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

the yang lee edge singularity in spin models on connected and non connected rings renormalization group arguments based on a 3 field theory lead us to expect a certain universal behavior for the density of partition function zeros in spin models with short range interaction such universality has been tested analytically and numerically in different d 1 and higher dimensional spin models in d 1 one finds usually the critical exponent 1 2 recently we have shown in the d 1 blume emery griffiths beg model on a periodic static lattice one ring that a new critical behavior with 2 3 can arise if we have a triple degeneracy of the transfer matrix eigenvalues here we define the d 1 beg model on a dynamic lattice consisting of connected and non connected rings non periodic lattice and check numerically that also in this case we have mostly 1 2 while the new value 2 3 can arise under the same conditions of the static lattice triple degeneracy which is a strong check of universality of the new value of we also show that although such conditions are necessary they are not sufficient to guarantee the new critical behavior

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

non borel summable φ4 theory in zero dimension a toy model for testing numerical and analytical methods the exact analytical solution for the generating functional of the zero dimensional 4 theory with degenerate minima is obtained in the whole complex coupling parameter plane for testing purposes the efficiency and precision of different computing tools proposed in non borel summable field theories to obtain approximate solutions in both perturbative and nonperturbative regimes are analyzed furthermore a new resummation approach is proposed in order to successfully deal with factorially divergent series it provides a representation of the generating function in terms of an unambiguously defined laplace borel integral on the other hand a recent approach called the generalized borel transform is shown to be an accurate and robust technique to capture non perturbative contributions in the coupling parameter an extension of this approach to path integrals is proposed

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

quenched random graphs spin models on quenched random graphs are related to many important optimization problems we give a new derivation of their mean field equations that elucidates the role of the natural order parameter in these models

on the distribution of the effective field for a spin glass on a cayley tree at zero temperature an ansatz for the probability distribution of the effective field for a spin glass on a cayley tree at zero temperature is proposed the continuous part of the distribution is not differentiable at the origin in contrast to an ansatz by katsura and in qualitative agreement with numerical results of de oliveira possible implications are discussed

differential games with ergodic payoff we address a zero sum differential game with ergodic payoff we study this problem via the viscosity solutions of an associated hamilton jacobi isaacs equation under certain condition we establish the existence of a value and prove certain representation formulae

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

well posedness of higher order camassa holm equations abstract we consider higher order camassa holm equations describing exponential curves of the manifold of smooth orientation preserving diffeomorphisms of the unit circle in the plane we establish the existence of global weak solutions we also present some invariant spaces under the action of the equation moreover we prove a weak equals strong uniqueness result

on the absence of breakdown of symmetry for the plane rotator model with long range random interaction we study the plane rotator model with hamiltonian frac 1 2 sum limits x ne y frac jxy cos mathcal o x mathcal o y left x y right 3 mathcal e where jxy for different pair x y are independent symmetric unbounded random variables it is proved that for almost all j all gibbs states p j are invariant by rotation

spin glass order parameter on a bethe lattice an order parameter function for the spin glass state on a bethe lattice of coordination number z is calculated by the use of hierarchical boundary conditions the order parameter is discrete for finite z and becomes a continuous function in the limit z to infinity numerical calculations are performed at zero temperature and zero external field for several values of z

on the mean field ising model in a random external field we use a method developed by van hemmen to obtain the free energy of the mean field ising model in a random external magnetic field some results of previous mean field calculations are confirmed and generalized the tricritical point in the global phase diagram is discussed in detail we also consider different probability distributions of the random fields and provide some proofs regarding the conditions for the existence of a tricritical point

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

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

the statistics of random directed graphs the analysis of random graphs developed by the author principally as a model for polymerization processes is extended to the case of directed random graphs with models of neural nets in mind the principal novelty of the directed case is the representation of the partition function by a complex rather than a real integral and the replacement of simple maxima in asymptotic evaluations by an interesting form of saddle point

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

stability of dynamical systems with competitive controls the degenerate case abstract we study the stability of a dynamical system with two competitive controls to a general closed set the stability is proven to be equivalent to the continuity on such set of the value function of a differential game with maximum cost functional the existence of a suitably defined and continuous lyapunov function is proven to imply the stability our lyapunov function satisfies a differential inequality in the sense of the theory of viscosity solutions

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

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

multicomponent spin models with transitive symmetry groups i general theory abstract a general theory for multicomponent spin models with transitive symmetry groups is developed these symmetry groups are shown to belong to a certain class of permissible groups a number of theorems concerning these groups is proved and large classes of them are explicitly constructed the statistical mechanics of this type of model is discussed a detailed derivation for the phase transitions of these models on cayley trees is given it is shown that the ising potts ashkin teller and clock models are special cases of larger classes of models of this type as completely new models the so called f models are defined as models with a nonabelian regular permissible symmetry group for this type of model the set of all pair energy functions is shown not to form an algebra which results in unique properties not found for the well known models and their generalizations which all possess associated algebras

lyapunov functions for infinite dimensional systems abstract we study lyapunov functions for infinite dimensional dynamical systems governed by general maximal monotone operators we obtain a characterization of lyapunov pairs by means of the associated hamilton jacobi partial differential equations whose solutions are meant in the viscosity sense as evolved in works of tataru and crandall lions our approach also leads to a new sufficient condition for lyapunov pairs generalizing a classical result of pazy

low regularity solutions of the periodic modified two component camassa holm equation this paper studies low regularity periodic solutions of the modified two component camassa holm equation with initial value we prove the existence and c0 well posedness of solutions

second order phase transitions and tricritical points in the random field ising models abstract a method that applies to a wide class of the random field ising models is used to investigate the influence of random field on the critical phenomena the 3d random field ising models with nearest neighbour interactions exhibit tricritical behavior the obtained results express rather large deviations from the mean field theory for particular cases the formulation gives some well known exact results

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

tri hamiltonian duality between solitons and solitary wave solutions having compact support a simple scaling argument shows that most integrable evolutionary systems which are known to admit a bi hamiltonian structure are in fact governed by a compatible trio of hamiltonian structures we demonstrate how their recombination leads to integrable hierarchies endowed with nonlinear dispersion that supports compactons solitary wave solutions having compact support or cusped and or peaked solitons a general algorithm for effecting this duality between classical solitons and their nonsmooth counterparts is illustrated by the construction of dual versions of the modified korteweg de vries equation the nonlinear schr odinger equation the integrable boussinesq system used to model the two way propagation of shallow water waves and the ito system of coupled nonlinear wave equations these hierarchies include a remarkable variety of interesting integrable nonlinear differential equations textcopyright 1996 the american physical society

the yang lee edge singularity on feynman diagrams we investigate the yang lee edge singularity on non planar random graphs which we consider as the feynman diagrams of various d 0 field theories in order to determine the value of the edge exponent we consider the hard dimer model on and random graphs to test the universality of the exponent with respect to coordination number and the ising model in an external field to test its temperature independence the results here for generic thin random graphs provide an interesting counterpoint to the discussion by staudacher of these models on planar random graphs

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

multicomponent spin models with transitive symmetry groups ii special cases on a cayley tree l 2 l 4 k 4 f 6 abstract some of the multicomponent spin models with transitive symmetry groups derived in the first paper of this series are solved in detail on the cayley tree pseudo lattice the symmetry groups treated are i the group of the cube l 2 l 4 ii the klein group k 4 equivalent to the ashkin teller model and iii the group f 6 as an example of a nonabelian regular permissible group three dimensional phase diagrams in the ferromagnetic unit cube are presented for all cases the connection between the small field cayley tree phase transitions and those predicted for real lattices by the bethe approximation is clarified for the general case

well posedness and blow up solution for a new coupled camassa holm equations with peakons a new camassa holm system with two component admitting peakon solitons is proposed the local well posedness for the system is established a criterion and a condition on the initial data guaranteeing the development of singularities in finite time for strong solutions are obtained and an existence result for a class of local weak solution is also given

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 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

a note on the non perturbative zero dimensional λϕ4 model abstract we exhibit the partition function of the zero dimensional 4 model as a simple exact expression in terms of the macdonald function for re 0 then by analytic continuation we obtain an expression defined in the complex coupling constant plane for arg consequently the partition function understood as an analytic continuation is defined for all values of except for a branch cut along the real negative axis this shows that at least in zero dimension the partition function can be defined for negative coupling constant where the integral is formally divergent provided it has a non vanishing imaginary part we also evaluate the partition function on perturbative grounds using the borel summation technique and we found that in the common domain of validity for re 0 it coincides precisely with the exact expression furthermore a connection between the non perturbative zero dimensional solution and the ultralocal 4 model in arbitrary dimension d is presented

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

differential games and zubov s method in this paper we provide generalizations of zubov s equation to differential games without the isaacs condition we show that both generalizations of zubov s equation which we call the min max and max min zubov equation respectively possess unique viscosity solutions which characterize the respective controllability domains as a consequence we show that under the usual isaacs condition the respective controllability domains as well as the local controllability assumptions coincide

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

statistical mechanics of z m models on cayley trees abstract the problem of small field phase transitions for z m models on cayley trees is solved in detail phase diagrams for zero field are obtained for m 2 3 4 5 and 6 as special cases potts models are also considered and all phases not only those in zero field are identified the m limit the planar rotator model is also solved completely for the zero field case the relevance of the results especially of the phase diagrams for the problems associated with z m models on real lattices is discussed

a converse lyapunov theorem for almost sure stabilizability we prove a converse lyapunov theorem for almost sure stabilizability and almost sure asymptotic stabilizability of controlled diffusions given a stochastic system a s stochastic open loop stabilizable at the origin we construct a lower semicontinuous positive definite function whose level sets form a local basis of viable neighborhoods of the equilibrium this result provides with the direct lyapunov theorems proved in a companion paper a complete lyapunov like characterization of the a s stabilizability

a viscosity approach to infinite dimensional hamilton jacobi equations arising in optimal control with state constraints we consider nonlinear optimal control problems with state constraints and nonnegative cost in infinite dimensions where the constraint is a closed set possibly with empty interior for a class of systems with a maximal monotone operator and satisfying certain stability properties of the set of trajectories that allow the value function to be lower semicontinuous we prove that the value function is a viscosity solution of the bellman equation and is in fact the minimal nonnegative supersolution

almost sure stabilizability of controlled degenerate diffusions we develop a direct lyapunov method for the almost sure open loop stabilizability and asymptotic stabilizability of controlled degenerate diffusion processes the infinitesimal decrease condition for a lyapunov function is a new form of hamilton jacobi bellman partial differential inequality of second order we give local and global versions of the first and second lyapunov theorems assuming the existence of a lower semicontinuous lyapunov function satisfying such an inequality in the viscosity sense an explicit formula for a stabilizing feedback is provided for affine systems with smooth lyapunov function several examples illustrate the theory

singular solutions of a modified two component camassa holm equation the camassa holm ch equation is a well known integrable equation describing the velocity dynamics of shallow water waves this equation exhibits spontaneous emergence of singular solutions peakons from smooth initial conditions the ch equation has been recently extended to a two component integrable system ch2 which includes both velocity and density variables in the dynamics although possessing peakon solutions in the velocity the ch2 equation does not admit singular solutions in the density profile we modify the ch2 system to allow a dependence on the average density as well as the pointwise density the modified ch2 system mch2 does admit peakon solutions in the velocity and average density we analytically identify the steepening mechanism that allows the singular solutions to emerge from smooth spatially confined initial data numerical results for the mch2 system are given and compared with the pure ch2 case these numerics show that the modification in the mch2 system to introduce the average density has little short time effect on the emergent dynamical properties however an analytical and numerical study of pairwise peakon interactions for the mch2 system shows a different asymptotic feature namely besides the expected soliton scattering behavior seen in overtaking and head on peakon collisions the mch2 system also allows the phase shift of the peakon collision to diverge in certain parameter regimes

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

the yang lee zeros of the 1d blume capel model on connected and non connected rings we carry out a numerical and analytic analysis of the yang lee zeros of the 1d blume capel model with periodic boundary conditions and its generalization on feynman diagrams for which we include sums over all connected and non connected rings for a given number of spins in both cases for a specific range of the parameters the zeros originally on the unit circle are shown to depart from it as we increase the temperature beyond some limit the curve of zeros can bifurcate and become two disjoint arcs as in the 2d case we also show that in the thermodynamic limit the zeros of both blume capel models on the static connected ring and on the dynamical feynman diagrams lattice tend to overlap in the special case of the 1d ising model on feynman diagrams we can prove for arbitrary number of spins that the yang lee zeros must be on the unit circle the proof is based on a property of the zeros of legendre polynomials

inclusion of short range order in a mean field theory for the disordered magnetic lattice gas the disordered magnetic lattice gas dmlg as a unifying description of many simpler random spin systems has been investigated in an attempt to devise a mean field theory which goes beyond the infinitely long ranged model by incorporating short range order sro the authors have shown rigorously that the local thermodynamic properties of the dmlg on a cayley tree of finite coordination number z are identical to the thermodynamic properties of the dmlg in a pair approximation obtained by using the method of the distribution function further a modified pair approximation for the dmlg is presented which is exactly solvable it is formulated for general random bond distribution functions and is then examined for the special case of gaussian distributions

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

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

comment on ising models on hyperbolic graphs new proofs are given for propositions 1 and 2 of c c wu j stat phys 50 251 1996 the propositions involved upper and lower bounds on the critical temperature for these models besides being more direct than the previous proofs the new proofs improve both bounds

replica symmetry breaking in finite connectivity systems a large connectivity expansion at finite and zero temperature parisi replica symmetry breaking is extended to random bond systems with finite fixed connectivity m 1 the free energy f is explicitly calculated for large m within the first stage of symmetry breaking at finite temperature t the 1 m expansion is found to diverge as t to o indeed direct evaluation at zero temperature shows that the large m expansion is in powers of 1 m introduction of symmetry breaking brings the value of f close approximately 1 for 10 m 20 to numerical estimates of banavar et al the same techniques apply to systems with average finite connectivity

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

higher order susceptibilities of the regular and the random ising model on the cayley tree i explicit expressions for the fourth order susceptibility 4 the fourth derivative of thebulk free energy with respect to the external field are given for the regular and the random bond ising model on the cayley tree in the thermodynamic limit at zero external field the fourth order susceptibility for the regular system diverges at temperature tc 4 2kb 1j ln 1 2 z 1 3 4 1 confirming a result obtained by muller hartmann and zittartz phys rev lett 33 893 1974 herez is the coordination number of the lattice j is the exchange integral andkb is the boltzmann constant the temperatures at which 4 and the ordinary susceptibility 2 diverge are given also for the random bond and the random site ising model and for diluted ising models

yang lee zeros of the two and three state potts model defined on phi3 feynman diagrams we present both analytical and numerical results on the position of partition function zeros on the complex magnetic field plane of the q 2 state ising and the q 3 state potts model defined on phi 3 feynman diagrams thin random graphs our analytic results are based on the ideas of destructive interference of coexisting phases and low temperature expansions for the case of the ising model an argument based on a symmetry of the saddle point equations leads us to a nonperturbative proof that the yang lee zeros are located on the unit circle although no circle theorem is known in this case of random graphs for the q 3 state potts model our perturbative results indicate that the yang lee zeros lie outside the unit circle both analytic results are confirmed by finite lattice numerical calculations

the random bond ising model on the bethe lattice the paper studies the random bond ising model of a spin glass of a tree of coordination number q 3 the model is studied as an example of the method of ring recurrence a set of recursion relations between the moments of the probability distribution of the order parameter are produced it is shown that solutions of the recursion relations exist which form an infinite bifurcating set below a critical temperature tau c the nature of the first such solution is investigated in detail and the critical exponent analogous to beta is found

graph bipartitioning and statistical mechanics the problem of bipartitioning a random graph of fixed finite local valence connectivity so as to minimise the number of cross edges is studied by the application of monte carlo simulation of thermodynamics and annealing this problem is np complete and is relevant as an idealisation of several practical organisation problems evidence is presented for the existence of a multiplicity of metastable states and for non self averaging and ultrametricity in the space of overlaps albeit with a possible critical valence an empirical formula is presented for the optimal cost in excellent accord with values obtained by simulation and those known exactly

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

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

well posedness and blow up solution for a modified two component periodic camassa holm system with peakons considered herein is a modified two component periodic camassa holm system with peakons the local well posedness and low regularity result of solutions are established the precise blow up scenarios of strong solutions and several results of blow up solutions with certain initial profiles are described in detail and the exact blow up rate is also obtained

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