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

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

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

eigenvalue and eigenfunction asymptotics for regular sturm liouville problems abstract in this paper we obtain asymptotic formulas for eigenvalues eigenfunctions and the reciprocals of the eigenfunction norms for eigenvalue problems associated with the general sturm liouville equation pu r q u 0 having regular endpoints the method is based on a conversion to liouville normal form and an iterative procedure of solving the associated volterra integral equation producing an asymptotic expansion of the solution in higher powers of 1 1 2 as

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

asymptotics of eigenvalues for regular sturm liouville problems with eigenvalue parameter in the boundary condition abstract in this paper we obtain asymptotic estimates of eigenvalues for regular sturm liouville problems having the eigenparameter in the boundary condition the method is based on an iterative procedure solving the associated riccati equation and producing an asymptotic expansion of the solution in the higher powers of 1 1 2 as

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

time dependent schrodinger equations having isomorphic symmetry algebras ii symmetry algebras coherent and squeezed states using the transformations from paper i we show that the schrodinger equations for 1 systems described by quadratic hamiltonians 2 systems with time varying mass and 3 time dependent oscillators all have isomorphic lie space time symmetry algebras the generators of the symmetry algebras are obtained explicitly for each case and sets of number operator states are constructed the algebras and the states are used to compute displacement operator coherent and squeezed states some properties of the coherent and squeezed states are calculated the classical motion of these states is demonstrated

displacement operator squeezed states i time dependent systems having isomorphic symmetry algebras in this paper we use the lie algebra of space time symmetries to construct states which are solutions to the time dependent schrodinger equation for systems with potentials v x g 2 x2 g 1 x g 0 we describe a set of number operator eigenstates states n x that form a complete set of states but which however are usually not energy eigenstates from the extremal state 0 and a displacement squeeze operator derived using the lie symmetries we construct squeezed states and compute expectation values for position and momentum as a function of time we prove a general expression for the uncertainty relation for position and momentum in terms of the squeezing parameters specific examples all corresponding to choices of v x and having isomorphic lie algebras will be dealt with in the following paper ii

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

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

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

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

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

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

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

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

time dependent schrodinger equations having isomorphic symmetry algebras i classes of interrelated equations in this paper we focus on a general class of schrodinger equations that are time dependent and quadratic in x and p we transform schrodinger equations in this class via a class of time dependent mass equations to a class of solvable time dependent oscillator equations this transformation consists of a unitary transformation and a change in the time variable we derive mathematical constraints for the transformation and introduce two examples

symmetry of time dependent schrodinger equations i a classification of time dependent potentials by their maximal kinematical algebras potentials for the time dependent schrodinger equation 1 2 xx v x t x t i t x t are classified according to their space time or kinematical algebras in a search for exactly solvable time dependent models in addition it is shown that their dynamical algebras are isomorphic to their kinematical algebras on the solution space of the schrodinger equation

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

symmetry of time dependent schrodinger equations ii exact solutions for the equation xx 2i t 2g2 t x2 2g1 t x 2g0 t ψ x t 0 the maximal kinematical algebra of the schrodinger equation xx 2i t 2g2 t x2 2g1 t x 2 g0 t x t 0 is known to be the schrodinger algebra s1 the kinematical symmetries are realized as first order differential operators in the space and time variables a subalgebra g of s1 is chosen and from g and its invariants a complete set of commuting observables are constructed the solution space of the schrodinger equation is identified with the appropriate irreducible representation space of g the wave functions simultaneous eigenvectors of the compatible observables are computed as explicit functions of space and time the properties of a system with a potential v x t g2 t x2 g1 t x g0 t are discussed

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

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

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