the relation between amplitudes and critical exponents in finite size scaling the relation between critical exponents and the amplitude of the correlation length divergence at the critical point of two dimensional systems as a function of finite size is investigated and generalised in two ways correlations more general than those of order order type are included a form appropriate for anisotropic systems is proposed the authors present a exact results for the ising and gaussian models and b numerical results for the symmetric eight vertex baxter continuous q state potts and continuous n component cubic models

precise critical exponents for the basic contact process we calculated some of the critical exponents of the directed percolation universality class through exact numerical diagonalizations of the master operator of the one dimensional basic contact process perusal of the power method together with finite size scaling allowed us to achieve a high degree of accuracy in our estimates with relatively little computational effort a simple reasoning leading to the appropriate choice of the microscopic time scale for time dependent simulations of markov chains within the so called quantum chain formulation is discussed our approach is applicable to any stochastic process with a finite number of absorbing states

quantum operators in classical probability theory iv quasi duality and thinnings of interacting particle systems duality has proved to be a powerful technique in the study of interacting particle systems ips this concept can be enlarged and a quasi duality defined between various pairs of ips previously thought unrelated consequently theorems of a similar style to those involving duality can be deduced the concept of quasi duality follows naturally from our previous studies into the use ofsingle site operators an idea borrowed from quantum physics in paper ii of this series it is shown that a necessary condition for quasi duality is that the eigenvalues of the corresponding two site infinitesimal generators be the same and using this observation a number of quasi dual pairs have been found and studied it is further shown that if two different ips share a common dual then one can be considered as a thinning of the other

finite size scaling for directed bond percolation with and without cycles on a triangular lattice the authors apply finite size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic directed cyclic and undirected bond percolation on a triangular lattice the results are in good agreement with known estimates and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases as well as the distinction between locally directed and fully isotropic problems in addition the authors draw tentative comments on the extension to directed problems of the relation a pi eta where a is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point known to hold for various isotropic systems

conformal invariance and universality in finite size scaling the universal relation between critical exponents and the amplitude of the correlation length divergence as a function of finite size at the critical point of two dimensional systems is shown to be a consequence of conformal invariance both periodic and free boundary conditions are considered

finite size scaling and the two dimensional xy model shows that the finite size scaling assumption and particularly a conjectured relation involving the exponent eta are valid in the low temperature phase of the 2d xy model order by order in a perturbative expansion of the two point correlation function

universal finite size scaling functions for critical systems with tilted boundary conditions we calculate finite size scaling functions fssf s of binder parameter g and magnetization distribution function p m for the ising model on l 1 ifmmode times else texttimes fi l 2 square lattices with periodic boundary conditions in the horizontal l 1 direction and tilted boundary conditions in the vertical l 2 direction such that the i mathrm th site in the first row is connected with the mathrm mod i cl 1 l 1 mathrm th site in the l 2 row of the lattice where 1l il l 1 for fixed sets of a c with a l 1 l 2 the fssf s of g and p m are universal and in such cases a c 2 a 2 1 is an invariant for percolation on lattices with fixed a the fssf of the existence probability also called spanning probability is not affected by c

the interaction representation in the self focusing theory abstract the general scheme of dividing the solution of the nonlinear schrodinger equation 2d case into the focused singular part of the beam called the peak and the rest nonsingular part the background is presented the peak is compared with a manifold of self similar modes whose dynamics is analysed it is shown that if the beam power is less than critical the self similar beam is strictly localized if the beam power is more than critical the self similar solution is a quasi localized mode with tunnel emission of power from the paraxial region and the width of the peak tends to zero the parametric peak background interaction analysis proves that the limiting self focusing law a t t 0 t ln ln t 0 t 1 2 is determined only by the peak dynamics

generalized mittag leffler functions in the theory of finite size scaling for systems with strong anisotropy and or long range interaction the difficulties arising in the investigation of finite size scaling in d dimensional o n systems with strong anisotropy and or long range interaction decaying with the interparticle distance r as r d 0 2 are discussed some integral representations aiming at the simplification of the investigations are presented for the classical and quantum lattice sums that take place in the theory special attention is paid to a more general form allowing to treat both cases on an equal footing and in addition cases with strong anisotropic interactions and different geometries the analysis is simplified further by expressing this general form in terms of a generalization of the mittag leffler special functions this turned out to be very useful for the extraction of asymptotic finite size behaviours of the thermodynamic functions

anisotropic scaling and generalized conformal invariance at lifshitz points the behaviour of the 3d axial next nearest neighbor ising model at the uniaxial lifshitz point is studied using monte carlo techniques a new variant of the wolff cluster algorithm permits the analysis of systems far larger than in previous studies the lifshitz point critical exponents are ensuremath alpha phantom rule 0ex 0ex phantom rule 0ex 0ex 0 18 2 ensuremath beta phantom rule 0ex 0ex phantom rule 0ex 0ex 0 238 5 and ensuremath gamma phantom rule 0ex 0ex phantom rule 0ex 0ex 1 36 3 data for the spin spin correlation function are shown to be consistent with the explicit scaling function derived from the assumption of local scale invariance which is a generalization of conformal invariance to the anisotropic scaling at the lifshitz point

computer simulation of wave collapses in the nonlinear schro dinger equation abstract the self tuning numerical scheme for simulation of collapse phenomena in the framework of the nonlinear schrodinger equation is developed the scheme allows to reach extremely high field amplitudes 2 0 2 10 15 without lack of validity numerical results are compared with analytical predictions

directed percolation a finite size renormalisation group approach the finite size renormalisation group technique introduced by nightingale 1976 is applied to the directed percolation problem the decay of correlations is anisotropic in this model and finite size scaling is extended to treat such anisotropy precise estimates for critical exponents and percolation probabilities are obtained for site bond and site bond percolation on the square lattice with bonds directed along the positive axes both free boundary conditions for which the results converge linearly with 1 n as n to infinity and helical boundary conditions for which unexpectedly the results converge linearly with 1 n3 are considered

sharper criteria for the wave collapse abstract sharper criteria for three dimensional wave collapse described by the nonlinear schrodinger equation nlse are derived the collapse threshold corresponds to the ground state soliton which is known to be unstable thus for nonprefocusing distributions this represents the separatrix between collapsing and noncollapsing sectors numerical results support the theoretical results generalizations of the criteria for the nlse with arbitrary power nonlinearity are also presented

local structure of the self focusing singularity of the nonlinear schro dinger equation abstract for the cubic schrodinger equation in two dimensions we construct a family of singular solutions by perturbing slightly the dimension d 2 tod 2

the survival of nonattractive interacting particle systems on z we consider interacting particle systems on z which allow five types of pairwise interaction annihilation birth coalescence death and exclusion with corresponding rates a b c d e we show that whatever the values of a c d e if the birthrate is high enough there is a positive probability the particle system will survive starting from any finite occupied set in particular an ips with rates a b c d e has a positive probability of survival if b 4d 6a c a d e or b 7d 3a 3c 3e c a d e we create a suitable supermartingale by extending the method used by holley and liggett in their treatment of the contact process

on the symmetry of universal finite size scaling functions in anisotropic systems we postulate a symmetry of universal finite size scaling functions under a certain anisotropic scale transformation which connects the properties of a finite two dimensional 2d system at criticality with generalized aspect ratio 1 to a system with 1 the symmetry is formulated within a finite size scaling theory and expressions for several universal amplitude ratios are derived the predictions are confirmed within the exactly solvable weakly anisotropic 2d ising model and are checked within the strongly anisotropic 2d dipolar in plane ising model using monte carlo simulations

finite lattice methods in quantum hamiltonian field theory i o 2 and o 3 heisenberg models for pt i see ibid vol 14 p 241 57 1981 two efficient methods for finding the low lying states of hamiltonians on finite lattices are described the first involves constructing a finite representation of the hamiltonian using strong coupling eigenstates while the second is based on the lanczos recursion method the methods are used to determine the mass gap of the o 2 and o 3 heisenberg hamiltonians in 1 1 dimensions for a sequence of finite chains the critical behaviour of the infinite chain is then analysed by extrapolating the finite lattice estimates using finite size scaling a remarkably sensitive test is developed for the presence of a phase transition for the o 2 model data this test yields strong evidence for a phase transition with the weak coupling phase massless while in the o 3 case the test supports although more weakly the absence of any transition

universal amplitudes in finite size scaling generalisation to arbitrary dimensionality the relationship between the correlation length and critical exponents in finite width strips in two dimensions is generalised to cylindrical geometries of arbitrary dimensionality d for d not 2 these correspond however to curved spaces the result is verified for the spherical model

on the possibility of soft and hard turbulence in the complex ginzburg landau equation abstract we present analytical methods which predict the occurrence of both soft weak and hard strong turbulence in the complex ginzburg landau cgl equation a t ra 1 i a 1 i a a 2 on a periodic domain 0 1 d in d spatial dimensions hard turbulence is characterised by large fluctuations away from spatial and temporal averages with a cascade of energy to small scales this form of hard turbulence appears to occur not in 1d but only in 2d and 3d in parameter regions which are bounded by hyperbolic curves in the second and fourth quadrants of the planes where the system is modulationally unstable 1 2n f n n 1 a 2 n a 2n dx for n 0 for large times and large r upper bounds exist for the infinite set of fn s constructed from the hierarchy of differential inequalities fn 2nr cn a 2 fn bnf2n fn 1 for cn bn 0 f0 1 estimates for the bottom rung f2 give upper bounds for the whole ladder long time upper bounds on f2 and a 2 and hence all fn are well controlled in the soft region but become much larger in the hard region whereas spati al and temporal averages remain comparatively small when the nonlinearity is a a 2q the critical case qd 2 gives parallel results

lower bounds for critical values of a cancellative model we consider the following one dimensional discrete time cancellative model whose evolution is given by n 1 x n x 1 n x 1 mod 2 with probability p and n 1 0 with probability 1 p concerning critical probabilities pc and pc on a survival probability it is known that 0 706 pc pc 1 under a condition in this paper we give improved lower bounds of 0 771 and 0 781 on pc and pc respectively by finding suitable supermartingales for the model

on the stability of collapse in the critical case collapsing solutions in the cauchy problem of the nonlinear schrodinger equation i t 2 p 0 x rd are considered in the so called critical case pd 4 where d is the spatial dimension a stability theorem for radial collapse is presented which proves that the formation of the singularity remains close to the self similar collapsing solution with a spatial profile given by the ground state solitary wave provided the energy h 0 an analogous result is given for the scalar zakharov equations and a generalized korteweg de vries equation tu p 1 up xu 2 xu 0

quantum operators in classical probability theory ii the concept of duality in interacting particle systems duality has proved to be a powerful tool in the theory of interacting particle systems the approach in this paper is algebraic rather than via harris diagrams a form of duality is found which includes coalescing and annihilating duality as special cases this enables new results for the branching annihilating random walk and the biased annihilating branching process to be derived

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

finite size scaling studies of the o 2 heisenberg spin model in d 1 dimensions the finite size scaling behaviour of the o 2 heisenberg spin model in d 1 dimensions d 1 and 2 is computed in the low temperature region using the stochastic truncation monte carlo method the numerical results so obtained agree very well with effective lagrangian theory and with spin wave perturbation theory

phenomenological renormalisation of the self avoiding walk in two dimensions phenomenological renormalisation is used to calculate the exponent v and the connective constant of the self avoiding walk problem on a square lattice a transfer matrix technique is developed for the polymer problem the results indicate that flory s value v 0 75 is true in two dimensions to extremely high accuracy

a numerical scheme for the simulation of blow up in the nonlinear schrodinger equation we present a self adaptive multigrid version of a conservative finite difference scheme useful for the study of collapse processes in nonlinear schrodinger equations nlses as an example we study the character of the focusing singularity of the two dimensional critical nlse

the survival of branching annihilating random walk branching annihilating random walk is an interacting particle system on as time evolves particles execute random walks and branch and disappear when they meet other particles it is shown here that starting from a finite number of particles the system will survive with positive probability if the random walk rate is low enough relative to the branching rate but will die out with probability one if the random walk rate is high since the branching annihilating random walk is non attractive standard techniques usually employed for interacting particle systems are not applicable instead a modification of a contour argument by gray and griffeath is used

improved upper bounds for the contact process critical value the best known upper bound for the critical value c of the basic one dimensional contact process is 2 most techniques for finding bounds on critical values have the property that they can be modified in order to obtain improved bounds this seemed not to be the case for the approach which yielded c 2 for the basic contact process in this paper we propose a technique for generating better bounds in this context to illustrate its use we carry out the full program in one case with the conclusion that c 1 942

a method for finding bounds on critical values for non attractive interacting particle systems the method is to use a computer to find a function for the process that always has negative expectation by considering all relevant configurations of 0 s and 1 s at the boundary of a finite process in one dimension it is shown that a branching annihilating random walk will die out if the diffusion parameter is greater than 0 176 the method may also be applied to attractive processes and is used for the contact process in one dimension obtaining the same values as ziezold and grillenberger 1988 for up to 10 places in from the boundaries

critical behaviour and logarithmic corrections of a quantum model with three spin interaction extended finite size scaling and series expansion methods are used to determine the critical properties of the one dimensional quantum ising model with three spin interaction by supposing the presence of logarithmic corrections the universality class of the model is investigated the results obtained by different methods support the conjecture that the model belongs to the same universality class as the four state potts model

symmetry reductions of a generalized cylindrical nonlinear schrodinger equation in this work symmetry reductions for a generalized cylindrical nonlinear schrodinger equation are presented these are obtained using an extension of the direct method originally developed by clarkson and kruskal 1989 which involves no group theoretic techniques

similarity structure of wave collapse abstract similarity transformations of the cubic schrodinger equation cse are investigated the transformations are used to remove the explicit time variation in the cse and reduce it to differential equations in the spatial variables only two different methods for similarity reduction are employed and the significance of similarity in the evolution of a collapsing wave packet is investigated numerical solutions in radial symmetry demonstrate that the similarity behaviour is local in space and time and that some similarity solutions must be classified as improper solutions the nature of the collapsing singularity is reexamined

the 1 1 do 2 model a finite lattice analysis finite size methods and the theory of conformal invariance are applied to the 1 1 do 2 or x y planar model the critical point is found to be xc approximately 2 0 the value sigma 0 501 or 0 005 is obtained for the index governing the critical divergence of the correlation length and the conformal anomaly is confirmed to be c 1 the kosterlitz thouless features of a low temperature phase with zero spontaneous magnetisation but algebraic decay of correlations together with a standard high temperature phase are observed the critical exponent eta is determined as a function of coupling constant and compared with theoretical expectations

finite size scaling of the quantum ising chain with periodic free and antiperiodic boundary conditions the authors give exact results for the energy spectrum of a chain of n ising spins in a transverse field with periodic free and antiperiodic boundary conditions the dependence of the energy gaps on boundary conditions is compatible with predictions of conformal invariance for correlation lengths in two dimensional strips the feasibility of calculating surface critical indices using phenomenological renormalisation with free boundary conditions and the convergence for large n are discussed