antiferromagnetic potts model in a magnetic field a finite size scaling study abstract the finite size scaling method is used to investigate the phase diagrams of the q state antiferromagnetic potts model in an external magnetic field on a square lattice critical lines where the system undergoes second order phase transitions between ordered and disordered phases are found which are in complete qualitative agreement with known results for q 3 also new unexpected features are found for q 4 where the transition vanishes in the negative field region while in the positive field region as q increases the critical line flattens out with increasing slopes both at the t 0 h h c point where h c is the critical field at zero temperature and at the t 0 h 0 point thermal exponents y t are also calculated for arbitrary q along the critical lines

linear fuzzy graphs we introduce the concept of linear fuzzy graph and give its matrix and functional representations this allows a generalization of the notions of semiorder and pseudo order

modulated phase of a potts model with competing binary interactions on a cayley tree we study the phase diagram for potts model on a cayley tree with competing nearest neighbor interactions j1 prolonged next nearest neighbor interactions jp and one level next nearest neighbor interactions jo vannimenus proved that the phase diagram of ising model with jo 0 contains a modulated phase as found for similar models on periodic lattices but the multicritical lifshitz point is at zero temperature later mariz et al generalized this result for ising model with jo 0 and recently ganikhodjaev et al proved similar result for the three state potts model with jo 0 we consider potts model with jo 0 and show that for some values of jo the multicritical lifshitz point be at non zero temperature we also prove that as soon as the same level interactionjo is nonzero the paramagnetic phase found at high temperatures for jo 0 disappears while ising model does not obtain such property to perform this study an iterative scheme similar to that appearing in real space renormalization group frameworks is established it recovers as particular case previous work by ganikhodjaev et al for jo 0 at vanishing temperature the phase diagram is fully determined for all values and signs of j1 jp and jo at finite temperatures several interesting features are exhibited for typical values of jo j1

the thermodynamic limit on bethe lattices the author emphasises the fact that to obtain the exact solution of hamiltonian models on bethe lattices one can apply explicitly the formal method of rigorous statistical mechanics i e the thermodynamic limit of probability measures this approach solves the well known dichotomy between clashing alternative solutions and provides a very simple analytic solution for a large class of hamiltonian models

an exact solution on the ferromagnetic face cubic spin model on a bethe lattice the lattice spin model with q component discrete spin variables restricted to have orientations orthogonal to the faces of q dimensional hypercube is considered on the bethe lattice the recursive graph which contains no cycles the partition function of the model with dipole dipole and quadrupole quadrupole interaction for arbitrary planar graph is presented in terms of double graph expansions the latter is calculated exactly in case of trees the system of two recurrent relations which allows to calculate all thermodynamic characteristics of the model is obtained the correspondence between thermodynamic phases and different types of fixed points of the rr is established using the technique of simple iterations the plots of the zero field magnetization and quadrupolar moment are obtained analyzing the regions of stability of different types of fixed points of the system of recurrent relations the phase diagrams of the model are plotted for q 2 the phase diagram of the model is found to have three tricritical points whereas for q 2 there are one triple and one tricritical points pacs number s 05 50 q 05 70 fh

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

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

field behaviour of the xy chiral model on a cayley tree the authors present a study of the xy chiral model on a cayley tree in the presence of an external uniform magnetic field they calculate the phase diagram at zero temperature from a corresponding 1d mapping ferromagnetic commensurate and incommensurate modulated phases as well as chaotic structures are present in the zero temperature limit in this case they also obtain a devil s staircase and determine the route to chaos which is in agreement with feigenbaum s scenario for finite temperatures the phase diagrams are obtained from a 2d mapping the chaotic behaviour is present only at low temperatures

some ab percolation problems in the antiferromagnetic potts model abstract the ab correlated site random bond percolation problem in a q state antiferromagnetic potts model on bethe lattices is solved we find the analytic expression of the ab percolation characteristic functions in terms of the temperature the external field and the active bond concentration p b the ab threshold and the phase boundary of the system coincide at zero temperature and at most in two other points for every constant p b 1 the properties of the bethe lattice allow us to find the temperature dependent p b which defines the ab droplets i e those special ab clusters which diverge with thermal exponents along the phase boundary

metastability in the potts model on the cayley tree metastability in the ferromagneticp state potts model defined on the cayley tree is discussed it is shown that the sign of the boundary fieldhs determines the order of the transition as well as the stability of the low temperature phase lowering the temperature withhs 0 a system withp 2 will display a second first order transition to a metastable stable phase forhs 0 a second first order transition to a metastable stable phase occurs ifp 2 p 2 in this case the system also has a residual entropy which is negative forp 2

the potts model on the bethe lattice abstract the approach developed by thompson for the ising model is used to calculate the free energy of the potts model on the bethe lattice this is accomplished by introducing a simplification in obtaining the recursion relationship for the local magnetization and a generalization of the hyperbolic functions

julia sets associated with the potts model on the bethe lattice and other recursively solved systems the q state potts model on the bethe lattice has been investigated by a number of individuals over the last several decades with some of the preceding studies taking a dynamical systems perspective however other than for the special case of the q 2 potts model i e the ising model the julia sets of the discrete dynamical systems associated with the potts model on the bethe lattice have been largely ignored we look at these sets and find an interesting connection between the phase transition temperature and the dimension of the julia set in particular we find the dimension of the julia set to be a minimum at a phase transition furthermore we show this property is not restricted to potts models on bethe lattices this adds to a number of other special properties present at temperatures where a phase transition occurs

critical behaviour in three state potts antiferromagnets on a bethe lattice abstract a rigorous method recently introduced for the solution of spin models on bethe lattices is used to solve the three state antiferromagnetic potts model in an external field and to find its phase diagram critical lines where the system undergoes first order or second order phase transitions are found which are in complete qualitative agreement with recent results by racz and vicsek also new unexpected ordered phases are found

kernel systems on l ordered sets abstract this paper presents fuzzy versions of kernel operators and kernel systems on l ordered sets and establishes a one to one correspondence between them

phase diagrams of the q state potts model on bethe lattices abstract we use a rigorous recently developed method of solution of spin models on bethe lattices cayley trees it allows us to detect all the phases of the ferromagnetic and antiferromagnetic q state potts models in a field fpm and apm respectively the phase diagrams of the fpm are characterized by three critical lines of first order transitions in the apm the ordered phases which appear always for h 0 and under certain conditions for h 0 and or h

traces of fuzzy binary relations abstract in this paper the right and left trace of a fuzzy binary relation is defined and investigated basic properties of a fuzzy relation such as reflexivity different kinds of transitivity linearity etc are characterized by means of its traces

phase diagrams of competing quadruple and binary interactions on cayley tree like lattice triangular chandelier abstract we study the phase diagrams for the ising model on a cayley tree like lattice called triangular chandelier with competing nearest neighbour interactions j 1 prolonged next nearest neighbour interactions j p and one level next nearest neighbour quadruple interactions j l 1 the phase diagrams display the multicritical points the lifshitz points that are at nonzero temperature and many modulated phases to perform this study an iterative scheme similar to that appearing in real space renormalization group frameworks is established it recovers as particular case previous work of vannimenus extension result given by ganikhodjaev and uguz for k 3 at vanishing temperature the phase diagram is fully determined for all values and signs of j 1 j p and j l 1 at finite temperatures several interesting features are exhibited for typical values of j l 1 j 1 and j p j 1

the clone relation of a binary relation abstract in a recent paper de baets et al introduced the clone relation of a strict order relation two elements of a poset are said to be a pair of clones or to be clones if every other element that is greater resp smaller than one of them is also greater resp smaller than the other one this clone relation played a key role in the characterization of the l fuzzy tolerance relations and the l fuzzy equivalence relations that a strict order relation is compatible with in this paper we extend the notion of clone relation to any binary relation although the definition of such extension is trivial the corresponding properties significantly differ from those of the clone relation of a strict order relation we analyse the most important ones among these properties paying particular attention to a partition of the clone relation in terms of three different types of pairs of clones

phase diagram of the ising model on a cayley tree in the presence of competing interactions and magnetic field we study the phase diagram for the ising model on a cayley tree with competing nearest neighbor interactionsj1 and next nearest neighbor interactionsj2 andj3 in the presence of an external magnetic field to perform this study an iterative scheme similar to that appearing in real space renormalization group frameworks is established it recovers as particular cases previous works by vannimenus and by inawashiroet al at vanishing temperature the phase diagram is fully determined for all values and signs ofj2 j1 andj3 j2 in particular we verify that values ofj3 j2 high enough favor the paramagnetic phase at finite temperatures several interesting features evolution of reentrances separation of the modulated region into two disconnected pieces etc are exhibited for typical values ofj2 j1 andj3 j2

global bethe lattice consideration of the spin 1 ising model the spin 1 ising model with bilinear and biquadratic exchange interactions and single ion crystal field is solved on the bethe lattice using exact recursion equations the general procedure of investigation of critical properties is discussed and a full set of phase diagrams are constructed for both positive and negative biquadratic couplings a comparison is made with the results of other approximation schemes

one dimensional mapping modulated phases and lyapunov exponent for the antiferromagnetic q state potts and multi site exchange interaction ising models abstract we describe the bifurcation structure period doubling and chaos for the antiferromagnetic q state potts model on the bethe lattice and three site interaction ising model on husimi one in a magnetic field by using the recursion relation technique a chaotic behavior of the magnetic susceptibility for the models is observed at low temperatures the resulting one dimensional rational mapping has a positive lyapunov exponent in the region of the chaotic regime for the antiferromagnetic q state potts q 2 and three site interaction ising models we discuss modulated phases for the antiferromagnetic q state potts q 2 and q 2 and three site interaction ising model at low temperatures the q state potts model q 2 has only one modulated phase with 1 2 pinching corresponding to the 2 cycle the q state potts q 2 and three site interaction ising models have an infinite number of modulated phases with different pinching numbers we construct the first modulated phase after the first bifurcation point

new order parameters in the potts model on a cayley tree for the q state potts model new order parameters projecting on a group of spins instead of a single spin are introduced on a cayley tree this allows the physical interpretation of the potts model at noninteger values q0 of the number of states the model can be solved recursively this recursion exhibits chaotic behaviour changing qualitatively at critical values of q0 using an additional order parameter belonging to a group of zero extrapolated size the additional ordering is related to a percolation problem this percolation distinguishes different phases and explains the critical indices of percolation class occuring at the peierls temperature

on the structure of f indistinguishability operators abstract some properties concerning the structure of f indistinguishability operators are analyzed it is shown that any of such operators is generated by a family of fuzzy subsets this result since it gives the way to construct f indistinguishabilities facilitates new applications of fuzzy relations the links between f indistinguishability operators and a kind of generalized metrics in the unit interval which are also explored are used to define the canonical generators of an f indistinguishability operator that is the fuzzy partition associated with the operator

on the transitivity property abstract in this short note general transitivity properties of fuzzy binary relations are discussed it is shown that under some weak conditions any fuzzy binary relation can be regarded as a transitive one with respect to a properly chosen t norm a representation theorem for such relations is formulated

on contour arguments for the three state potts model with competing interactions on a semi infinite cayley tree we consider the potts model with competing two step interactions and spin values 1 2 3 on a semi infinite cayley tree of order 2 we describe ground states and verify the peierls condition for the model using a contour argument we show the existence of three different gibbs measures associated with translation invariant ground states

diluted ising model with competing interactions abstract we introduce aperiodic but deterministic dilution of bonds in the ising model with competing ferro and antiferromagnetic interactions between first and second neighbors along the branches of a cayley tree the problem is formulated as a non linear dissipative map whose attractors correspond to solutions deep inside the tree we use a scheme of successive periodic approximations to obtain the main modulated structures of the phase diagrams the paramagnetic lines as well as some other features of the phase diagrams can be obtained from closed expressions

phase diagrams of ising models on husimi trees i pure multisite interaction systems lattice spin systems with multisite interactions have rich and interesting phase diagrams we present some results for such systems involving ising spins 1 using a generalization of the bethe lattice approximation first we show that our approach yields good approximations for the phase diagrams of some recently studied multisite interaction systems second a multisite interaction system with competing interactions is investigated and a strong connection with results from the theory of dynamical systems is made we exhibit a full bifurcation diagram chaos period 3 windows etc for the magnetization of the base site of this system

potts models on feynman diagrams we investigate numerically and analytically potts models on thin random graphs generic feynman diagrams using the idea that such models may be expressed as the limit of a matrix model the thin random graphs in this limit are locally tree like in distinction to the fat random graphs that appear in the planar feynman diagram limit more familiar from discretized models of two dimensional gravity the interest of the thin graphs is that they give mean field theory behaviour for spin models living on them without infinite range interactions or the boundary problems of genuine tree like structures such as the bethe lattice q state potts models display a first order transition in the mean field for q 2 so the thin graph potts models provide a useful test case for exploring discontinuous transitions in mean field theories in which many quantities can be calculated explicitly in the saddle point approximation such discontinuous transitions also appear in multiple ising models on thin graphs and may have implications for the use of the replica trick in spin glass models on random graphs

phase diagram of the one state potts model on the cayley tree the phase diagram of the one state potts model on the closed asymmetric cayley tree with branching ratior 2 is obtained from the bethe peierls map the route to chaos via the period doubling cascade is obtained by considering the antiferromagnetic coupling limit the connection of the potts model with the percolation problem is shown by calculating the order parameter its susceptibility the internal energy and the specific heat as well as their asymptotic behavior at the paramagnetic ferromagnetic critical point due to the type of the lattice and to the polynomial character of the map this is the simplest known example of a mckay berker kirkpatrick spin glass

on mean field equations for spin glasses in this paper we study rigorously the random ising model on a cayley tree in the limit of infinite coordination numberz 8 an iterative scheme is developed relating mean magnetizations and mean square magnetizations of successive shells far removed from the surface of the lattice in this way we obtain local properties of the model in the thermodynamic limit of an infinite number of shells when the coupling constants are independent gaussian random variables the sk expressions emerge as stable fixed points of our scheme and provide a valid local mean field theory of spin glasses in which negative local entropy at low temperatures while perfectly possible mathematically may still perhaps be physically undesirable finally we examine the tap equations and show that if the average over bond disorder and the limitz 8 are actually performed one recovers our iterative scheme and hence the sk equations also in the thermodynamic limit

transitive fuzzy orderings of fuzzy numbers abstract in this paper we introduce a generalization of the baas kwakernaak index by replacing the min operation in their definition by a t norm some properties of the thus defined induced fuzzy ordering are established in particular it is shown that restrictions of the induced fuzzy ordering on some special classes of fuzzy numbers are reflexive fuzzy orders

a formula for calculating some schur indices abstract we describe a method for calculating the schur indices of certain characters of finite groups this method helps in particular in the calculation of the schur indices of the irreducible characters of the special linear groups

ising model with third neighbour interactions on the cayley tree the authors formulate the ising model with competing first second and third neighbour interactions along the branches of a cayley tree in the infinite coordination limit as a three dimensional non linear mapping problem the phase diagrams display a lifshitz point and many modulated phases the introduction of third neighbour interactions affects mainly the low temperature region of the phase diagrams

simple ising models still thrive a review of some recent progress abstract a selection of recent work by various authors concerning spin 1 2 ising modls with pair couplings is reviewed topics addressed include the universality of corrections to scaling and nonlinear scaling fields in planar ising models aspects of many particle correlation functions the testing and verification of scaling hypotheses for local magnetizations in the presence of walls and boundaries including the singular nature of the critical region scaling functions and the decay of the influence of a far wall a soluble model of a surface roughening transition nonuniversal critical effects of one dimensional perturbations in planar ising models the effects of spatially anisotropic further neighbor interactions in particular annni models in d 2 dimensions and soluble mock annni models in 2 and 3 dimensions

on the realization of maximal simple types and epsilon factors of pairs let g be the group of rational points of a general linear group over a non archimedean local field f we show that certain representations of open compact mod centre subgroups of g the maximal simple types of bushnell and kutzko can be realized as concrete spaces in the level zero case our result is essentially due to gel fand this allows us for a supercuspidal representation pi of g to compute a distinguished matrix coefficient of pi by integrating we obtain an explicit whittaker function for pi we use this to compute the varepsilon factor of pairs for supercuspidal representations pi 1 pi 2 of g when pi 1 and the contragredient of pi 2 differ only at the tame level more precisely pi 1 and check pi 2 contain the same simple character we do this by computing both sides of the functional equation defining the epsilon factor using the definition of jacquet piatetskii shapiro shalika we also investigate the behavior of the varepsilon factor under twisting of pi 1 by tamely ramified quasi characters our results generalize the special case pi 1 check pi 2 totally wildly ramified due to bushnell and henniart

strict preference relations based on weak t norms abstract in this paper we introduce and investigate weak t norms and related concepts in order to give a general representation of valued strict preference relations associated with a given valued preference relation all the well known forms of valued strict preference relations are particular cases of our results moreover when some types of rationality assumptions are made our unified approach seems to be the only possible one

local properties of an ising model on a cayley tree the ising model on a cayley tree displays a peculiar continuous order phase transition with zero long range order at all finite temperatures when one studies expection values of spins far removed from the surface which contains a finite fraction of the total number of spins in the thermodynamic limit however one obtains the so called bethe approximation here we study such a local description by setting up a simple recurrence relation for successive shell magnetizations far removed from the surface in the ferromagnetic case the local magnetization is a fixed point of the iterative transformation while in the antiferromagnetic case the fixed point bifurcates to a two cycle of the transformation for low temperatures and fields giving rise to local sublattice magnetizations in both cases local thermodynamical properties are obtained by integration

probability measures and hamiltonian models on bethe lattices i properties and construction of mrt probability measures the properties of one step markov rotationally and m step m 1 or 2 translationally invariant mrt probability measures on q state site qss bethe lattices are studied a theorem is proven which completely defines such measures in terms of m q2 q fundamental probabilities these are explicitly calculated for any mrt qss hamiltonian model as a consequence of our approach the dychotomy between alternative solutions of hamiltonian models on bethe lattices is solved

description of the translation invariant splitting gibbs measures for the potts model on a cayley tree for the q state potts model on a cayley tree of order k ge 2 it is well known that at sufficiently low temperatures there are at least q 1 translation invariant gibbs measures which are also tree indexed markov chains such measures are called translation invariant splitting gibbs measures tisgms in this paper we find all tisgms and show in particular that at sufficiently low temperatures their number is 2 q 1 we prove that there are q 2 where a is the integer part of a critical temperatures at which the number of tisgms changes and give the exact number of tisgms for each intermediate temperature for the binary tree we give explicit formulae for the critical temperatures and the possible tisgms while we show that these measures are never convex combinations of each other the question which of these measures are extremals in the set of all gibbs measures will be treated in future work

strange attractor in the ising model with competing interactions on the cayley tree on formule le modele d ising avec interactions de competition sur un arbre de cayley comme une application non lineaire a 2 dimensions

a categorical isomorphism between injective stratified fuzzy t_ bm 0 spaces and fuzzy continuous lattices for a frame l as the truth value table we apply fuzzy domain theory for the study of injective objects in the category of stratified l t 0 spaces we show that every fuzzy continuous lattice equipped with the fuzzy scott topology is an injective stratified l t 0 space and conversely the specialization l ordered set of an injective stratified l t 0 space is a fuzzy continuous lattice these two transformations form a categorical isomorphism between injective stratified l t 0 spaces and fuzzy continuous lattices

some exact results from the mean field renormalization group abstract the mean field renormalization group is applied to the study of the directed self avoiding random walk in d dimensions and to the spin ferromagnetic ising model on the bethe lattice in both cases the known exact results are obtained

a note on cactus trees variational vs recursive approach in this article we deal with the variational approach to cactus trees husimi trees and the more common recursive approach that are in principle equivalent for finite systems we discuss in detail the conditions under which the two methods are equivalent also in the analysis of infinite self similar cactus trees usually investigated to the purpose of approximating ordinary lattice systems such issue is hardly ever considered in the literature we show on significant test models that the phase diagram and the thermodynamic quantities computed by the variational method when they deviates from the exact bulk properties of the cactus system generally provide a better approximation to the behavior of a corresponding ordinary system generalizing a property proved by kikuchi we also show that the numerical algorithm usually employed to perform the free energy minimization in the variational approach is always convergent

potts model on a cayley tree and logistic equation the q state potts model on a cayley tree can be solved by a recursion formula depending on the properties at the surface the model on a bethe lattice is obtained by extrapolating the interior of a cayley tree sufficiently far from the surface in order to have a stable fixed point of the recursion for q 2 we find a second order transition of percolation type at the bethe peierls temperature and a first order transition at a higher temperature for coordination number z 3 the recursion extrapolated to q 1 is identical to the logistic equation the feigenbaum route to chaos appears for antiferromagnetic coupling of the potts model the first period doubling corresponds to a multicritical point in the phase diagram of the potts model

relational compositions in fuzzy class theory we present a method for mass proofs of theorems of certain forms in a formal theory of fuzzy relations and classes the method is based on formal identification of fuzzy classes and inner truth values with certain fuzzy relations which allows transferring basic properties of sup t and inf r compositions to a family of more than 30 composition related operations including sup t and inf r images pre images cartesian products domains ranges resizes inclusion height plinth etc besides yielding a large number of theorems on fuzzy relations as simple corollaries of a few basic principles the method provides a systematization of the family of relational notions and generates a simple equational calculus for proving elementary identities between them thus trivializing a large part of the theory of fuzzy relations

complete devil s staircase in an ising model with competing interactions an ising model on a regular cayley tree with competing ferro and antiferromagnetic nearest neighbour interactions can be formulated as a discrete two dimensional mapping the authors use this mapping to obtain sequences of modulated phases at low temperatures associated with complete devil s staircases the fractal dimensionality of the staircases increase with temperature at high temperatures the incommensurate phases may occupy regions of finite measure of the phase diagram

first order transitions in percolation models abstract we describe the analytic solution of a potts correlated site random bond percolation model which exhibits a first order percolative transition at the potts critical point the subject is discussed in connection with the droplet theory in lattice spin models

relations in fuzzy class theory this paper studies fuzzy relations in the graded framework of fuzzy class theory fct this includes i rephrasing existing work on graded properties of binary fuzzy relations in the framework of fuzzy class theory and ii generalizing existing crisp results on fuzzy relations to the graded framework our particular aim is to demonstrate that fuzzy class theory is a powerful and easy to use instrument for handling fuzzified properties of fuzzy relations this paper does not rephrase the whole theory of fuzzy relations instead it provides an illustrative introduction showing some representative results with a strong emphasis on fuzzy preorders and fuzzy equivalence relations

on fuzzy strict preference indifference and incomparability relations abstract the existence of fuzzy models for strict preference indifference and incomparability satisfying all classical properties is established a particular class of such models is completely described

ising model with competing interactions on a cayley tree an iterative scheme is developed for a renormalized effective nearest neighbor couplingk r and effective field per sitek r for spins in therth shell of a cayley tree with nearest neighborj and next nearest neighborj interactions between ising spins on the lattice in addition to the expected paramagnetic ferromagnetic and antiferromagnetic phases we find an intermediate range ofj j 0 values wherex r and kr iterate to a continuous or quasicontinuous attractor in thex k plane in this range the local magnetization is mainly chaotic with oscillatory glasslike behavior embedded in the chaos however are regions of periodic and commensurate phases

discrete anchoring bulk substrate coupling in lattice models abstract we formulate the anchoring problem for discrete state lattice models anchoring is the selection of a bulk equilibrium state from a degenerate set of equivalent equilibrium states in semi infinite samples in contact with a substrate a phenomenon widely discussed in the context of liquid crystalline displays as a concrete example we consider this problem for the three state potts model employing two different approximations viz a layered mean field approximation and a bethe lattice approach the anchoring behaviour of the model is shown to be completely determined by the symmetry properties of the hamiltonian

equivalence of ferromagnetic spin models on trees and random graphs recently we have found identical behaviour for various spin models on thin random graphs feynman diagrams and the corresponding bethe lattices in this paper we observe that the ratios of the saddle point equations in the random graph approach are identical to the fixed point s of the recursion relations which are used to solve the models on the bethe lattice the loops in the random graphs thus have no influence on the thermodynamic limit for such ferromagnetic spin models we consider the correspondence explicitly for ising and q state potts models and also note that multispin interaction models on cacti admit a similar correspondence with a randomized version of the cacti graphs which contain loops

an iterative scheme for the 2d annni model the 2d annni model is studied using a new iterative method the method of ring recurrence the authors show that the method produces an iterative scheme which enables the phase diagram and behaviour of the wavevector to be obtained using low order matrices with minimal computational effort they find a one dimensional attractor associated with the incommensurate phase

modulated structures of an ising model with competing nearest neighbour interactions the authors use a two dimensional mapping to analyse an ising model with competing ferromagnetic and antiferromagnetic nearest neighbour interactions on a cayley tree the phase diagram which had some resemblance to the case of an ising spin glass displays some tricritical points and many modulated phases

behavior of the effective fields for a regular ising model on the cayley tree a regular ising model with nearest neighbor interactions ofj and j j 0 on a cayley tree of coordination number 3 is investigated for the behavior of effective fields in a uniform external field the effective fields show periodic and also aperiodic structures in the temperature field plane at absolute zero temperature the equations determining effective fields are reduced to a nonlinear one dimensional iterative equation arithmetic furcations of period and a screening of the furcations are observed

a potts ising correspondence on thin graphs we note that it is possible to construct a bond vertex model that displays q state potts criticality on an ensemble of 3 random graphs of arbitrary topology which we denote as thin random graphs in contrast to the fat graphs of the planar diagram expansion since the four vertex model in question also serves to describe the critical behaviour of the ising model in field the formulation reveals an isomorphism between the potts and ising models on thin random graphs on planar graphs a similar correspondence is present only for q 1 the value associated with percolation

probability measures and hamiltonian models on bethe lattices ii the solution of thermal and configurational problems in a previous paper we introduced a method for the construction of rotationally and translationally invariant probability measures generated by one step markov hamiltonian models on q state site bethe lattices here the corresponding thermal problems are solved by finding the relative free energy which gives complete information on the properties of the models under study configurational problems also can be solved with the present tools as an example the solution of polychromatic correlated site random bond percolation models is found