interfacial adsorption phenomena of the three dimensional three state potts model the authors study the interfacial adsorption phenomena of the three state ferromagnetic potts model on the simple cubic lattice by the monte carlo method finite size scaling analyses of the net adsorption yield evidence of the phase transition being of first order and kbtc j 1 8166 2

graphical representations and cluster algorithms i discrete spin systems abstract graphical representations similar to the fk representation are developed for a variety of spin systems in several cases it is established that these representations have fkg monotonicity properties which enables characterization theorems for the uniqueness phase and the low temperature phase of the spin system certain systems with intermediate phases and or first order transitions are also described in terms of the percolation properties of the representations in all cases these representations lead in a natural fashion to swendsen wang type algorithms hence at least in the above mentioned instances these algorithms realize the program described by kandel and domany phys rev b 43 1991 8539 8548 all of the algorithms are shown to satisfy a li sokal bound which at least for systems with a divergent specific heat implies critical slowing down however the representations also give rise to invaded cluster algorithms which may allow for the rapid simulation of some of these systems at their transition points

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

finite size scaling of the three state potts model on a simple cubic lattice finite size scaling is studied for the three state potts model on a simple cubic lattice we show that the specific heat and the magnetic susceptibility scale accurately as the volume the correlation length exhibits behaviors expected for a genuine first order transition the one extracted from the unsubtracted correlation function shows a characteristic finite size behavior whereas the physical correlation length that characterizes the first excited state stays at a finite value and is discontinuous at the transition point

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

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

finite size effects for the potts model with weak boundary conditions using the pirogov sinai theory we study finite size effects for the ferromagnetic q state potts model in a cube with boundary conditions that interpolate between free and constant boundary conditions if the surface coupling is about half of the bulk coupling and q is sufficiently large we show that only small perturbations of the ordered and disordered ground states are dominant contributions to the partition function in a finite but large volume this allows a rigorous control of the finite size effects for these weak boundary conditions in particular we give explicit formulae for the rounding of the infinite volume jumps of the internal energy and magnetization as well as the position of the maximum of the finite volume specific heat while the width of the rounding window is of order l d the same as for periodic boundary conditions the shift is much larger of order l 1 for strong boundary conditions the surface coupling is either close to zero or close to the bulk coupling the finite size effects at the transition point are shown to be dominated by either the disordered or the ordered phase respectively in particular it means that sufficiently small boundary fields lead to the disordered and not to the ordered gibbs state this gives an explicit proof of a van enter s result that the phase transition in the potts model is not robust

critical behaviour of the two dimensional potts model with a continuous number of states a finite size scaling analysis we investigate the critical behaviour of the two dimensional q state potts model using finite size scaling and transfer matrix methods for the continuous transition range 0 these results for continuous q were obtained from a transfer matrix constructed for a generalized whitney polynomial representing the potts models an effective algorithm to compute the dominant eigenvalues of this essentially nonsymmetric transfer matrix is developed

algorithmic pirogov sinai theory we develop an efficient algorithmic approach for approximate counting and sampling in the low temperature regime of a broad class of statistical physics models on finite subsets of the lattice mathbb z d and on the torus mathbb z n mathbb z d our approach is based on combining contour representations from pirogov sinai theory with barvinok s approach to approximate counting using truncated taylor series some consequences of our main results include an fptas for approximating the partition function of the hard core model at sufficiently high fugacity on subsets of mathbb z d with appropriate boundary conditions and an efficient sampling algorithm for the ferromagnetic potts model on the discrete torus mathbb z n mathbb z d at sufficiently low temperature

volumes of balls in large riemannian manifolds we prove two lower bounds for the volumes of balls in a riemannian manifold if m n g is a complete riemannian manifold with lling radius at least r then it contains a ball of radius r and volume at least n r n if m n hyp is a closed hyperbolic manifold and if g is another metric on m with volume no greater than n vol m hyp then the universal cover of m g contains a unit ball with volume greater than the volume of a unit ball in hyperbolic n space

series analysis of the 3 state potts model in 2 1 dimensions both high temperature and low temperature series are used to locate and characterize the first order transition in the 3 state potts model in 2 1 dimensions on both square and triangular lattices estimates are presented for the vacuum energy latent heat magnetization susceptibility and mass gap at the transition the spontaneous magnetization and latent heat appear to display an approximate universality at this weak first order transition

continuous phase transitions with a convex dip in the microcanonical entropy the appearance of a convex dip in the microcanonical entropy of finite systems usually signals a first order transition however a convex dip also shows up in some systems with a continuous transition as for example in the baxter wu model and in the four state potts model in two dimensions we demonstrate that the appearance of a convex dip in those cases can be traced back to a finite size effect the properties of the dip are markedly different from those associated with a first order transition and can be understood within a microcanonical finite size scaling theory for continuous phase transitions results obtained from numerical simulations corroborate the predictions of the scaling theory

monte carlo study of the triangular blume capel model under bond randomness the effects of bond randomness on the universality aspects of a two dimensional d 2 blume capel model embedded in the triangular lattice are discussed the system is studied numerically in both its first and second order phase transition regimes by a comprehensive finite size scaling analysis for a particularly suitable value of the disorder strength we find that our data for the second order phase transition emerging under random bonds from the second order regime of the pure model are compatible with the universality class of the two dimensional 2d random ising model furthermore we find evidence that the second order transition emerging under bond randomness from the first order regime of the pure model belongs again to the same universality class although the first finding reinforces the scenario of strong universality in the 2d ising model with quenched disorder the second is in difference from the critical behavior emerging under randomness in the cases of the ex first order transitions of the potts model finally our results verify previous renormalization group calculations on the blume capel model with disorder in the crystal field coupling

invaded cluster algorithm for potts models the invaded cluster algorithm a method for simulating phase transitions is described in detail theoretical albeit nonrigorous justification of the method is presented and the algorithm is applied to potts models in two and three dimensions the algorithm is shown to be useful for both first order and continuous transitions and evidently provides an efficient way to distinguish between these possibilities the dynamic properties of the invaded cluster algorithm are studied numerical evidence suggests that the algorithm has no critical slowing for ising models s1063 651x 96 09208 2

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

small volume of balls large volume entropy and the margulis constant in his seminal work about bounded cohomology gromov showed that under some topological conditions every closed riemannian manifold of small volume has large volume entropy in this article we strengthen some aspects of this result using an alternative approach more precisely we prove that under some similar yet different topological assumptions every closed riemannian manifold whose volume of balls is small has large volume entropy along the proof of this result we establish a new systolic inequality involving the commutator systole and a new curvature free estimate relating the filling radius to the margulis constant

finite size effects and phase transition in the three dimensional three state potts model the three state potts model in three dimensions is studied by monte carlo and finite size scaling techniques using a histogram method recently proposed by ferrenberg and swendsen the finite size dependence for the maximum of the specific heat is found to scale with the volume of the system indicating that the phase transition is of first order the value of the latent heat per spin and the correlation length at the transition are estimated

interfacial adsorption in the two dimensional blume capel model new monte carlo mc results on the interfacial adsorption in the vicinity of the tricritical point of the two dimensional blume capel model are reported the net adsorption w0 is found to diverge as w0 approximately tc w with omega 0 44 or 0 02 as the tricritical point is approached along a path asymptotically not parallel to the phase boundary this behaviour as well as previous mc results on the interfacial adsorption at the critical line and the first order transition of the model are explained using analytic and scaling arguments in addition finite size effects at the first order transition are analysed and it is shown that the critical adsorption scales as lperpendicular to f l sub lperpendicular to 2 where l sub and lperpendicular to are the lengths of the system in the directions parallel and perpendicular to the interface the scaling function f is evaluated by performing mc simulations on systems with various length to width ratios

a first order phase transition in the three dimensional four state antiferromagnetic potts model studies the four state antiferromagnetic potts model with next nearest neighbour ferromagnetic interactions on the simple cubic lattice by the monte carlo method the ratio j2 j1 of the strength of the next nearest neighbour interactions to that of the nearest neighbour interactions is to be 10 the author analyses the finite size effects of the energy the specific heat the fourth order cumulant the magnetization the susceptibility and the effective transition temperatures with the help of the finite size scaling theory the author concludes that the transition of the model is assigned to be of first order

short geodesic loops on complete riemannian manifolds with a finite volume in this paper we will show that on any complete noncompact riemannian manifold with a finite volume there exist uncountably many geodesic loops of arbitrarily small length

monte carlo simulation study of the thicknesses of wetting layers in pure and impure quenched random field environments abstract wetting layer thicknesses have been measured in monte carlo simulations of a three dimensional solid on solid model of adsorption in a pure environment the average thickness l is found to follow the expected logarithmic law as a function of the undersaturation or bulk field h b at small h b the results are consistent with a cross over to a power law l h 1 b predicted on the basis of finite size effects in an impure quenched random field environment a logarithmic law is found at relatively large h b as for the pure system when h b is lowered the simulations show pinning of the interface by the random fields implying slowing down of the relaxation strong dependence on the starting configuration and strong sensitivity to the random field environment in addition to the cross over due to the finite system size these problems have made the expected power law behaviour l h 1 2 b inextricable

crossover finite size scaling at first order transitions in a recent paper we developed a method which allows one to control rigorously the finite size behavior in long cylinders near first order phase transitions at low temperature here we apply this method to asymmetric transitions with two competing phases and to theq state potts model as a typical model of a temperature driven transition whereq low temperature phases compete with one high temperature phase we obtain the finite size scaling of the firstn eigenvalues wheren is the number of competing phases of the transfer matrix in a periodic box of volumel l t and as a corollary the finite size scaling of the shape of the order parameter in a hypercubic box t l the infinite cylinder t and the crossover regime from hypercubic to cylindrical scaling for the two phase case n 2 we find that the crossover length l is given by o lw exp lv where is the inverse temperature is the surface tension and w 1 2 if v 1 2 whilew 0 if v 1 2 for the standard ising model we also consider free boundary conditions showing that l exp lv o lv 1 for any dimension v 1 2 for v 1 2 we finally discuss a class of boundary conditions which interpolate between free corresponding to the interpolating parameter g 0 and periodic boundary conditions corresponding to g 1 finding that l o lw exp lv withw 0 forg 0 andw 1 2 for 0 g 1

finite size analysis of two dimensional u 1 lattice gauge theories we analyse the finite size scaling of two dimensional u 1 lattice gauge theories with a theta term interaction the finite size corrections to the specific heat binder landau and u4 cumulants agree with the expected asymptotic behaviour for first order phase transitions however we find that the leading correction to the position of the extremal points of these quantities is not universal on the other hand the finite size corrections to the mass gap behave as for second order phase transitions in particular the curves corresponding to different size approximations do not cross in the vicinity of the transition points the feature is associated to the existence of a divergent correlation length and holds for a wider class of models

tight bounds for mixing of the swendsen wang algorithm at the potts transition point we study two widely used algorithms for the potts model on rectangular subsets of the hypercubic lattice mathbb z d heat bath dynamics and the swendsen wang algorithm and prove that under certain circumstances the mixing in these algorithms is torpid or slow in particular we show that for heat bath dynamics throughout the region of phase coexistence and for the swendsen wang algorithm at the transition point the mixing time in a box of side length l with periodic boundary conditions has upper and lower bounds which are exponential in ld 1 this work provides the first upper bound of this form for the swendsen wang algorithm and gives lower bounds for both algorithms which significantly improve the previous lower bounds that were exponential in l log l 2

growth of balls in the universal cover of surfaces and graphs in this paper we prove uniform lower bounds on the volume growth of balls in the universal covers of riemannian surfaces and graphs more precisely there exists a constant 0 such that if m hyp is a closed hyperbolic surface and h another metric on m with area m h area m hyp then for every radius r 1 the universal cover of m h contains an r ball with area at least the area of an r ball in the hyperbolic plane this positively answers a question of l guth for surfaces we also prove an analog theorem for graphs

correlation length of the three state potts model in three dimensions a monte carlo study is carried out for the three state potts model in three dimensions the correlation length is shown to remain finite and to be discontinuous at the transition point it is also shown that the transition exhibits genuine first order characteristics in all aspects

automated modeling of complex systems to answer prediction questions abstract a question about the behavior of a complex physical system can be answered by simulating the system the challenge is building a model of the system that is appropriate for answering the question if the model omits relevant aspects of the system the predicted behavior may be wrong if on the other hand the model includes many aspects that are irrelevant to the question it may be difficult to simulate and explain the leading approach to automated modeling compositional modeling constructs a simplest adequate model for a question from building blocks model fragments that are designed by knowledge engineers this paper presents a new compositional modeling algorithm that constructs models from simpler building blocks the individual influences among system variables and addresses important modeling issues that previous programs left to the knowledge engineer in the most rigorous test of a modeling algorithm to date we implemented our algorithm applied it to a large knowledge base for plant physiology and asked a domain expert to evaluate the models it produced

layering transitions at an interface in the blume capel model an interface in the three dimensional blume capel model is studied using low temperature series and mean field theory by calculating the leading order diagrams in the low temperature series expansion to all orders using a transfer matrix technique the authors are able to show that the interface wets through an infinite sequence of layering transitions which become quasicontinuous as the bulk phase boundary is approached the mean field approximation agrees well with the low temperature series results

the layered ising model mean field and interfacial approximations the critical temperature for layered ising models on the s c lattice is calculated using the recently proposed transfer matrix version of a mean field approximation for two layered case the present results tc j 3 20755 5 and 3 62279 2 for the open and periodic boundary conditions respectively improve considerably on the previous estimation obtained by the high temperature series expansion this model is also studied using the interfacial approximation i e the muller hartmann zittartz method we show that in this method the behaviour of interface in the weak interlayer coupling limit is similar to that of interacting quantum particles for the strong interlayer interaction limits some results for the two layered case are also obtained

model simplification by asymptotic order of magnitude reasoning abstract one of the hardest problems in reasoning about a physical system is finding an approximate model that is mathematically tractable and yet captures the essence of the problem this paper describes an implemented program aom which automates a powerful simplification method aom is based on two domain independent ideas self consistent approximations and asymptotic order of magnitude reasoning the basic operation of aom consists of five steps 1 assign order of magnitude estimates to terms in the equations 2 find maximal terms of each equation i e terms that are not dominated by any other terms in the same equation 3 consider all possible n term dominant balance assumptions 4 propagate the effects of the balance assumptions and 5 remove partial models based on inconsistent balance assumptions aom also exploits constraints among equations and submodels we demonstrate its power by showing how the program simplifies difficult fluid models described by coupled nonlinear partial differential equations with several parameters we believe the derivation given by aom is more systematic and easily understandable than those given in published papers

critical phenomena with convergent series expansions in a finite volume linked cluster expansions are generalized from an infinite to a finite volume they are performed to 20th order in the expansion parameter to approach the critical region from the symmetric phase a new criterion is proposed to distinguish first from second order transitions within a finite size scaling analysis the criterion applies also to other methods for investigating the phase structure such as monte carlo simulations our computational tools are illustrated with the example of scalar o n models with four and six point couplings forn 1 andn 4 in three dimensions it is shown how to localize the tricritical line in these models we indicate some further applications of our methods to the electroweak transition as well as to models for superconductivity

interfaces in the potts model i pirogov sinai theory of the fortuin kasteleyn representation we develop a new analysis of the order disorder transition in ferromagnetic potts models for large numberq of spin states we use the pirogov sinai theory which we adapt to the fortuin kasteleyn representation of the models this theory applies in a rather direct way in our approach and leads to a system of non interacting contours with small activities as a consequence simpler and more natural techniques are found allowing us to recover previous results on the bulk properties of the model which then extend to non integer values ofq and to deal with non translation invariant boundary conditions this will be applied in a second part of this work to study the behaviour of the interfaces at the transition point

task dependent qualitative domain abstraction automated problem solving for engineered devices is based on models that capture the essential aspects of the behavior in this paper we deal with the problem of automatically abstracting behavior models such that their level of granularity is as coarse as possible but still sufficiently detailed to carry out a given behavioral prediction or diagnostic task a task is described by a behavior model as composed from a library a specified granularity of the possible observations and a specified granularity of the desired results the goal of task dependent qualitative domain abstraction is to determine maximal partitions for the variables domains termed qualitative values that retain all the necessary distinctions we present a formalization of this problem within a relational constraint based framework and devise solutions to automatically determine qualitative values for a device model the results enhance the ability to use a behavior model of a device as a common basis to support different tasks along its life cycle

finite size scaling for first order phase transitions rigorous results abstract recent rigorous results on finite size effects for lattice models are reviewed we consider symmetric and asymmetric first order phase transitions in cubic and cylindrical geometries and discuss the determination of transition points from finite size data

determinant of the potts model transfer matrix and the critical point abstract by using a decomposition of the transfer matrix of the q state potts model on a three dimensional m n n simple cubic lattice its determinant is calculated exactly by using the calculated determinants a formula is conjectured that approximates the critical temperature for a d dimensional hypercubic lattice

a unified approach to phase diagrams in field theory and statistical mechanics we construct the phase diagram of any system which admits a low temperature polymer or cluster expansion such an expansion turns the system into a hard core interacting contour model with small but not necessarily positive activities the method uses some of zahradnik s ideas z1 but applies equally well to systems with complex interactions we give two applications first to low temperaturep 2 models with complex couplings and second to a computation of asymptotics of partition functions in periodic volumes if the index of a supersymmetric field theory is known the second application would help determine the number of phases in infinite volume

interface properties of the two dimensional blume emery griffiths model the authors study the interface behaviour of the two dimensional blume emery griffiths model 1971 which is described by the hamiltonian h j sigma ij sisj k sigma ij si2sj 2 d sigma isi2 si 1 0 1 an interface is introduced into the system by fixing the spins on opposite boundaries in two different states 1 and 1 they pay particular attention to the appearance of an excess of the third state 0 in the vicinity of the interface the net adsorption of the non boundary state is studied near critical first order and tricritical transitions two methods have been used to attack this problem the monte carlo technique and a modified version of the interface free energy approximation of muller hartmann and zittartz 1977 which is seen to give a surprisingly good description of interface properties in three state systems

necklace solid on solid models of interfaces and surfaces using the exact generating functions for solid on solid models the authors construct necklace partition functions for wetting and prewetting at a sticky wall in two bulk dimensions three component generalisations of these models constructed to study interfacial wetting are introduced and solved based on these models they discuss the types of interfacial critical behaviour to be expected in three component systems in two dimensions

surface induced finite size effects for first order phase transitions we consider classical lattice models describing first order phase transitions and study the finite size scaling of the magnetization and susceptibility in order to model the effects of an actual surface in systems such as small magnetic clusters we consider models with free boundary conditions for a field driven transition with two coexisting phases at the infinite volume transition pointh h t we prove that the low temperature finite volume magnetizationmfree l h per site in a cubic volume of sizel d behaves like m free l h frac m m 2 frac m m 2 tanh left frac m m 2 l d h h chi l right o left frac 1 l right whereh x l is the position of the maximum of the finite volume susceptibility andm are the infinite volume magnetizations ath h t 0 andh h t 0 respectively we show thath x l is shifted by an amount proportional to 1 l with respect to the infinite volume transition pointh t provided the surface free energies of the two phases at the transition point are different this should be compared with the shift for periodic boundary conditions which for an asymmetric transition with two coexisting phases is proportional only to 1 l2d one can consider also other definitions of finite volume transition points for example the positionh u l of the maximum of the so called binder cumulantufree l h whileh u l is again shifted by an amount proportional to 1 l with respect to the infinite volume transition pointh t its shift with respect toh l is of the much smaller order 1 l2d we give explicit formulas for the proportionality factors and show that in the leading 1 l2d term the relative shift is the same as that for periodic boundary conditions