the numerical analysis of ordinary differential equations runge kutta and general linear methods mathematical and computational introduction the euler method and its generalizations analysis of runge kutta methods general linear methods bibliography

logarithmic corrections and finite size scaling in the two dimensional 4 state potts model we analyze the scaling and finite size scaling behavior of the two dimensional 4 state potts model we find new multiplicative logarithmic corrections for the susceptibility in addition to the already known ones for the specific heat we also find additive logarithmic corrections to scaling some of which are universal we have checked the theoretical predictions at criticality and off criticality by means of high precision monte carlo data

persistence properties of a system of coagulating and annihilating random walkers we study a d dimensional system of diffusing particles that on contact either annihilate with probability 1 q ensuremath 1 or coagulate with probability q ensuremath 2 q ensuremath 1 in one dimension the system models the zero temperature glauber dynamics of domain walls in the q state potts model we calculate overline p m t the probability that a randomly chosen lattice site contains a particle whose ancestors have undergone exactly m ensuremath 1 coagulations using perturbative renormalization group analysis for dl2 we show that if the number of coagulations m is much less than the typical number m t then overline p m t ensuremath sim m ensuremath zeta d t ensuremath ensuremath theta with ensuremath theta dq q q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 ensuremath zeta 2q ensuremath 1 ensuremath epsilon 2q ensuremath 1 q ensuremath 1 1 2 aq ensuremath epsilon 2 o ensuremath epsilon 3 where q q ensuremath 1 q ensuremath epsilon 2 ensuremath d and a ensuremath 0 006 dots m t is shown to scale as m t ensuremath sim t d 2 ensuremath ensuremath delta where ensuremath delta d 1 ensuremath q q ensuremath 1 q ensuremath 1 2 ensuremath epsilon o ensuremath epsilon 2 in two dimensions we show that overline p m t ensuremath sim mathrm ln t q 3 ensuremath 2q mathrm ln m 2q ensuremath 1 2 t ensuremath 2q for m ensuremath ll t 2q ensuremath 1 we also derive an exact nonperturbative relation between the exponents namely ensuremath delta q ensuremath theta 1 ensuremath q the one dimensional results corresponding to ensuremath epsilon 1 are compared with results from monte carlo simulations

percolation transitions in two dimensions we investigate bond and site percolation models on several two dimensional lattices numerically by means of transfer matrix calculations and monte carlo simulations the lattices include the square triangular honeycomb kagome and diced lattices with nearest neighbor bonds and the square lattice with nearest and next nearest neighbor bonds results are presented for the bond percolation thresholds of the kagome and diced lattices and the site percolation thresholds of the square honeycomb and diced lattices we also include the bond and site percolation thresholds for the square lattice with nearest and next nearest neighbor bonds we find that corrections to scaling behave according to the second temperature dimension xt2 4 predicted by the coulomb gas theory and the theory of conformal invariance in several cases there is evidence for an additional term with the same exponent but modified by a logarithmic factor only for the site percolation problem on the triangular lattice does such a logarithmic term appear to be small or absent the amplitude of the power law correction associated with xt2 4 is found to be dependent on the orientation of the lattice with respect to the cylindrical geometry of the finite systems

diffusion limited aggregation processes with three particle elementary reactions a diffusion limited aggregation process in which clusters coalesce by means of 3 particle reaction a a a a is investigated in one dimension we give a heuristic argument that predicts logarithmic corrections to the mean field asymptotic behavior for the concentration of clusters of mass m at time t c m t m 1 2 log t t 3 4 for 1 infty we also investigate the problem with a localized steady source of monomers and find that the steady state concentration c r scales as r 1 log r 1 2 r 1 and r 1 log r 1 2 respectively for the spatial dimension d equal to 1 2 and 3 the total number of clusters n t grows with time as log t 3 2 t 1 2 and t log t 1 2 for d 1 2 and 3 furthermore in three dimensions we obtain an asymptotic solution for the steady state cluster mass distribution c m r sim r 1 log r 1 phi z with the scaling function phi z z 1 2 exp z and the scaling variable z m sqrt log r

continuous variable stepsize explicit pseudo two step rk methods abstract the aim of this paper is to apply a class of constant stepsize explicit pseudo two step runge kutta methods of arbitrarily high order to nonstiff problems for systems of first order differential equations with variable stepsize strategy embedded formulas are provided for giving a cheap error estimate used in stepsize control continuous approximation formulas are also considered for use in an eventual implementation of the methods with dense output by a few widely used test problems we compare the efficiency of two pseudo two step runge kutta methods of orders 5 and 8 with the codes dopri5 dop853 and pirk8 this comparison shows that in terms of evaluations on a parallel computer these two pseudo two step runge kutta methods are a factor ranging from 3 to 8 cheaper than dopri5 dop853 and pirk8 even in a sequential implementation mode fifth order new method beats dopri5 by a factor more than 1 5 with stringent error tolerances

dynamical critical behaviours of the ising spin chain swendsen wang and wolff algorithms we study the zero temperature ising chain evolving according to the swendsen wang dynamics we determine analytically the domain length distribution and various historical characteristics e g the density of unreacted domains is shown to scale with the average domain length as l with 3 2 for the q state potts model 1 q 1 we also compute the domain length distribution for the ising chain endowed with the zero temperature wolff dynamics

reaction diffusion models in one dimension with disorder we study a large class of one dimensional reaction diffusion models with quenched disorder using a real space renormalization group method rsrg which yields exact results at large time particles e g of several species undergo diffusion with random local bias sinai model and may react upon meeting we obtain a detailed description of the asymptotic states i e attractive fixed points of the rsrg such as the large time decay of the density of each specie their associated universal amplitudes and the spatial distribution of particles we also derive the spectrum of nontrivial exponents which characterize the convergence towards the asymptotic states for reactions which lead to several possible asymptotic states separated by unstable fixed points we analyze the dynamical phase diagram and obtain the critical exponents characterizing the transitions we also obtain a detailed characterization of the persistence properties for single particles as well as more complex patterns we compute the decay exponents for the probability of no crossing of a given point by respectively the single particle trajectories ensuremath theta or the thermally averaged packets overline ensuremath theta the generalized persistence exponents associated to n crossings are also obtained specifying to the process a stackrel ensuremath rightarrow a ensuremath emptyset or a with probabilities r 1 ensuremath r we compute exactly the exponents ensuremath delta r and ensuremath psi r characterizing the survival up to time t of a domain without any merging or with mergings respectively and the exponents ensuremath delta a r and ensuremath psi a r characterizing the survival up to time t of a particle a without any coalescence or with coalescences respectively overline ensuremath theta ensuremath psi and ensuremath delta obey hypergeometric equations and are numerically surprisingly close to pure system exponents though associated to a completely different diffusion length the effect of additional disorder in the reaction rates as well as some open questions are also discussed

anomalous mean field behavior of the fully connected ising model although the fully connected ising model does not have a length scale we show that the critical exponents for thermodynamic quantities such as the mean magnetization and the susceptibility can be obtained using finite size scaling with the scaling variable equal to n the number of spins surprisingly the mean value and the most probable value of the magnetization are found to scale differently with n at the critical temperature of the infinite system and the magnetization probability distribution is not a gaussian even for large n similar results inconsistent with the usual understanding of mean field theory are found at the spinodal we relate these results to the breakdown of hyperscaling and show that hyperscaling can be restored by increasing n while holding the ginzburg parameter rather than the temperature fixed or by doing finite size scaling at the pseudocritical temperature where the susceptibility is a maximum for a given value of n we conclude that finite size scaling for the fully connected ising model yields different results depending on how the mean field limit is approached

dynamic critical behavior of the swendsen wang algorithm the two dimensional three state potts model revisited we have performed a high precision monte carlo study of the dynamic critical behavior of the sweden wang algorithm for the two dimensional three state potts model we find that the li sokal bound int const cn is almost but not quite sharp the ratio int cn seems to diverge either as a small power 0 08 or as a logarithm

efficient block predictor corrector methods with a small number of corrections recently various classes of predictor corrector methods have been proposed as being suitable for solving nonstiff ordinary differential equations in a parallel environment this paper shows that methods based on a low order predictor and a runge kutta corrector are not efficient and that if predictor corrector methods are to be used efficiently for solving nonstiff problems in parallel then high order predictors are required examples of methods with high order predictors are given and their efficiency properties are studied in terms of stability and local error theory

aggregation kinetics for a one dimensional zero degree kelvin model of spinodal decomposition we study analytically the approach to equilibrium in a simple zero temperature model for phase separation in a binary alloy in which nearest neighbor interchange can occur only if the portion of ab bonds is thereby decreased the approach to equilibrium is found analytically because of the existence of infinitely many possible stationary states the asymptotic distribution of ab pairs depends on the details of the initial state and must be obtained by a recursion method

an asphericity conjecture and kaplansky problem on zero divisors suppose a group representationh a r is aspherical x a w a x is a word in alphabet a x 1with nonzero sum of exponents onx and the grouphnaturally embeds ing a x r w a x it is conjectured that the presentationg a x r w a x is aspherical if and only ifgis torsion free it is proven that if this conecture is false andg a x r w a x is a counterexample then the integral group ring z g of torsion free groupgwill contain zero divisors some special cases when this conjecture holds are also indicated

parallel iteration of symmetric runge kutta methods for nonstiff initial value problems abstract this paper discusses parallel iteration schemes for collocation based symmetric runge kutta srk methods for solving nonstiff initial value problems our main result is the derivation of four a stable srk corrector methods of orders 4 6 8 and 10 that optimize the rate of convergence when iterated by means of the highly parallel fixed point iteration process the resulting pisrk method parallel iterated srk method shows considerably increased efficiency when compared with the fixed point iteration process applied to gauss legendre correctors

trivial unit conjecture and homotopy theory abstract a homotopy theoretic description is given for trivial unit conjecture in the group ring z g

domain number distribution in the nonequilibrium ising model we study domain distributions in the one dimensional ising model subject to zero temperature glauber and kawasaki dynamics the survival probability of a domain s t t and an unreacted domain q1 t t are characterized by two independent nontrivial exponents we develop an independent interval approximation that provides close estimates for many characteristics of the domain length and number distributions including the scaling exponents

monte carlo methods beyond detailed balance abstract monte carlo algorithms are nearly always based on the concept of detailed balance and ergodicity in this paper we focus on algorithms that do not satisfy detailed balance we introduce a general method for designing non detailed balance algorithms starting from a conventional algorithm satisfying detailed balance this approach is first applied to a very simple model which shows the basic viability of the method then we apply it to the ising model where we find that the method is an improvement compared to the standard metropolis algorithm be it with a modest gain of a factor 2 3

mixing times of critical two dimensional potts models we study dynamical aspects of the q state potts model on an n n box at its critical c q heat bath glauber dynamics and cluster dynamics such as swendsen wang that circumvent low temperature bottlenecks are all expected to undergo critical slowdowns in the presence of periodic boundary conditions the inverse spectral gap which in the subcritical regime is o 1 should at criticality be polynomial in n for 1 4 in accordance with the predicted discontinuous phase transition this was confirmed for q 2 the ising model by the second author and sly and for sufficiently large q by borgs et al here we show that the following holds for the critical potts model on the torus for q 3 the inverse gap of glauber dynamics is no 1 for q 4 it is at most no log n and for every q 4 in the phase coexistence regime the inverse gaps of both glauber dynamics and swendsen wang dynamics are exponential in n for free or monochromatic boundary conditions and large q we show that the dynamics at criticality is faster than on the torus unlike the ising model where free periodic boundary conditions induce similar dynamical behavior at all temperatures the inverse gap of swendsen wang dynamics is exp no 1 2017 wiley periodicals inc

parallel iteration of high order runge kutta methods with stepsize control abstract this paper investigates iterated runge kutta methods of high order designed in such a way that the right hand side evaluations can be computed in parallel using stepsize control based on embedded formulas a highly efficient code is developed on parallel computers the 8th order mode of this code is more efficient than the dopr18 implementation of the formulas of prince and dormand the 10th order mode is about twice as cheap for comparable accuracies

statistical dynamics of interacting kinks ii abstract a statistical mechanical theory is developed for a gas of kinks or one dimensional domain walls interacting with exponential attractive forces which models long time behaviors of the bistable tdgl field equation employing a factorization approximation in the bbgky like hierarchical equations for the size distribution functions of consecutive domains we obtain a kinetic equation for the size distribution function of a single domain the scaling solution of thr latter yields the logarithmic growth law of domain sizes with the coefficient related to the width of the size distribution scattering structure function is also obtained the results are compared with those of molecular dynamics and experiments on layered magnets

rigorous lower bound on the dynamic critical exponents of the swendsen wang algorithm we prove the rigorous lower bound z sw v for the dynamic critical exponent of the swendsen wang algorithm for two dimensional q state potts models with q 2 3 4 this implies z sw 0 2 5 1 we present numerical data indicating that z sw 0 55 0 03 0 89 0 05 for q 3 4 95 confidence limits statistical errors only the discrepancy for q 4 appears to be caused by multiplicative logarithmic corrections

coarsening in the 1d ising model evolving with swendsen wang dynamics an unusual scaling we consider a simple model of domain growth the zero temperature 1d ising model evolving according to the swendsen wang dynamics we find that in the long time limit the pair correlation function scales with a characteristic length increasing as the square of the average domain size in that limit a few large domains occupy almost all the space with many small domains between them in contrast to the usual picture of coarsening the average domain size here is not a characteristic length of the growth problem instead one finds a power law distribution for the sizes of large domains with a cut off at a length which grows as the square of the average size of the domains

asphericity and zero divisors in group algebras abstract s v ivanov recently discovered a surprising connection between asphericity of certain spaces and the existence of zero divisors in group algebras we give a different proof of a more general result

eigenvalue analysis of an irreversible random walk with skew detailed balance conditions an irreversible markov chain monte carlo mcmc algorithm with skew detailed balance conditions originally proposed by turitsyn et al is extended to general discrete systems on the basis of the metropolis hastings scheme to evaluate the efficiency of our proposed method the relaxation dynamics of the slowest mode and the asymptotic variance are studied analytically in a random walk on one dimension it is found that the performance in irreversible mcmc methods violating the detailed balance condition is improved by appropriately choosing parameters in the algorithm

zeros of the partition function and pseudospinodals in long range ising models the relation between the zeros of the partition function and spinodal critical points in ising models with long range interactions is investigated we find that the spinodal is associated with the zeros of the partition function in four dimensional complex temperature magnetic field space the zeros approach the real temperature magnetic field plane as the range of interaction increases

local equivalence of reversible and general markov kinetics a b s t r a c t we consider continuous time markov kinetics with a finite number of states and a positive equilibrium p this class of systems is significantly wider than the systems with detailed

on the coupling time of the heat bath process for the fortuin kasteleyn random cluster model we consider the coupling from the past implementation of the random cluster heat bath process and study its random running time or coupling time we focus on hypercubic lattices embedded on tori in dimensions one to three with cluster fugacity at least one we make a number of conjectures regarding the asymptotic behaviour of the coupling time motivated by rigorous results in one dimension and monte carlo simulations in dimensions two and three amongst our findings we observe that for generic parameter values the distribution of the appropriately standardized coupling time converges to a gumbel distribution and that the standard deviation of the coupling time is asymptotic to an explicit universal constant multiple of the relaxation time perhaps surprisingly we observe these results to hold both off criticality where the coupling time closely mimics the coupon collector s problem and also at the critical point provided the cluster fugacity is below the value at which the transition becomes discontinuous finally we consider analogous questions for the single spin ising heat bath process

two step by two step pirk type pc methods based on gauss legendre collocation points abstract this paper concerns with parallel predictor corrector pc iteration methods for solving nonstiff initial value problems ivps for systems of first order differential equations the predictor methods are based on adams type formulas the corrector methods are constructed by using coefficients of s stage collocation gauss legendre runge kutta rk methods based on c 1 c s and the 2 s stage collocation rk methods based on c 1 c s 1 c 1 1 c s at n th integration step the stage values of the 2 s stage collocation rk methods evaluated at t n 1 c 1 h t n 1 c s h can be used as the stage values of the collocation gauss legendre rk method for n 2 th integration step by this way we obtain the corrector methods in which the integration processes can be proceeded two step by two step the resulting parallel pc iteration methods which are called two step by two step tbt parallel iterated rk type pirk type pc methods based on gauss legendre collocation points two step by two step pirkg methods or tbtpirkg methods give us a faster integration process fixed step size applications of these tbtpirkg methods to the three widely used test problems reveal that the new parallel pc iteration methods are much more efficient when compared with the well known parallel iterated rk methods pirk methods and sequential codes odex dopri5 and dop853 available from the literature

a new test for asphericity and diagrammatic reducibility of group presentations we present a new test for studying asphericity and diagrammatic reducibility of group presentations our test can be applied to prove diagrammatic reducibility in cases where the classical weight test fails we use this criterion to generalize results of j howie and s m gersten on asphericity of lots and of adian presentations and derive new results on solvability of equations over groups we also use our methods to investigate a conjecture of s v ivanov related to kaplansky s problem on zero divisors we strengthen ivanov s result for locally indicable groups and prove a weak version of the conjecture

exact finite size scaling corrections to the critical two dimensional ising model on a torus we analyse the finite size corrections to the energy and specific heat of the critical two dimensional spin ising model on a torus we extend the analysis of ferdinand and fisher to compute the correction of order l 3 to the energy and the corrections of order l 2 and l 3 to the specific heat we also obtain general results on the form of the finite size corrections to these quantities only integer powers of l 1 occur unmodified by logarithms except of course for the leading log l term in the specific heat and the energy expansion contains only odd powers of l 1 in the specific heat expansion any power of l 1 can appear but the coefficients of the odd powers are proportional to the corresponding coefficients of the energy expansion

efficient simulation of the random cluster model the simulation of spin models close to critical points of continuous phase transitions is heavily impeded by the occurrence of critical slowing down a number of cluster algorithms usually based on the fortuin kasteleyn representation of the potts model and suitable generalizations for continuous spin models have been used to increase simulation efficiency the first algorithm making use of this representation suggested by sweeny in 1983 has not found widespread adoption due to problems in its implementation however it has been recently shown that it is indeed more efficient in reducing critical slowing down than the more well known algorithm due to swendsen and wang here we present an efficient implementation of sweeny s approach for the random cluster model using recent algorithmic advances in dynamic connectivity algorithms

domain statistics in coarsening systems we study the domain number and size distributions in the one dimensional ising and q state potts models subject to zero temperature glauber dynamics the survival probability of a domain s t t 2c and an unreacted domain q1 t t 2d are characterized by two independent nontrivial exponents for the ising case we find c50 126 and d51 27 using numerical simulations we develop an independent interval approximation that predicts the qualitative behavior of the domain distribution and provides good estimates for the exponents exact results for the domain distribution are also obtained in several solvable cases s1063 651x 97 00110 4

duality and the universality class of the three state potts antiferromagnet on plane quadrangulations we provide a criterion based on graph duality to predict whether the three state potts antiferromagnet on a plane quadrangulation has a zero or finite temperature critical point and its universality class the former case occurs for quadrangulations of self dual type and the zero temperature critical point has central charge c 1 the latter case occurs for quadrangulations of non self dual type and the critical point belongs to the universality class of the three state potts ferromagnet we have tested this criterion against high precision computations on four lattices of each type with very good agreement we have also found that the wang swendsen koteck algorithm has no critical slowing down in the former case and critical slowing down in the latter

irreversible monte carlo algorithms for efficient sampling abstract equilibrium systems evolve according to detailed balance db this principle guided the development of monte carlo sampling techniques of which the metropolis hastings mh algorithm is the famous representative it is also known that db is sufficient but not necessary we construct irreversible deformation of a given reversible algorithm capable of dramatic improvement of sampling from known distribution our transformation modifies transition rates keeping the structure of transitions intact to illustrate the general scheme we design an irreversible version of metropolis hastings imh and test it on an example of a spin cluster standard mh for the model suffers from critical slowdown while imh is free from critical slowdown

twostep by twostep pirk type pc methods with continuous output formulas this paper deals with parallel predictor corrector pc iteration methods based on collocation runge kutta rk corrector methods with continuous output formulas for solving nonstiff initial value problems ivps for systems of first order differential equations at nth step the continuous output formulas are used not only for predicting the stage values in the pc iteration methods but also for calculating the step values at n 2 th step in this case the integration processes can be proceeded twostep by twostep the resulting twostep by twostep tbt parallel iterated rk type pirk type methods with continuous output formulas twostep by twostep pirkc methods or tbtpirkc methods give us a faster integration process fixed stepsize applications of these tbtpirkc methods to a few widely used test problems reveal that the new pc methods are much more efficient when compared with the well known parallel iterated rk methods pirk methods parallel iterated rk type pc methods with continuous output formulas pirkc methods and sequential explicit rk codes dopri5 and dop853 available from the literature

mean field behavior of cluster dynamics the dynamic behavior of cluster algorithms is analyzed in the classical mean field limit rigorous analytical results below mathit t mathit c establish that the dynamic exponent has the value mathit z mathrm sw 1 for the swendsen wang algorithm and mathit z mathit w 0 for the wolff algorithm an efficient monte carlo implementation is introduced adapted for using these algorithms for fully connected graphs extensive simulations both above and below mathit t mathit c demonstrate scaling and evaluate the finite size scaling function by means of a rather impressive collapse of the data textcopyright 1996 the american physical society

on locally indicable groups a group is said to be locally indicable if each of its non trivial finitely generated subgroups has the infinite cyclic group as a homomorphic image such groups were studied by higman 9 in connection with the zero divisor and unit problems for group rings more recently they have arisen i 2 12 in the study of equations over groups in 12 i proved a freiheitsatz for locally indicable groups this has been proved independently by brodskii 2 and short 22 the present paper arises from an investigation of a question put to me by s j pride whether torsionfree 1 relator groups are locally indicable the question was raised originally by baumslag i 1 problem 19 and an affirmative solution has recently been announced by brodski 2 as a consequence the group algebra rg of a torsion free 1 relator group g over an integral domain r has no non trivial zero divisors and no non trivial units using higman s results 9 the first fact was also proved by lewin and lewin 16 by embedding rg in a division ring the second appears to be new as was pointed out to me by k a brown a second consequence is that no 1 relator group has a non trivial finitely generated perfect subgroup which answers 1 problem 7 and 10 question 1 in fact using the freiheitsatz and the tower method described in 12 it is possible to prove the following general version of brodskii s theorem

dynamic critical behavior of the chayes machta algorithm for the random cluster model i two dimensions we study via monte carlo simulation the dynamic critical behavior of the chayes machta dynamics for the fortuin kasteleyn random cluster model which generalizes the swendsen wang dynamics for the q state potts ferromagnet to non integer q 1 we consider spatial dimension d 2 and 1 25 q 4 in steps of 0 25 on lattices up to 10242 and obtain estimates for the dynamic critical exponent z cm we present evidence that when 1 q 1 95 the ossola sokal conjecture z cm is violated though we also present plausible fits compatible with this conjecture we show that the li sokal bound z cm is close to being sharp over the entire range 1 q 4 but is probably non sharp by a power as a byproduct of our work we also obtain evidence concerning the corrections to scaling in static observables

universal scaling function for domain growth in the glauber ising chain the equal time correlation function is calculated at t 0 for the one dimensional ising model with glauber dynamics random initial conditions appropriate to a sudden quench from non zero temperature are imposed averaging over initial conditions yields the scaling form c r t f r2 t the scaling function is given by f x integral 01 dy pi y 1 y 1 2 exp x 4y and is universal i e independent of the probability distribution for the initial conditions

the self dual point of the two dimensional random cluster model is critical for q 1 we prove a long standing conjecture on random cluster models namely that the critical point for such models with parameter q 1 on the square lattice is equal to the self dual point p sd q sqrt q 1 sqrt q this gives a proof that the critical temperature of the q state potts model is equal to log 1 sqrt q for all q 2 we further prove that the transition is sharp meaning that there is exponential decay of correlations in the sub critical phase the techniques of this paper are rigorous and valid for all q 1 in contrast to earlier methods valid only for certain given q the proof extends to the triangular and the hexagonal lattices as well

variance reduction using nonreversible langevin samplers a standard approach to computing expectations with respect to a given target measure is to introduce an overdamped langevin equation which is reversible with respect to the target distribution and to approximate the expectation by a time averaging estimator as has been noted in recent papers 30 37 61 72 introducing an appropriately chosen nonreversible component to the dynamics is beneficial both in terms of reducing the asymptotic variance and of speeding up convergence to the target distribution in this paper we present a detailed study of the dependence of the asymptotic variance on the deviation from reversibility our theoretical findings are supported by numerical simulations

dynamic critical behavior of a swendsen wang type algorithm for the ashkin teller model we study the dynamic critical behavior of a swendsen wang type algorithm for the ashkin teller model we find that the li sokal bound on the autocorrelation time int const xc h holds along the self dual curve of the symmetric ashkin teller model and is almost but not quite sharp the ratio int c h appears to tend to infinity either as a logarithm or as a small power 0 05 p 0 12 in an appendix we discuss the problem of extracting estimates of the exponential autocorrelation time

distribution of domain sizes in the zero temperature glauber dynamics of the one dimensional potts model for the zero temperature glauber dynamics of the q state potts model we calculate the exact distribution of domain sizes by mapping the problem on an exactly soluble one species coagulation model a1a a in the long time limit this distribution is universal and from its complicated exact expression we extract its behavior in various regimes our results are tested in a simulation and compared to the predictions of a simple approximation proposed recently considering the dynamics of domain walls as a reaction diffusion model a1a a with probability q22 q21 and a1a f with probability 1 q21 we calculate the pair correlation function in the long time regime s1063 651x 96 12809 9

persistence in cluster cluster aggregation persistence is considered in one dimensional diffusion limited cluster cluster aggregation when the diffusion coefficient of a cluster depends on its size s as d s approximately s gamma the probabilities that a site has been either empty or covered by a cluster all the time define the empty and filled site persistences the cluster persistence gives the probability of a cluster remaining intact the empty site and cluster persistences are universal whereas the filled site depends on the initial concentration for gamma 0 the universal persistences decay algebraically with the exponent 2 2 gamma for the empty site case the exponent remains the same for gamma 0 but the cluster persistence shows a stretched exponential behavior as it is related to the small s behavior of the cluster size distribution the scaling of the intervals between persistent regions demonstrates the presence of two length scales the one related to the distances between clusters and that between the persistent regions

graphical representations and cluster algorithms ii we continue the study initiated in part i of graphical representations and cluster algorithms for various models in or related to statistical mechanics for certain models e g the blume emery griffths model and various generalizations we develop fortuin kasteleyn type representations which lead immediately to swendsen wang type algorithms for other models e g the random cluster model that are defined by a graphical representation we develop cluster algorithms without reference to an underlying spin system in all cases phase transitions are related to percolation or incipient percolation in the graphical representation which via the ic algorithm allows for the rapid simulation of these systems at the transition point pertinent examples include the continuum widom rowlinson model the restricted 1 step solid on solid model and the xy model

the pivot algorithm a highly efficient monte carlo method for the self avoiding walk the pivot algorithm is a dynamic monte carlo algorithm first invented by lal which generates self avoiding walks saws in a canonical fixed n ensemble with free endpoints heren is the number of steps in the walk we find that the pivot algorithm is extraordinarily efficient one effectively independent sample can be produced in a computer time of ordern this paper is a comprehensive study of the pivot algorithm including a heuristic and numerical analysis of the acceptance fraction and autocorrelation time an exact analysis of the pivot algorithm for ordinary random walk a discussion of data structures and computational complexity a rigorous proof of ergodicity and numerical results on self avoiding walks in two and three dimensions our estimates for critical exponents are 0 7496 0 0007 ind 2 and 0 592 0 003 ind 3 95 confidence limits based on saws of lengths 200 n 10000 and 200 n 3000 respectively

lifted worm algorithm for the ising model we design an irreversible worm algorithm for the zero field ferromagnetic ising model by using the lifting technique we study the dynamic critical behavior of an energylike observable on both the complete graph and toroidal grids and compare our findings with reversible algorithms such as the prokof ev svistunov worm algorithm our results show that the lifted worm algorithm improves the dynamic exponent of the energylike observable on the complete graph and leads to a significant constant improvement on toroidal grids