conformal invariance and critical dynamics the author discusses the behaviour of dynamic correlation functions under conformal transformations in particular the exact form of the dynamic scaling function at the critical point for an arbitrary isotropic two dimensional model is obtained with either conserved or nonconserved order parameter

universal configurational structure in two dimensional scalar models monte carlo methods are used to explore the universal configurational structure of two dimensional spin 1 2 spin 1 and border phi 4 models comparison of spin 1 2 and spin 1 data provides evidence that the magnetisation distribution effectively the helmholtz free energy function and its coupling derivative effectively the internal energy function constitute readily accessible signatures of a universality class it is shown that when allowance is made for relatively large corrections to scaling effects the behaviour of the border phi 4 model may be satisfactorily matched to that of the other two models substantiating the view that the border model does indeed belong to the ising universality class

cuntz krieger algebras of infinite graphs and matrices we show that the cuntz krieger algebras of infinite graphs and infinite 0 1 matrices can be approximated by those of finite graphs we then use these approximations to deduce the main uniqueness theorems for cuntz krieger algebras and to compute their k theory since the finite approximating graphs have sinks we have to calculate the k theory of cuntz krieger algebras of graphs with sinks and the direct methods we use to do this should be of independent interest

limitations on universality in the continuous spin ising model the authors give by explicit counterexample numerical evidence that critical exponent universality fails for the two dimensional continuous spin ising model this result is contrary to what has previously been assumed

on universality between two dimensional ising like systems questions concerning the universality between different two dimensional ising like systems have recently been raised with evidence for the existence of several different subclasses of ising like systems being presented some new facts concerning similarity between border model and low temperature spin 1 ising model series analyses are noted for the spin 1 case an alternative determination of the critical temperature determines unequivocally that this model is in the ising universality class and it is suggested that a similar independent result could lead to a similar conclusion in the border model case the possibility of an analytic correction to scaling in the spin 1 model is proposed

monte carlo computation of cluster free energies in the ising model a test for the validity of the capillarity approximation a new monte carlo technique is proposed to compute directly the free energy difference fn 1 fn of two clusters with size n 1 and n the algorithm is applied to the two dimensional square ising model with nearest neighbour interactions results are given for 1 9 but for smaller cluster sizes the obtained energies are larger than predicted by the capillarity model the deviation exhibits no systematic size dependence but a strong temperature dependence the existence of a size independent term in fn is confirmed by the values of the internal energy un obtained as a by product of the calculation

series expansion analysis of corrections to scaling in the three state potts model several extant series for the d 2 q 3 potts model are analysed with a method that explicitly accounts for the effect of confluent corrections to scaling excellent agreement with the conjectured exponents for this model is found as well as confirmation of renormalisation group estimates of the sub leading eigenvalue

universality of amplitude combinations in two dimensional percolation we generated extended low concentration series for the second moment of distances between cluster points a 2 in site and bond percolation on the square lattice and used them to calculate series for the correlation length we analysed these and other series for their critical amplitudes and compared them with results from monte carlo simulations of triangular site square site and square bond percolation the values obtained from the different models for the amplitude combination b 2 0 2 2 0 2 and b are the amplitudes of the second moment of the cluster size distribution m 2 and the strength of the infinite cluster p respectively are all within the range 2 23 0 10 confirming universality

universal features and tail analysis of the order parameter distribution of the two dimensional ising model an entropic sampling monte carlo study we present a numerical study of the order parameter probability density function pdf of the square ising model for lattices with linear sizes l 80 140 a recent efficient entropic sampling scheme combining the wang landau and broad histogram methods and based on the high levels of the wang landau process in dominant energy subspaces is employed we find that for large lattices there exists a stable window of the scaled order parameter in which the full ansatz including the pre exponential factor for the tail regime of the universal pdf is well obeyed this window is used to estimate the equation of state exponent and to observe the behavior of the universal constants implicit in the functional form of the universal pdf the probability densities are used to estimate the universal privman fisher coefficient and to investigate whether one could obtain reliable estimates of the universal constants controlling the asymptotic behavior of the tail regime

monte carlo calculations of the conformal charge the conformal charge is an important quantity which characterizes the nature of the two dimensional phase transition we report a first attempt to use a new numerical method to calculate the conformal charge in this paper we apply our method to the 2 dimensional 4 continuous spin ising model by varying the parameters in the hamiltonian one can change continuously from the known gaussian limit to the ising limit it is well known that the critical points for these two systems are not in the same universality class we study this behavior for the gaussian model the single well 4 model the border model and the double well 4 model for a large lattice our results while giving a good general picture are not so far sufficient to differentiate whether the non gaussian cases studied belong to the ising model universality class or not further studies of other lattice sizes should serve to improve greatly our conclusions

low temperature series expansions for the spin 1 ising model the finite lattice method of series expansion has been used to extend low temperature series for the partition function order parameter and susceptibility of the spin 1 ising model on the square lattice a new formalism is described which uses two distinct transfer matrix approaches in order to significantly reduce computer memory requirements and which permits the derivation of the series to 79th order subsequent analysis of the series clearly confirms that the spin 1 model has the same dominant critical exponents as the spin 1 2 ising model accurate estimates for both the critical temperature and non physical singularities are obtained in addition evidence for a non analytic confluent correction with exponent delta 1 1 1 or 0 1 is found

af embeddability of 2 graph algebras and quasidiagonality of k graph algebras abstract we characterise quasidiagonality of the c algebra of a cofinal k graph in terms of an algebraic condition involving the coordinate matrices of the graph this result covers all simple k graph c algebras in the special case of cofinal 2 graphs we further prove that af embeddability quasidiagonality and stable finiteness of the 2 graph algebra are all equivalent

finite size scaling analysis of the φ 4 field theory on the square lattice monte carlo calculations are performed for the model hamiltonian i r 2 2 i u 4 gf4 i c 2 i j 2 for various values of the parametersr u c in the crossover region from the ising limit r u to the displacive limit r 0 the variable i is a scalar continuous spin variable which can lie in the range i for each lattice site i i is a priori selected proportional to the single site probability in our monte carlo algorithm the critical line is obtained in very good agreement with other previous approaches a decrease of apparent critical exponents deduced from a finite size scaling analysis is attributed to a crossover toward mean field values at the displacive limit the relation of this model to the coarse grained landau ginzburg wilson free energy functional of ising models is discussed in detail and by matching local moments 2 i 4 i to corresponding averages of subsystem blocks of ising systems with linear dimensionsl 5 tol 15 an explicit construction of this coarse grained free energy is attempted self consistency checks applied to this matching procedure show qualitatively reasonable behavior but quantitative difficulties remain indicating that higher order terms are needed for a quantitatively satisfactory description

graphs groupoids and cuntz krieger algebras we associate to each locally finite directed graphgtwo locally compact groupoidsgandg the unit space ofgis the space of one sided infinite paths ing andg is the reduction ofgto the space of paths emanating from a distinguished vertex we show that under certain conditions theirc algebras are morita equivalent the groupoidc algebrac g is the cuntz krieger algebra of an infinite 0 1 matrix defined byg and that the algebrasc g contain thec algebras used by doplicher and roberts in their duality theory for compact groups we then analyse the ideal structure of these groupoidc algebras using the general theory of renault and calculate theirk theory

nonexistence for the kassoy problem in dimensions 1 and 2 modele d ignition pour l explosion thermique a grande energie d activation d un combustible solide

series expansions of the percolation probability for directed square and honeycomb lattices we have derived long series expansions of the percolation probability for site and bond percolation on directed square and honeycomb lattices for the square bond problem we have extended the series from 41 terms to 54 for the square site problem from 16 terms to 37 and for the honeycomb bond problem from 13 terms to 36 analysis of the series clearly shows that the critical exponent beta is the same for all the problems confirming expectations of universality for the critical probability and exponent we find in the square bond case qc 0 3552994 or 0 0000010 beta 0 27643 or 0 00010 in the square site case qc 0 294515 or 0 000005 beta 0 2763 or 0 0003 and in the honeycomb bond case qc 0 177143 or 0 000002 beta 0 2763 or 0 0002 in addition we have obtained accurate estimates for the critical amplitudes in all cases we find that the leading correction to scaling term is analytic i e the confluent exponent delta 1

universal functions in finite size scaling a numerical study in the ising and potts q 3 universality class the universality of the correlation length order parameter and susceptibility amplitudes is studied in the vicinity of the critical point of two dimensional models on strips with periodic boundary conditions for the ising and potts q 3 universality class the results confirm privman fisher conjecture 1984

analytical calculation of two leading exponents of the dilute potts model a potts model on a square lattice with two and four spin interaction and site and bond dilution is shown to be dual to itself the model is mapped onto a vertex problem which in turn is equivalent to a solid on solid model by means of these mappings the dilute potts model can be written as a gaussian like model with staggered and direct periodic fields the leading and next to leading exponents of the potts model are calculated subject to the validity of certain assumptions

non perturbative renormalization using dimensional regularization applications to the epsilon expansion the authors give a prescription for the one loop renormalisation of the imaginary parts of vertex functions in g phi 4 which are generated when g 0 within the dimensional regularisation scheme this involves another non perturbative renormalisation over and above the usual one loop perturbative renormalisation the authors use an extended minimal subtraction scheme so that the resulting renormalisation group functions have a simple dependence on epsilon 4 d in contrast to the usual case however both the beta function and the renormalised coupling gr are in general complex for real g and the fixed point for epsilon gr 0 is non perturbative in epsilon this fixed point determines the imaginary parts of the critical exponents which are generated when epsilon 0 and allows the determination of the high order behaviour of the perturbation series in epsilon for these exponents the generalisation of these ideas to the o n symmetric g phi 2 2 model is also given

series expansions of the percolation probability on the directed triangular lattice we have derived long series expansions of the percolation probability for site bond and site bond percolation on the directed triangular lattice for the bond problem we have extended the series from order 12 to 51 and for the site problem from order 12 to 35 for the site bond problem which has not been studied before we have derived the series to order 32 our estimates of the critical exponent are in full agreement with results for similar problems on the square lattice confirming expectations of universality for the critical probability and exponent we find in the site case qc d 0 404 352 8 0 000 001 0 and d 0 276 45 0 000 10 in the bond case qc d 0 521 98 0 000 01 and d 0 2769 0 0010 and in the site bond case qc d 0 264 173 0 000 003 and d 0 2766 0 0003 in addition we have obtained accurate estimates for the critical amplitudes in all cases we find that the leading correction to scaling term is analytic i e the confluent exponent 1 d 1 in an earlier paper jensen and guttmann 1995 we reported on the derivation and analysis of long series for the percolation probability of site and bond percolation on the directed square and hexagonal lattices in this paper we extend this work to site bond and site bond percolation on the directed triangular lattice we refer to our earlier paper for a more general introduction to directed percolation and its role in the modelling of physical systems in directed site percolation each site is either present with probability p or absent with probability q d 1 p independent of all other sites on the lattice similarly for bond percolation each bond is absent or present independently of other bonds finally in site bond percolation both sites and bonds may be absent or present with equal probability but again with no dependency on any other sites or bonds two sites in the various models are connected if one can find a path respecting the directions indicated in figure 1 through occupied sites bonds or sites and bonds respectively from one to the other when p is smaller than a critical value pc all clusters of connected sites remain finite while for p pc there is an infinite cluster spanning the lattice in the preferred direction the order parameter of the system is the percolation probability p p that a given site belongs to the infinite cluster this quantity is strictly zero when p p c the behaviour of p p in the vicinity of pc may be described by a critical exponent p p p pc p p c c 1

separable representations kms states and wavelets for higher rank graphs letbe a strongly connected finite higher rank graph in this paper we construct faithful representations of c on certain separable hilbert spaces of the form l 2 x by introducing the notion of a semibranching function system a generalization of the semibranching function systems studied by marcolli and paolucci in particular we obtain a faithful representation of c on l 2 1 m where m is the perron frobenius probability measure on the infinite path space 1 recently studied by an huef laca raeburn and sims we also show how a semibranching function system gives rise to kms states for c for the higher rank graphs of robertson and steger we also obtain a faithful representation of c on l 2 x where x is a fractal subspace of 0 1 by embedding 1 into 0 1 as a fractal subset x of 0 1 in this latter case we additionally show that there exists a kms state for c whose inverse temperature is equal to the hausdorff dimension of x finally we construct a wavelet system for l 2 1 m by generalizing the work of marcolli and paolucci from graphs to higher rank graphs

droplet structure in ising and potts models the structure of a recently introduced droplet in the q state potts model is analysed the authors derive exact relations from which it follows that the incipient infinite droplet at the critical temperature is a self similar fractal made of links and blobs just as recently found in random percolation the number of links nlinks between two points separated by a distance of the order b is given by nlinks varies as b1 nu b q where nu b q is the connectedness length exponent when t tc q and pb is used as an independent variable this result and the available calculations of nu b q indicate that the number of links decreases as q increases an analysis of the structure of the usual clusters made of nearest neighbour sites in a given potts site configuration is also made in particular it shows that for d 2 the incipient infinite cluster is made only of blobs and no links

on asymptotic forms of reactive diffusive runaway different aspects of reactive diffusive runaway are discussed in this paper it is shown that a wide range of reactive nonlinearities can be analysed in essentially the same way leading to the conclusion that there appear to be only two distinct non singular self similar or approximately self similar forms of description for non homogeneous blowup these exactly self similar and asymptotically self similar descriptions have previously been recognized but are examined here in some detail for many different types of nonlinearity using asymptotic and numerical techniques it is confirmed that exactly self similar descriptions only behave non pathologically in fairly extreme discrete cases that are very close to a reactive diffusive or steady state balance under three or more dimensions of symmetry quantitative identification of these cases indicates that they may be of dubious practical relevance it is then shown that only one class of non singular asymptotically self similar descriptions can be found using coordinate perturbation techniques resulting solutions are derived to a high asymptotic order and are found to reveal a fairly universal structure for symmetric blowup a simple analysis reveals that non symmetric blowup also fits in with this structure at least to leading order

affine buildings tiling systems and higher rank cuntz krieger algebras to an r dimensional subshift of finite type satisfying certain special properties we associate a c algebra a this algebra is a higher rank version of a cuntz krieger algebra in particular it is simple purely infinite and nuclear we study an example if is a group acting freely on the vertices of an a2 building with finitely many orbits and if is the boundary of that building then c o is the algebra associated to a certain two dimensional subshift

isotropic majority vote model on a square lattice the stationary critical properties of the isotropic majority vote model on a square lattice are calculated by monte carlo simulations and finite size analysis the critical exponents and are found to be the same as those of the ising model and the critical noise parameter is found to beqc 0 075 0 001

time coarse graining near critical points the authors investigate the statistical properties of local variables coarse grained in time in systems undergoing continuous phase transitions a general theory is developed which suggests universal scaling behaviour in the limit of large coarse graining times the predictions are substantiated with monte carlo studies of two dimensional scalar models a simple fractal model of the time profile of the ordering variable is developed which captures and illuminates its timescale invariant character the theory is used to study the critical behaviour of local resonance lineshapes in the two dimensional case studied explicitly the critical slowing down is shown to drive a crossover to a split slow motion lineshape

droplet theory of correlations in ordered phases the explicit droplet theory of low dimensional phase transitions developed recently is extended to yield a description of the pair correlation function for the universality class of q state potts models in the ordered phase and in zero external field it is shown that in d 1 epsilon dimensions the short distance behaviour of the correlation function and in particular the exponent eta are controlled by nested nearly spherical droplets in contrast it is argued that the spatial dependence of the large distance behaviour in d 2 or above is controlled by highly anisotropic droplets thus illuminating the widom relation sigma xi d 1 approximately constant linking the dimensionless surface tension sigma and the bulk correlation length xi in the particular case of d 2 the statistical weight of the relevant droplets is determined as that of two appropriately interacting strings in the spirit of recent independent arguments by abraham and by fisher a non ornstein zernike correlation function prefactor and the result of sigma xi 1 2 follow in accord with exact results for the q 2 ising case and with implications for the q to 1 percolation problem

the ising model below tc calculation of nonuniversal amplitudes using a primitive droplet model using recent exact results for the surface free energies of ising droplets on a square lattice and applying a primitive droplet theory due to langer 1967 the author obtains an expression for the principal nonuniversal amplitude appearing in the imaginary part of the analytic continuation of the free energy across the coexistence line for the square lattice ising model below tc the significance of this calculation is the known dependence on this amplitude of the large order terms in the expansion of the physical free energy in powers of h thus allowing a direct numerical comparison with existing series data to be made after carrying out a simple renormalisation of the droplet volume term the author finds excellent numerical agreement

spectral triples and wavelets for higher rank graphs abstract in this paper we present a new way to associate a finitely summable spectral triple to a higher rank graph via the infinite path space of moreover we prove that this spectral triple has a close connection to the wavelet decomposition of which was introduced by farsi gillaspy kang and packer in 2015 we first introduce the concept of stationary k bratteli diagrams in order to associate a family of ultrametric cantor sets and their associated pearson bellissard spectral triples to a finite strongly connected higher rank graph we then study the zeta function abscissa of convergence and dixmier trace associated to the pearson bellissard spectral triples of these cantor sets and show these spectral triples are regular in the sense of pearson and bellissard we obtain an integral formula for the dixmier trace given by integration against a measure and show that is a rescaled version of the measure m on which was introduced by an huef laca raeburn and sims finally we investigate the eigenspaces of a family of laplace beltrami operators associated to the dirichlet forms of the spectral triples we show that these eigenspaces refine the wavelet decomposition of l 2 m which was constructed by farsi et al

universal local dynamics the authors suggest that near a critical point the statistical behaviour of local variables averaged over times long on microscopic scales is universal reflecting large scale partial structure the contention is supported with monte carlo studies of two dimensional scalar models

blowup in a partial differential equation with conserved first integral a reaction diffusion equation with a nonlocal term is studied the nonlocal term acts to conserve the spatial integral of the unknown function as time evolves such equations give insight into biological and chemical problems where conservation properties predominate the aim of the paper is to understand how the conservation property affects the nature of blowup the equation studied has a trivial steady solution that is proved to be stable existence of nontrivial steady solutions is proved and their instability established numerically blowup is proved for sufficiently large initial data by using a comparison principle in fourier space the nature of the blowup is investigated by a combination of asymptotic and numerical calculations

finite size scaling study of the equilibrium cluster distribution of the two dimensional ising model the authors use a very fast and efficient algorithm to study by monte carlo methods the equilibrium cluster distribution cl l and the mean number of clusters per lattice site containing l particles in a square lattice of l2 sites of the two dimensional ising model at the critical point finite size scaling theory is then used to analyse the scaling ansatz cl l l tau f ls l tau and s being critical exponents the second moment of the cluster distribution p2 l sigma l2cl is shown to behave as p2 l approximately ltheta with theta 1 895 or 0 010 the effect of corrections to scaling is also discussed

simulation of static critical phenomena in nonideal fluids with the lattice boltzmann method a fluctuating nonideal fluid at its critical point is simulated with the lattice boltzmann method it is demonstrated that the method employing a ginzburg landau free energy functional correctly reproduces the static critical behavior associated with the ising universality class a finite size scaling analysis is applied to determine the critical exponents related to the order parameter compressibility and specific heat a particular focus is put on finite size effects and issues related to the global conservation of the order parameter

scaling at the percolation threshold above six dimensions the fractal dimensionality of the infinite cluster at the percolation threshold for dimensionalities d 6 is shown to be d 4 rather than the naive finite size scaling prediction d d 2 similarly the conductivity of a sample of size l scales as l d rather than l 6 this anomalous behaviour is related to a dangerous irrelevant variable associated with the probability to have vertices of three bonds the crossover to the homogeneous behaviour occurs at length scales which are short compared with the correlation length the links and blobs picture is confirmed for d 6 and the size of the latter is estimated

von neumann algebras of strongly connected higher rank graphs we investigate the factor types of the extremal kms states for the preferred dynamics on the toeplitz algebra and the cuntz krieger algebra of a strongly connected finite k graph for inverse temperatures above 1 all of the extremal kms states are of type mathrm i infty at inverse temperature 1 there is a dichotomy if the k graph is a simple k dimensional cycle we obtain a finite type mathrm i factor otherwise we obtain a type iii factor whose connes invariant we compute in terms of the spectral radii of the coordinate matrices and the degrees of cycles in the graph

primitive ideal space of higher rank graph c algebras and decomposability abstract in this paper we describe primitive ideal space of the c algebra c associated to any locally convex row finite k graph to do this we will apply the farthing s desourcifying method on a recent result of carlsen kang shotwell and sims we also characterize certain maximal ideals of c furthermore we study the decomposability of c we apply the description of primitive ideals to show that if i is a direct summand of c then it is gauge invariant and isomorphic to a certain k graph c algebra so we may characterize decomposable higher rank c algebras by giving necessary and sufficient conditions for the underlying k graphs moreover we determine all such c algebras which can be decomposed into a direct sum of finitely many indecomposable c algebras

monte carlo and series study of corrections to scaling in the baxter wu model abstract we compare the results of extensive monte carlo simulations with the exact solution and the series expansion for the baxter wu model analysis of the series and of the monte carlo simulations show the leading corrections to the critical magnetization and energy to be nonanalytic and compatible with the exact solution an in depth discussion of the uniqueness of parameter determination for both cases is given

synergistic development of differential approximants and the finite lattice method in lattice statistics several decades of parallel developments in the calculation and analysis of series expansions for lattice statistics have led to many new insights into critical phenomena these studies have centered on the use of the finite lattice method for series expansions in lattice statistics and the use of differential approximants in analysing such series one of these strands of research ultimately led to the result that a number of unsolved lattice statistics problems cannot be expressed as d finite functions somewhat ironically given power and success of differential approximants in analysing series neither the assumed functional form nor any finite generalisation thereof can fit such cases exactly

metric properties of fractal lattices the authors study the connectivity dimension d of fractal lattices viewed as networks graphs of sites and constant length bond two examples are investigated in detail the 2d sierpinski gasket and 2d infinite percolation clusters on square and triangular lattices in the first case d is shown to coincide with the fractal dimension whereas d 1 72 or 0 02 appears as a universal exponent for percolation clusters in two dimensions considered as an intrinsic parameter the connectivity dimension d is compared with other intrinsic and extrinsic characteristic parameters of fractal lattices in particular the authors argue that d or d or d holds on fractal lattices in general d fractal dimension d spectral dimension

droplet theory in low dimensions ising systems in zero field the authors develop a theory of the universal configurational physics underlying critical point phenomena in the ising universality class the theory is formally justifiable in d 1 epsilon dimensions and may be regarded as the natural continuation of the kink based theory of one dimension which it incorporates as a limiting case in d 1 epsilon the configurational building block is the droplet the typical droplet is not spherical and the many droplet assembly is not dilute the implied problems are handled with renormalisation group methods it is found that droplet shape fluctuation effects control the correlation length exponent nu while the nesting of droplets within droplets control the order parameter exponent beta the exponents nu and beta thus effectively define respectively the fractal dimensions of the droplet surface and the droplet volume the theory is used to determine and illuminate the critical behaviour of further quantities including the free energy the susceptibility the droplet number distribution and the distribution of the intra droplet order

confluent singularities in directed bond percolation existing series for directed bond percolation on the square honeycomb and face centred cubic lattices are analysed with a view to detecting their confluent singularities the authors compare our results with the confluent singularities in reggeon field theory which has been shown to be in the same universality class

the two dimensional spin 1 ising system and related models although the two dimensional spin 1 2 ising model was solved in zero field in 1944 no exact results are yet available for the spin 1 model this model should have the same dominant critical exponents as the spin 1 2 model but appears to exhibit non analytic corrections to scaling as expected for a phi 4 model and unlike the spin 1 2 model in this paper new 45 term low temperature series for the magnetisation specific heat and susceptibility of the spin 1 model on the square lattice are presented and analysed a new 24 term staggered susceptibility series for the hard square model is presented and the extant order parameter series for this model which is also in the phi 4 universality class are also considered for both models a non analytic confluent correction with exponent 1 0 delta 1 1 3 is found the validity of this result is enhanced by a comparison with the s 1 2 case

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

anisotropic voter model a majority vote model subject to anisotropic voting rules is studied in two dimensions using a first order mean field approximation and monte carlo simulations the critical behavior is consistent with the 2d ising universality class

universality in the two dimensional continuous spin model monte carlo methods have been used to analyse the critical behaviour of the d 2 border phi 4 model which according to a recent series expansion study violates expectations based upon universality the results suggest that the model does in fact fall into the ising universality class at least as regards those parameters which characterise the critical point configurations of the ordering variable

transfer matrix calculation of the relative noise exponent in a two dimensional percolating network a new method is given for calculating the relative noise exponent describing the size dependence of macroscopic fluctuations due to microscopic resistance noise of a random resistor network at the percolation threshold pc the macroscopic fluctuation is calculated exactly using the transfer matrix idea in a given network the asymptotic exponent b is found by finite size scaling

kms states on the c algebra of a higher rank graph and periodicity in the path space we study the kms states of the c algebra of a strongly connected finite k graph we find that there is only one 1 parameter subgroup of the gauge action that can admit a kms state the extreme kms states for this preferred dynamics are parameterised by the characters of an abelian group that captures the periodicity in the infinite path space of the graph we deduce that there is a unique kms state if and only if the k graph c algebra is simple giving a complete answer to a question of yang when the k graph c algebra is not simple our results reveal a phase change of an unexpected nature in its toeplitz extension

probability distribution of magnetization in the one dimensional ising model effects of boundary conditions finite size scaling functions are investigated both for the mean square magnetization fluctuations and for the probability distribution of the magnetization in the one dimensional ising model the scaling functions are evaluated in the limit of the temperature going to zero t 0 the size of the system going to infinity n while n 1 tanh j kbt is kept finite j being the nearest neighbour coupling exact calculations using various boundary conditions periodic antiperiodic free block demonstrate explicitly how the scaling functions depend on the boundary conditions we also show that the block small part of a large system magnetization distribution results are identical to those obtained for free boundary conditions

droplet theory in low dimensions potts models and percolation a recently developed droplet theory of low dimensional ising systems is generalised to describe the q state potts universality class and thence the percolation problem realised in the q to 1 limit an integral equation for the potts model free energy is derived and used to obtain the generating function for the cluster size probability distribution in the percolation problem above the percolation threshold a closed equation for the distribution is obtained and shown to yield scaling behaviour the form of the distribution is determined analytically in the regimes of small and large cluster sizes yielding respectively power law decay and kunz souillard exponential decay the form of the distribution in the regime of intermediate cluster sizes is determined numerically in space dimension d 2

universal ratio of magnetization moments in two dimensional ising models the authors calculate the universal critical point ratios of the square of the second and the fourth moment of the magnetization for ferromagnetic ising models on the square and on the triangular lattices periodic boundary conditions are used in accordance with the four fold and six fold rotational symmetries of the respective lattices these results which are obtained by means of an analysis of finite size data computed with a transfer matrix technique have an accuracy of the order of one millionth this analysis is also applied to rectangular systems with arbitrary aspect ratios

the self avoiding walk on the honeycomb lattice the author analyses the chain generating function for the self avoiding walk problem on the honeycomb lattice she uses the exact critical point value of nienhuis and finds exponent values of gamma approximately 1 344 and delta 1 approximately 0 93 for the dominant and first confluent exponents the former value is in agreement with nienhuis exact result second and third confluent exponents delta 2 approximately 1 2 and delta 3 approximately 1 5 are also identified these exponents being consistent with monte carlo results and nienhuis exact irrelevant eigenvalue respectively

the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources abstract we catalogue the primitive ideals of the cuntz krieger algebra of a row finite higher rank graph with no sources each maximal tail in the vertex set has an abelian periodicity group of finite rank at most that of the graph the primitive ideals in the cuntz krieger algebra are indexed by pairs consisting of a maximal tail and a character of its periodicity group the cuntz krieger algebra is primitive if and only if the whole vertex set is a maximal tail and the graph is aperiodic

stability of the isotropic fixed point near one dimension within the context of the l epsilon expansion and the near planar interface hamiltonian the isotropic fixed point is found to be stable to a wide class of perturbations in particular the universality class includes all non singular forms of anisotropic surface tension

two three state potts models the universality of confluent corrections to scaling the authors present a study of an extended 35 power low temperature series for the susceptibility of the ordinary two site three state potts model on the square lattice as well as reanalyses of several extant series for the three site three state potts model on the triangular lattice dominant and confluent singularities are investigated and the exponents of both are found to be universal and in agreement with known and conjectured exact values

critical temperatures of the d 3 s 1 2 ising model the effect of confluent corrections to scaling much attention has been paid to the d 3 s 1 2 ising model on the bcc lattice in recent years because the best high temperature series expansions were available for this lattice however in order to compare series estimates of critical exponents and temperatures with independent evaluations via the monte carlo method lattices of lower coordination number must be considered since monte carlo studies are usually made on these a study of extant susceptibility series on the fcc sc diamond and tetrahedron lattices is reported on and some comments are made on aspects of recent monte carlo analyses of the d 3 s 1 2 ising model on the sc and diamond lattices

droplet theory of low dimensional potts models formulates a generalisation of the droplet theory of ising systems to droplet systems containing more than two phases valid near one dimension and based on the configurational energy of surface tension thermal and magnetic exponents for these multi phase systems are compared with the results for a q state potts model in the external field the potts exponents are obtained using a migdal recursion formula and are shown to be of very similar structure

extrapolation procedure for low temperature series for the square lattice spin 1 ising model the finite lattice method of series expansions has been combined with a new extrapolation procedure to extend the low temperature series for the specific heat spontaneous magnetization and susceptibility of the spin 1 ising model on the square lattice the extended series were derived by directly calculating the series to order 99 in the variable and using the new extrapolation procedure to calculate an additional 13 14 terms

droplet theory in low dimensions ising systems in an ordering field the authors extend the microscopic droplet theory of ising systems developed recently to incorporate the effects of an ordering field the theory yields a free energy which is the solution of a renormalisation group equation describing droplet nesting and which has a full scaling form the behaviour near the coexistence curve is investigated and found to display the essential singularity suggested by primitive droplet models but with parameters renormalised in a physically intelligible way the droplet population function implied by the theory is parametrised in a simple way in space dimension d 2 to yield predictions for various thermodynamic properties including the equation of state which are found to be in fair accord with series based results the structure of the field dependent droplet number distributions is investigated and contrasted with the forms assumed in phenomenological droplet theories the domain of validity of the theory is assessed it is concluded that the theory fails to describe sufficiently large droplets in the presence of a finite field and is thus in principle trustworthy only in a region close to the coexistence curve

a generalized cuntz krieger uniqueness theorem for higher rank graphs we present a uniqueness theorem for k graph c algebras that requires neither an aperiodicity nor a gauge invariance assumption specifically we prove that for the injectivity of a representation of a k graph c algebra it is sufficient that the representation be injective on a distinguished abelian c subalgebra a crucial part of the proof is the application of an abstract uniqueness theorem which says that such a uniqueness property follows from the existence of a jointly faithful collection of states on the ambient c algebra each of which is the unique extension of a state on the distinguished abelian c subalgebra

finite size scaling of the free energy levels in the ising and potts q 3 universality the finite size scaling of the free energy levels of privman and fisher 1984 is tested numerically on two dimensional lattice strips in the ising and q 3 potts universality the universal amplitudes and the free energy per site are obtained

are critical finite size scaling functions calculable from knowledge of an appropriate critical exponent critical finite size scaling functions for the order parameter distribution of the two and three dimensional ising model are investigated within a recently introduced classification theory of phase transitions the universal part of critical finite size scaling functions has been derived by employing a scaling limit which differs from the traditional finite size scaling limit in this paper the analytical predictions are compared with monte carlo simulation results we find good agreement between the analytical expression and the simulation results the agreement is consistent with the possibility that the functional form of the critical finite size scaling function for the order parameter distribution is determined uniquely by only a few universal parameters most notably the equation of state exponent

the mean size of ising clusters a confluent singularity analysis the authors re examine using a confluent singularity analysis existing low temperature series for the mean size of finite clusters for the pure spin 1 2 nearest neighbour ising model in two dimensions although the triangular lattice is problematical results for the honeycomb and square lattices may be interpreted as indicating a dominant exponent theta 1 87 or 0 04 consistent with the theoretical prediction theta gamma beta together with competing analytic and non analytic delta 1 1 35 or 0 25 correction to scaling terms