary dimensions the time for the computation for ann state model on anl d lattice scales like n l d 1 nl d i show examples of the use of this method by computing the partition function of the 2d ising and 3 state potts models for maximum lattice sizes 10 10 and 8 8 respectively the critical exponentsv and and the critical temperature one obtains from these are very near the exactly known values the distribution of zeros of the partition function of the potts model leads to the conjecture that the ratio of the amplitudes of the specific heat below and above the critical temperature is unity

bmo estimates for the p laplacian we prove bmo estimates of the inhomogeneous p laplace system given by div jruj p 2 ru divf we show that f 2 bmo impliesjruj p 2 ru2 bmo which is the limiting case of the nonlinear calder on zygmund theory this extends the work of dibenedetto and manfredi dm93 which was restricted to the super quadratic case p 2 to the full case 1 p 1 and even more general growth moreover we prove that a ru inherits the campanato and vmo regularity of f

percolation phase diagrams for multi phase models built on the overlapping sphere model abstract the overlapping sphere os percolation model gives a two phase microstructure matrix plus inclusions that is useful for testing composite material ideas and other applications as well as serving as a paradigm of overlapping object percolation and phase transitions real materials often have more than two phases so it is of interest to extend the applicability of the os model a flexible variant of the os model can be constructed by randomly assigning the spheres different phase labels according to a uniform probability distribution as they are inserted one by one into the matrix the resulting three or more phase models can have different amounts of percolating and non percolating phases depending on the volume fraction of each phase and the total os volume fraction a three dimensional digital image approach is used to approximately map out the percolation phase diagram of such models explicitly up to four phases one matrix plus three spherical inclusion phases and implicitly for n 4 phases for the three phase model it was found that a single os sub phase has a percolation threshold that ranges from about a volume fraction of 0 16 when the matrix volume fraction is about 0 01 to about 0 30 at a matrix volume fraction of about 0 7 the approximate analytical dependence of this sub phase percolation threshold on the defining model parameters serves to guide the building of the percolation phase diagram for the n phase model and is used to determine the maximum value of n n 6 at which all n phases can be simultaneously percolated

finite size corrections for ground states of the xxz heisenberg chain the method of de vega and woynarovitch 1985 is used to calculate finite size corrections to the ground state energy in different sectors for the xxz heisenberg chain finite size scaling amplitudes and correction to scaling exponents in the critical region are derived using conformal invariance a scaling dimension x pi gamma 2 pi is extracted corresponding to the electric field operator in the 8 vertex model this confirms a conjecture of baxter and kelland 1974 finite size scaling properties near the kosterlitz thouless critical point delta 1 are discussed

transfer matrix for the restricted canonical and microcanonical ensembles the numerical transfer matrix for the partition function of discrete lattice models is generalized to allow the calculation of the density of states ensuremath omega e and the restricted density of states ensuremath omega e m given ensuremath omega e m the partition function is expressed as a polynomial in the variables x mathit e mathrm ensuremath beta mathit h and y mathit e mathrm ensuremath mathrm ensuremath beta these algorithms are illustrated with calculations for the ising model on finite square lattices the zeros of the partition function are examined in both the complex x and y planes finite size scaling analysis of the zeros leads to very accurate estimates for the critical temperature and critical exponents

finite time scaling via linear driving application to the two dimensional potts model we apply finite time scaling to the q state potts model with q 3 and 4 on two dimensional lattices to determine its critical properties this consists in applying to the model a linearly varying external field that couples to one of its q states to manipulate its dynamics in the vicinity of its criticality and that drives the system out of equilibrium and thus produces hysteresis and in defining an order parameter other than the usual one and a nonequilibrium susceptibility to extract coercive fields from the finite time scaling of the order parameter the coercivity and the hysteresis area and its derivative we are able to determine systematically both static and dynamic critical exponents as well as the critical temperature the static critical exponents obtained in general and the magnetic exponent ensuremath delta in particular agree reasonably with the conjectured ones the dynamic critical exponents obtained appear to confirm the proposed dynamic weak universality but unlikely to agree with recent short time dynamic results for q 4 our results also suggest an alternative way to characterize the weak universality

vicious walkers and young tableaux i without walls we rederive previously known results for the number of star and watermelon configurations by showing that these follow immediately from standard results in the theory of young tableaux and integer partitions in this way we provide a proof of a result previously only conjectured for the total number of stars

fast flat histogram method for generalized spin models we present a monte carlo method that efficiently computes the density of states for spin models having any number of interaction per spin by combining a random walk in the energy space with collective updates controlled by the microcanonical temperature our method yields dynamic exponents close to their ideal random walk values reduced equilibrium times and very low statistical error in the density of states the method can host any density of states estimation scheme including the wang landau algorithm and the transition matrix method our approach proves remarkably powerful in the numerical study of models governed by long range interactions where it is shown to reduce the algorithm complexity to that of a short range model with the same number of spins we apply the method to the q state potts chains 3 ensuremath leqslant q ensuremath leqslant 12 with power law decaying interactions in their first order regime we find that conventional local update algorithms are outperformed already for sizes above a few hundred spins by considering chains containing up to 2 16 spins which we simulated in fairly reasonable time we obtain estimates of transition temperatures correct to five figure accuracy finally we propose several efficient schemes aimed at estimating the microcanonical temperature

exact solution and surface critical behaviour of an o n model on the honeycomb lattice the authors obtain the exact bethe ansatz solution of an o n model on the honeycomb lattice with open boundaries across a finite strip several quantities relevant to the surface critical behaviour of the o n model are derived from this solution the exact surface critical exponents are in agreement with those obtained by cardy 1987 and by duplantier and saleur 1986 based on conformal invariance arguments

the partition function zeros of the anisotropic ising model with multisite interactions on a zigzag ladder abstract it is shown that the spin 1 2 anisotropic ising model with multisite interactions on a zigzag ladder may be mapped into the one dimensional spin 1 2 axial next nearest neighbor ising annni model with multisite interactions the partition function zeros of the annni model with multisite interactions are investigated a comprehensive analysis of the partition function zeros of the annni model with and without three site interactions on a zigzag ladder is done using the transfer matrix method analytical equations for the distribution of the partition function zeros in the complex magnetic field yang lee zeros and temperature fisher zeros planes are derived the yang lee and fisher zeros distributions are studied numerically for a variety of values of the model parameters the densities of the yang lee and fisher zeros are studied and the corresponding edge singularity exponents are calculated it is shown that the introduction of three site interaction terms in the annni model leads to a simpler distribution of the partition function zeros for example the yang lee zeros tend to a circular distribution when increasing by modulus the three site interactions term coefficient it is found that the yang lee and fisher edge singularity exponents are universal and equal to each other 1 2

six vertex model with rotated boundary conditions we investigate the six vertex model on a square lattice rotated through an arbitrary angle with respect to the coordinate axes a model recently introduced by litvin and priezzhev auxiliary vertices are used to define an inhomogeneous system which leads to a one parameter family of commuting transfer matrices a product of commuting transfer matrices can be interpreted as a transfer matrix acting on zigzag walls in the rotated system using an equation for commuting transfer matrices we calculate their eigenvalues finite size properties of the model are discussed from the viewpoint of the conformal field theory

comment on equilibrium crystal shape of the potts model at the first order transition point we comment on the paper by fujimoto 1997 j phys a math gen 30 3779 where the exact equilibrium crystal shape ecs in the critical q state potts model on the square lattice was calculated and its equivalence with ecs in the ising model was established we confirm these results giving their alternative derivation applying the transformation properties of the one particle dispersion relation in the six vertex model it is shown that this dispersion relation is identical to that in the ising model on the square lattice

scaling dimensions and conformal anomaly in anisotropic lattice spin models the effect of anisotropic interactions on the eigenvalue spectrum of the row to row transfer matrix of critical lattice spin models is investigated it is verified that the predictions of conformal theory apply to anisotropic systems if one allows for spatial rescaling by incorporating an anisotropy factor zeta ay ax sin theta where ay and ax are lattice spacings and theta is an angle describing the anisotropy for exactly solvable models these anisotropy angles can be calculated analytically using corner transfer matrices this is done for the eight vertex model hard hexagons and interacting hard squares and it is found that theta pi u lambda 10u 3 and 5u respectively where u is the spectral parameter and lambda is the crossing parameter for each of these models the amplitude of the finite size corrections to the free energy at criticality is found to be of the form pi zeta c 6n2 where zeta is the anisotropy factor and the central charge or conformal anomaly is given by c 1 4 5 and 7 10 respectively this is an analytic result for the eight vertex model for the hard hexagon and square models the largest eigenvalues are found accurately by numerically solving their inversion identities for various anisotropies and strip widths up to n 48 finally the authors argue that the anisotropy angle for magnetic hard squares and the q state potts models is also given by theta pi u lambda so this result is quite general

determination of the dynamic and static critical exponents of the two dimensional three state potts model using linearly varying temperature we employ a successive monte carlo renormalization group procedure in the presence of a linearly varying temperature to study the three state q 3 potts model on square lattices by matching correlation functions of different lattice sizes at different renormalization levels a rate exponent r associated with the temperature sweep rate r and the correlation length exponent ensuremath nu are obtained the dynamic critical exponent z is then obtained by the scaling law z r ensuremath 1 ensuremath nu derived in our method the results of z 2 171 62 for q 3 in this work and the previously obtained z 2 15 13 for q 2 seem to support the extension of the weak universality hypothesis to dynamic critical behavior with these calculated exponents the dynamic scaling forms of both specific heat and order parameter at various r are presented and all the other static critical exponents are determined which verify our method discussions are made on how to improve the accuracy of the estimation of the dynamic critical exponent

finite lattice extrapolation algorithms two algorithms for sequence extrapolation due to van den broeck and schwartz 1979 and bulirsch and stoer 1964 are reviewed and critically compared applications to three state and six state quantum chains and to the 2 1 d ising model show that the algorithm of bulirsch and stoer is superior in particular if only very few finite lattice data are available

the effects of the distribution of bimodal bonds on phase transition in 2d potts model abstract we have studied the q 8 state potts model in two dimension to determine the effects of the distribution of bimodal bonds on the phase transition we have shown that by monitoring the energy histograms and autocorrelation times there exist a finite size dependent threshold value of the introduced quenched bond randomness for rounding the first order phase transition and this threshold value becomes smaller as the system size increased

finite size scaling of the density of zeros of the partition function in first and second order phase transitions the finite size scaling form for the density of zeros of the partition function in first and second order phase transitions is derived using the finite size scaling of the density of zeros the order of a phase transition can be easily determined and the order parameter calculated from finite size data we illustrate the scaling theory using exact values for the zeros of the partition function of the two dimensional ising model in the complex magnetic field plane

analytical results for random line networks applications to fracture networks and disordered fiber composites abstract we study the geometrical properties of a continuum model consisting of an ensemble of intersecting lines placed within a finite region in two dimensions we consider the model in the high density regime where it emulates fiber composites filtering elements and fracture networks we report exact results for the probability distribution of segment lengths produced by line intersections segment loop perimeter and area distributions are estimated numerically for the probability distribution of perimeters we found the interesting phenomenon of small pore perimeter power law repulsion for pores of four or more sides two subsidiary models with infinite line length are solved exactly and reveal the salient features of the finite line model important applications to fiber filter selectivity and optimal intersection trajectories in fracture networks are discussed

regularity and existence of solutions of elliptic equations with p q growth conditions for some positive constants m a4 and for exponents q 2 p 2 2 under 1 2 1 3 and some other assumptions by assuming also that the quotient q p is sufficiently close to one in dependence on n precisely if q p c n n 2 then we prove that every weak solution to 1 1 of class 4 4 0 is locally lipschitz continuous in q moreover there are positive constants 8 c and 0 2 1 such that

isomorphism of dimer configurations and spanning trees on finite square lattices one to one mappings of the close packed dimer configurations on a finite square lattice with free boundaries l onto the spanning trees of a related graph or two graph g are found the graph two graph g can be constructed from l by 1 deleting all the vertices of l with arbitrarily fixed parity of the row and column numbers 2 suppressing all the vertices of degree 2 except those of degree 2 in l 3 merging all the vertices of degree 1 into a single vertex g the matrix kirchhoff theorem reduces the enumeration problem for the spanning trees on g to the eigenvalue problem for the discrete laplacian on the square lattice l g g with mixed dirichlet neumann boundary conditions in at least one direction that fact explains some of the unusual finite size properties of the dimer model

six vertex model with domain wall boundary conditions variable inhomogeneities we consider the six vertex model with domain wall boundary conditions we choose the inhomogeneities as solutions of the bethe ansatz equations these equations have many solutions so we can consider a wide variety of inhomogeneities for certain choices of the inhomogeneities we study arrow correlation functions on the horizontal line going through the centre in particular we obtain a multiple integral representation for the emptiness formation probability that generalizes the known formulae for xxz antiferromagnets

fisher zeros of the q state potts model in the complex temperature plane for nonzero external magnetic field the microcanonical transfer matrix is used to study the distribution of the fisher zeros of the qg2 potts models in the complex temperature plane with nonzero external magnetic field h q unlike the ising model for h q ensuremath ne 0 which has only a nonphysical critical point the fisher edge singularity the qg2 potts models have physical critical points for h q l0 as well as the fisher edge singularities for h q g0 for h q l0 the crossover of the fisher zeros of the q state potts model into those of the q ensuremath 1 state potts model is discussed and the critical line of the three state potts ferromagnet is determined for h q g0 we investigate the edge singularity for finite lattices and compare our results with the high field low temperature series expansion of enting for 3l ql 6 we find that the specific heat magnetization susceptibility and the density of zeros diverge at the fisher edge singularity with exponents ensuremath alpha e ensuremath beta e and ensuremath gamma e which satisfy the scaling law ensuremath alpha e 2 ensuremath beta e ensuremath gamma e 2

new heuristic percolation criterion for continuum systems a heuristic criterion is introduced and applied for derivation of percolation thresholds in continuum systems the criterion is comprehensive in the sense that it applies to systems with diverse bonding criteria as well as to systems with a correlated distribution of sites its usefulness is manifested by the excellent quantitative agreement between the values obtained and available monte carlo data

precise determination of the critical threshold and exponents in a three dimensional continuum percolation model we present a large scale computer simulation of the prototypical three dimensional continuum percolation model consisting of a distribution of overlapping spatially uncorrelated spheres by using simulations of up to particles and studying the finite size scaling of various effective percolation thresholds we obtain a value of this value is significantly smaller than the values obtained for simulations that have been carried out using smaller systems employing this value of and systems of size l 160 relative to a sphere of unit radius we also obtain estimates of the critical exponents and for the continuum system and show that the values are different than those obtained using previous values of

evidences of the instability fixed points of first order phase transitions we study the dynamics of the first order phase transitions in the two dimensional potts model driven by a linearly varying temperature using a finite time scaling with extended dynamic monte carlo renormalization group method it is found that for sufficiently large lattice sizes the flows of apparent exponents of different temperature sweep rates upon renormalization show characteristics that are markedly distinct from those of continuous transitions and are argued to result from the instability fixed points involved and provide a method for estimating the associated instability exponents

effect of random impurities on fluctuation driven first order transitions we analyse the effect of quenched uncorrelated randomness coupling to the local energy density of a model consisting of n coupled two dimensional ising models for n 2 the pure model exhibits a fluctuation driven first order transition characterized by runaway renormalization group behaviour we show that the addition of weak randomness acts to stabilize these flows in such a way that the trajectories ultimately flow back towards the pure decoupled ising fixed point with the usual critical exponents apart from logarithmic corrections we also show by examples that in higher dimensions such transitions may either become continuous or remain first order in the presence of randomness

complex q zeros of the partition function of the potts model with long range interactions abstract the zeros of the partition function of the ferromagnetic q state potts model with long range interactions in the complex q plane are studied in the mean field case while preliminary numerical results are reported for the finite 1 d chains with power law decaying interactions in both cases at any fixed temperature the zeros lie on the arc shaped contours which cross the positive real axis at the value for which the given temperature is transition temperature for finite number of spins the positive real axis is free of zeros which approach to it in the thermodynamic limit the convergence exponent of the zero closest to the positive real q axis is found to have the same value as the temperature critical exponent 1

surface free energy of the critical six vertex model with free boundaries the bethe ansatz equations are derived for the six vertex model with general boundary weights on a lattice in a diagonal orientation these are solved in the thermodynamic limit finite size corrections to the free energy for a restricted class of boundary weights including those corresponding to a potts model with free boundaries are calculated in the critical region the first order term gives the surface free energy of the model the second order term is found to be pi okt tan pi nu 2 mu c 48n 2 where c 1 6 mu 2 pi 2 pi mu is the conformal anomaly this can be compared to pi kt sin pi nu mu c 6n 2 for a calculation on the lattice in the standard orientation and with periodic boundary conditions this difference can be explained geometrically using conformal invariance

the critical behaviour of self dual z n spin systems finite size scaling and conformal invariance this paper is concerned with the critical properties of a family of self dual two dimensional z n models whose bulk free energy is exactly known at the self dual point the author s analysis is performed by studying the finite size behaviour of the corresponding one dimensional quantum hamiltonians which also possess an exact solution at their self dual point by exploring finite size scaling ideas and the conformal invariance of the critical infinite system the author calculates for n up to 8 the critical temperature and critical exponents as well as the central charge associated with the underlying conformal algebra the results strongly suggest that the z n quantum field theory of zamolodchikov and fateev is the underlying field theory associated with these statistical mechanical systems the author also tests for the z 5 case the conjecture that these models correspond to the bifurcation points in the phase diagram of the general z n spin model where a massless phase originates

partial regularity for general systems of double phase type with continuous coefficients abstract we consider elliptic systems of double phase problems which satisfy a general structure with continuous coefficients and prove that weak solutions of such systems are holder continuous for any holder exponent 0 1 except for a measure zero set

renormalization group theory for cooling first order phase transitions in potts models we develop a dynamic field theoretic renormalization group rg theory for cooling first order phase transitions in the potts model it is suggested that the well known imaginary fixed points of the q state potts model for q 10 3 in the rg theory are the origin of the dynamic scaling found recently from numerical simulations apart from logarithmic corrections this indicates that the real and imaginary fixed points of the potts model are both physical and control the scalings of the continuous and discontinuous phase transitions respectively of the model our one loop results for the scaling exponents are already not far away from the numerical results further the scaling exponents depend on q only slightly consistent with the numerical results therefore the theory is believed to provide a natural explanation of the dynamic scaling including the scaling exponents and their scaling laws for various observables in the cooling first order phase transition of the potts model

illumination a new method for studying 3d percolation fronts in a concentration gradient a new method of studying percolation fronts is introduced in situations where the concentration is not constant gradient percolation approach it consists in an efficient way of determining subsets of the percolation cluster which are exactly at the percolation threshold the method is applied to both lattice and continuum percolation in order to determine the percolation threshold in 3d more accurately two variations of this method illumination and projection seem to give a similar scaling behaviour at low gradients for the cubic lattice case leading to a new estimate for the percolation threshold pc 0 31173 or 0 00007 for continuum percolation of overlapping spheres the new estimate for the critical volume fraction is 0 291 or 0 002 and the value of the critical exponent alpha sigma for the width of the front scaled with the concentration gradient is very close to the one for the lattice case

the correlation length of the potts model at the first order transition point the authors consider the q state two dimensional potts model for q g i e in the first order phase transition regime following a scheme given by p martin 1991 the authors prove an identity between the spectra of the transfer matrices of the potts model and the transfer matrices connecting the diagonals of a 6 vertex model by using a bethe ansatz for the latter they obtain an exact expression for the correlation length of the potts model at the transition point

hunting for the central charge of the virasoro algebra six and eight state spin models in two dimensions conformal invariance has important implications for the finite size scaling properties of the spectra of transfer matrices and quantum chains at the critical point some relations between the finite size scaling amplitudes are obtained which can be used as a test of conformal invariance and implicitly to distinguished between second order and first order phase transitions the numerical values of the scaling amplitudes give the scale dimensions of various operators six and eight state self dual quantum chains with cubic symmetries are considered at the critical point for three values of the coupling constant the systems are found to be conformal invariant and estimates for several critical exponents are obtained based on the approximate values of the critical exponent one tries to find the values of the corresponding central charges of the virasoro algebras

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

yang lee zeros of the q state potts model on recursive lattices the yang lee zeros of the q state potts model on recursive lattices are studied for noninteger values of q considering one dimensional 1d lattice as a bethe lattice with coordination number equal to 2 the location of yang lee zeros of 1d ferromagnetic and antiferromagnetic potts models is completely analyzed in terms of neutral periodical points three different regimes for yang lee zeros are found for qg1 and 0lql1 an exact analytical formula for the equation of phase transition points is derived for the 1d case it is shown that yang lee zeros of the q state potts model on a bethe lattice are located on arcs of circles with the radius depending on q and temperature for qg1 complex magnetic field metastability regions are studied for the qg1 and 0lql1 cases the yang lee edge singularity exponents are calculated for both 1d and bethe lattice potts models the dynamics of metastability regions for different values of q is studied numerically

auxiliary vertices method for kagome lattice eight vertex model we investigate a class of eight vertex models on a kagome lattice with the help of auxiliary vertices the kagome lattice eight vertex model kevm is related to an inhomogeneous system which leads to a one parameter family of commuting transfer matrices using an equation for commuting transfer matrices we determine their eigenvalues from calculated eigenvalues the correlation length of the kevm is derived with its full anisotropy there are two cases in the first case the anisotropic correlation length acl is the same as that of the triangular honeycomb lattice ising model by the use of an algebraic curve it is shown that the kagome lattice ising model the diced lattice ising model and the hard hexagon model also have essentially the same acl as the kevm in the second case we find that the acl displays 12fold rotational symmetry

dynamic critical behavior of a swendsen wang type algorithm for the ashkin teller model we study the dynamic critical behavior of a swendsen wang type algorithm for the ashkin teller model we find that the li sokal bound on the autocorrelation time int const xc h holds along the self dual curve of the symmetric ashkin teller model and is almost but not quite sharp the ratio int c h appears to tend to infinity either as a logarithm or as a small power 0 05 p 0 12 in an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time

quasiconvexity and partial regularity in the calculus of variations we prove partial regularity of minimizers of certain functionals in the calculus of variations under the principal assumption that the integrands be uniformly strictly quasiconvex this is of interest since quasiconvexity is known in many circumstances to be necessary and sufficient for the weak sequential lower semicontinuity of these functionals on appropriate sobolev spaces examples covered by the regularity theory include functionals with integrands which are convex in the determinants of various submatrices of the gradient matrix

non lipschitz minimizers of smooth uniformly convex functionals we construct non lipschitz minimizers of smooth uniformly convex functionals of type i u f du x dx our method is based on the use of null lagrangians

eight vertex model anisotropic interfacial tension and equilibrium crystal shape the anisotropic interfacial tension of the eight vertex model is found by a new method which introduces two inhomogeneous systems as the width of the system becomes large a doublet of the largest eigenvalues of the row row transfer matrix is asymptotically degenerate the anisotropic interfacial tension is calculated from their finite size correction terms in this limit by the use of the anisotropic interfacial tension the equilibrium crystal shape of the eight vertex model is derived via wulff s construction the equilibrium crystal shape is represented as a simple algebraic curve we discuss the close relation between the algebraic curve and the form of an elliptic function appearing in the expression of the interfacial tension

phases coexistence and surface tensions for the potts model theq states potts model exhibits a first order phase transition at some inverse temperature t between ordered and disordered phases forq large as proved in 1 in space dimension 2 we use theduality transformation as aninternal symmetry of the partition function at t to derive an estimate on the probability of a contour this enables us to prove the preceding result and the following new results i the discontinuity of the mass gap at t ii the existence of astrictly positive surface tension between two ordered phases up to t iii the existence of a non zero surface tension between an ordered and the disordered phase at t

spatial anisotropy and rotational invariance of critical hard squares we develop a new method to analyse solvable models defined on a square lattice rotated through an arbitrary angle phi with respect to the coordinate axes in the method we introduce auxiliary faces into a rotated system to relate it with an inhomogeneous one in the natural orientation phi 0 the inhomogeneous system is investigated by commuting a transfer matrix argument as an application of the method we consider interacting hard squares at the critical point we discuss relations between spatial anisotropy and rotational or conformal invariance of the model in calculations of finite size properties

equilibrium crystal shape of hard squares with diagonal attractions the author exactly calculates the anisotropic interfacial tension of a square lattice gas model with nearest neighbour exclusion and next nearest neighbour attractions this is achieved by a method which introduces the shift operator into the standard transfer matrix argument from the calculated anisotropic interfacial tension the equilibrium crystal shape ecs is derived by the use of the wulff construction the ecs is a closed curve in the x y plane represented as cosh lambda x y kbt a3cosh lambda x y kbt a4 2 0 with a scale factor lambda and constants a3 a4 he argues that this shape is a universal one which appears as the ecss of a wide class of models