critical behaviour of self avoiding walk in five or more dimensions we use the lace expansion to prove that in five or more dimensions the standard self avoiding walk on the hypercubic integer lattice behaves in many respects like the simple random walk in particular it is shown that the leading asymptotic behaviour of the number of step self avoiding walks is purely exponential that the mean square displacement is asymptotically linear in the number of steps and that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion some related facts are also proved these results are optimal according to the widely believed conjecture that the self avoiding walk behaves unlike the simple random walk in dimensions two three and four

critical exponents for the self avoiding random walk in three dimensions we compute by direct monte carlo simulation the main critical exponents 4 andv and the effective coordination number for the self avoiding random walk in three dimensions on a cubic lattice we find both hyperscaling relationsdv 2 anddv 2 4 0 satisfied ind 3

high dimensional incipient infinite clusters revisited the incipient infinite cluster iic measure is the percolation measure at criticality conditioned on the cluster of the origin to be infinite using the lace expansion we construct the iic measure for high dimensional percolation models in three different ways extending previous work by the second named author and jarai we show that each construction yields the same measure indicating that the iic is a robust object furthermore our constructions apply to spread out versions of both finite range and long range percolation models we also get estimates on structural properties of the iic such as the volume of the intersection between the iic and euclidean balls

algorithms for optimal discontinuous piecewise linear and constant l 2 fits to continuous functions with adjustable nodes in one and two dimensions in this paper a direct variational approach with nonstandard variations is used to generate algorithms to determine optimal discontinuous piecewise linear and piecewise constant l2 fits to a continuous function of one or two variables with adjustable nodes in the one variable case the algorithm is fast and robust the mesh cannot tangle and the resulting fits are continuous a e in the two variable case on an adjustable triangular grid the algorithm is less robust but gives good results for particular functions possessing a single steep feature the extension to higher dimensions is straightforward

periodic solutions of the ginzburg landau equation abstract spatially periodic solutions to the ginzburg landau equation are considered in particular we obtain criteria for primary and secondary bifurcation limit cycle solutions nonlinear dispersion relations relating spatial and temporal frequencies only relatively simple tools appear in the treatment and as a result a wide range of parameter cases are considered finally we briefly treat the case of spatial bifurcations

critical behaviour of the edwards random walk in two dimensions a case where the fractal and hausdorff dimensions are not equal the authors study by monte carlo simulations using two different algorithms the edwards walk in two dimensions and extract its critical exponents they establish that the fractal dimension computed in terms of critical exponents is different from the hausdorff dimension computed by measure theoretic methods

the scaling limit of self avoiding random walk in high dimensions the brydges spencer lace expansion is used to prove that the scaling limit of the finite dimensional distributions of self avoiding random walk in the d dimensional cubic lattice mathbb z d is gaussian if d is sufficiently large it is also shown that the critical exponent gamma for the number of self avoiding walks is equal to 1 if d is sufficiently large

gaussian behavior of loop erased self avoiding random walk in four dimensions une marche auto evitante de longueur n sur le reseau entier z d est une suite ordonnee de points x 0 0 x 1 x n avec x i x i 1 1 et x i x j pour i j on etudie le comportement d une telle marche a d 4 quand on applique le processus de boucle effacee

on the upper critical dimension of lattice trees and lattice animals we give a rigorous proof of mean field critical behavior for the susceptibility 1 2 and the correlation length v 1 4 for models of lattice trees and lattice animals in two cases i for the usual model with trees or animals constructed from nearest neighbor bonds in sufficiently high dimensions and ii for a class of spread out or long range models in which trees and animals are constructed from bonds of various lengths above eight dimensions this provides further evidence that for these models the upper critical dimension is equal to eight the proof involves obtaining an infrared bound and showing that a certain square diagram is finite at the critical point and uses an expansion related to the lace expansion for the self avoiding walk

the one dimensional edwards model for long polymer chains using the theory of dynamical systems the author investigates the t to infinity and g to 0 limits of edwards probability measure the author proves that there exists at least one limit measure and that this measure is the wiener measure

a note on the critical dimension for weakly self avoiding walks we consider a weakly self avoiding t step random walk on zdwith possible nonnearest neighbor jumps we prove that for d 3 the scaling limit of the end point is cauchy distributed as t

self avoiding random walks on the hexagonal lattice we use the algorithm recently introduce by a berretti and a d sokal to compute numerically the critical exponents for the self avoiding random walk on the hexagonal lattice we find 1 3509 0 0057 0 0023v 0 7580 0 0049 0 0046 0 519 0 082 0 077 where the first error is the systematic one due to corrections to scaling and the second is the statistical error for the effective coordination number we find 1 84779 0 00006 0 0017 the results support the nienhuis conjecture 43 32 and provide a rough numerical check of the hyperscaling relationdv 2 an additional analysis taking the nienhuis value of 2 21 2 1 2 for granted gives 1 3459 0 0040 0 0008

euclidean invariant phase dynamics for propagating pattern abstract the euclidean invariant phase dynamics for propagating spatially modulated structures is considered the phase dynamics equation of motion is obtained in a general way for sufficiently smooth deformation of a striped pattern the amplitude equation for a propagating pattern is also derived near a bifurcation threshold where the external parameter can contain a component oscillating in time the theory is applied to the oscillatory patterns described by the amplitude equation the new stability criterions for a propagating wave and a nonlinear standing wave are also presented

critical behavior and the limit distribution for long range oriented percolation i we consider oriented percolation on mathbb z d times mathbb z whose bond occupation probability is pd where p is the percolation parameter and d is a probability distribution on mathbb z d suppose that d x decays as x d for some 0 we prove that the two point function obeys an infrared bound which implies that various critical exponents take on their respective mean field values above the upper critical dimension d c 2 alpha wedge2 we also show that for every k the fourier transform of the normalized two point function at time n with a proper spatial scaling has a convergent subsequence to e c k alpha wedge2 for some c 0

self avoiding walk in 5 or more dimensions using an expansion based on the renormalization group philosophy we prove that for a t step weakly self avoiding random walk in five or more dimensions the variance of the endpoint is of order t and the scaling limit is gaussian as t

rigorous analysis of discontinuous phase transitions via mean field bounds we consider a variety of nearest neighbor spin models defined on the d dimensional hypercubic lattice z d our essential assumption is that these models satisfy the condition of reflection positivity we prove that whenever the associated mean field theory predicts a discontinuous transition the actual model also undergoes a discontinuous transition which occurs near the mean field transition temperature provided the dimension is sufficiently large or the first order transition in the mean field model is sufficiently strong as an application of our general theory we show that for d sufficiently large the 3 state potts ferromagnet on z d undergoes a first order phase transition as the temperature varies similar results are established for all q state potts models with q 3 the r component cubic models with r 4 and the o n nematic liquid crystal models with n 3

intersections of random walks a direct renormalization approach various intersection probabilities of independent random walks ind dimensions are calculated analytically by a direct renormalization method adapted from polymer physics this heuristic approach based on edwards continuum model leads to a straightforward derivation and also to refinements of lawler s results for the simultaneous intersections of two walks in 4 or three walks in 3 these results are generalized top walks in d d frac 2p p 1 p 2 ford 4 an infinite set of universal critical exponents l l 1 are derived they govern the asymptotic probability z l sim s sigma l thatl star walks in d with a common origin do not intersect before times the l s are calculated up to ordero e2 whered 4 e this information is used to calculate the probability z g that a set of independent random walks in d or d d 4 respectivelyd 3 form a given topological networks g of multiple intersection points in the absence of any other double point respectively triple point this is generalized to a network in d leqq frac 2p p 1 dimension with exclusion ofp tuple points the method is quite general and can be used to calculate any critical intersection probability and provides the probabilist with a large variety of exact results yet to be proven rigorously

on improving the effectiveness of periodic solutions of the nls and dnls equations a method for improving the effectiveness of one phase periodic solutions of integrable equations is suggested which is based on the explicit determination of the locus of the auxiliary spectrum point the method is applied to the nonlinear schrodinger equation and the derivative nonlinear schrodinger equation it is shown that the relations between polynomial p lambda defining the solution in the inverse transform method and its algebraic resolvents play an important role in the theory

mean field critical behaviour for correlation length for percolation in high dimensions extending the method of 27 we prove that the corrlation length of independent bond percolation models exhibits mean field type critical behaviour i e p p c p 1 2 asp p c in two situations i for nearest neighbour independent bond percolation models on ad dimensional hypercubic lattice d withd sufficiently large and ii for a class of spread out independent bond percolation models which are believed to belong to the same universality class as the nearest neighbour model in more than six dimensions the proof is based on and extends a method developed in 27 where it was used to prove the triangle condition and hence mean field behaviour of the critical exponents and 2 for the above two cases

non coincidence of quenched and annealed connective constants on the supercritical planar percolation cluster in this paper we study the abundance of self avoiding paths of a given length on a supercritical percolation cluster on mathbb z d more precisely we count z n the number of self avoiding paths of length n on the infinite cluster starting from the origin which we condition to be in the cluster we are interested in estimating the upper growth rate of z n limsup n rightarrow infty z n 1 n which we call the connective constant of the dilute lattice after proving that this connective constant is a s non random we focus on the two dimensional case and show that for every percolation parameter p in 1 2 1 almost surely z n grows exponentially slower than its expected value in other words we prove that limsup n rightarrow infty z n 1 n lim n rightarrow infty mathbb e z n 1 n where the expectation is taken with respect to the percolation process this result can be considered as a first mathematical attempt to understand the influence of disorder for self avoiding walks on a quenched dilute lattice our method which combines change of measure and coarse graining arguments does not rely on the specifics of percolation on mathbb z 2 so our result can be extended to a large family of two dimensional models including general self avoiding walks in a random environment

simple adaptive grids for 1 d initial value problems abstract in solving 1 d initial value problems by implicit finite difference methods it is advantageous to use an adaptive grid to provide local regions of high resolution and allow larger time steps to be taken a simple method of constructing such grids is presented and the benefits are demonstrated by calculations of sod s shock tube problem g a sod j comput phys 27 1 1978 and of a supernova explosion

random walks with short range interaction and mean field behavior we introduce a model of self repelling random walks where the short range interaction between two elements of the chain decreases as a power of the difference in proper time the model interpolates between the lattice edwards model and the ordinary random walk we show by means of monte carlo simulations in two dimensions that the exponentvmf obtained through a mean field approximation correctly describes the numerical data and is probably exact as long as it is smaller than the corresponding exponent for self avoiding walks we also compute the exponent and present a numerical study of the scaling functions

moving finite element least squares and finite volume approximations of steady and time dependent pdes in multidimensions abstract we review recent advances in galerkin and least squares methods for approximating the solutions of first and second order pdes with moving nodes in multidimensions these methods use unstructured meshes and minimise the norm of the residual of the pde over both solutions and nodal positions in a unified manner both finite element and finite volume schemes are considered as are transient and steady problems for first order scalar time dependent pdes in any number of dimensions residual minimisation always results in the methods moving the nodes with the often inconvenient approximate characteristic speeds for second order equations however the moving finite element mfe method moves the nodes usefully towards high curvature regions in the steady limit for pdes derived from a variational principle the mfe method generates a locally optimal mesh and solution this also applies to least squares minimisation the corresponding moving finite volume mfv method based on the l2 norm does not have this property however although there does exist a finite volume method which gives an optimal mesh both for variational principles and least squares

critical exponent for the loop erased self avoiding walk by monte carlo methods a monte carlo simulation was performed for loop erased self avoiding walks lesaw to ascertain the exponentv for the z2 and z3 lattices the estimated values were 2v 1 600 0 006 in two dimensions and 2v 1 232 0 008 in three dimensions leading to the conjecturev 4 5 for the two dimensional lesaw these results add to existing evidence that the loop erased self avoiding walks are not in the same universality class as self avoiding walks

proprietes d intersection des marches aleatoires i convergence vers le temps local d intersection we study intersection properties of multi dimensional random walks letx andy be two independent random walks with values in d d 3 satisfying suitable moment assumptions and letin denote the number of common points to the paths ofx andy up to timen the sequence in suitably normalized is shown to converge in distribution towards the intersection local time of two independent brownian motions results are applied to the proof of a central limit theorem for the range of a two dimensional recurrent random walk thus answering a question raised by n c jain and w e pruitt

a self avoiding walk with attractive interactions a self avoiding walk with small attractive interactions is described here the existence of the connective constant is established and the diffusive behavior is proved using the method of the lace expansion

amplitude equations at the critical points of unstable dispersive physical systems the amplitude equations that govern the motion of wavetrains near the critical point of unstable dispersive weakly nonlinear physical systems are considered on slow time and space scales t m e m t x m e m x m 1 2 such systems arise when the dispersion relation for the harmonic wavetrain is purely real and complex conjugate roots appear when a control parameter is varied at the critical point when the critical wavevector k c is non zero a general result for this general class of unstable systems is that the typical amplitude equations are either of the form t 1 c 1 x 1 t 1 c 2 x 1 a a ab t 1 c 2 x 1 b t 1 c 1 x 1 a 2 or of the form t 1 c 1 x 1 t 1 c 2 x 1 a a a a 2 the equations with the ab nonlinearity govern for example the two layer model for baroclinic instability and self induced transparency s i t in ultra short optical pulse propagation in laser physics the second equation occurs for the two layer kelvin helmholtz instability and a problem in the buckling of elastic shells this second type of equation has been considered in detail by weissman the ab equations are particularly important in that they are integrable by the inverse scattering transform and have a variety of multi soliton solutions they are also reducible to the sine gordon equation sin when a is real we prove some general results for this type of instability and discuss briefly their applications to various other examples such as the two stream instability examples in which dissipation is the dominant mechanism of the instability are also briefly considered in contrast to the dispersive type which operates on the t 1 time scale this type operates on the t 2 scale

moving finite elements i we present new and general numerical methods for dealing with problems whose solutions develop sharp transition layers or near shocks these methods allow many nodes automatically to concentrate in the critical regions and move with them for clarity of exposition we concentrate on the space of piecewise linear functions with movable nodes with burgers equation as our test equation but the generalization to much more general spaces and equations including even certain previous moving vorticity blobs of the first author and s doss for the navier stokes equations becomes clear in this paper we present the theoretical and computational details of our scheme along with computational trial runs for burgers equation showing that the nodes do concentrate sharply and move with the shocks as desired the conclusiveness of these preliminary numerical trials is marred somewhat by the fact that we never successfully debugged a newton s method for our implicit stiff ode solver and were thus limited to ver

an optimal triangulation for second order elliptic problems abstract let be a polygonal domain in r n h an associated triangulation and uh the finite element solution of a well posed second order elliptic problem on h let m mi p qi 1 be the set of nodes which defines the vertices of the triangulation h for each i m i x i l 1 l n in rn the object of this paper is to provide a computational tool to approximate the best set of positions m of the nodes and hence the best triangulation gt h which minimizes the solution error in the natural norm associated with the problem the main result of this paper are theorems which provide explicit expressions for the partial derivatives of the associated energy functional with respect to the coordinates xil 1 l n of each of the variable nodes mi i 1 p

the incipient infinite cluster for high dimensional unoriented percolation we consider bond percolation on mathbb z d at the critical occupation density p c for d 6 in two different models the first is the nearest neighbor model in dimension d 6 the second model is a spread out model having long range parameterized by l in dimension d 6 in the spread out case we show that the cluster of the origin conditioned to contain the site x weakly converges to an infinite cluster as x when d 6 and l is sufficiently large we also give a general criterion for this convergence to hold which is satisfied in the case d 6 in the nearest neighbor model by work of hara 12 we further give a second construction by taking p 6 the probability for invasion percolation to reach a site x is asymptotic to c x d 4 as x

design and application of a gradient weighted moving finite element code ii in two dimensions in part i the authors reported on the design of a robust and versatile gradient weighted moving finite element gwmfe code in one dimension and on its application to a variety of pdes and pde systems this companion paper does the same for the two dimensional 2d case these moving node methods are especially suited to problems which develop sharp moving fronts especially problems where one needs to resolve the fine scale structure of the fronts the many potential pitfalls in the design of gwmfe codes and the special features of the implicit one dimensional 1d and 2d codes which contribute to their robustness and efficiency are discussed at length in part i this paper concentrates on issues unique to the 2d case brief explanations are given of the variational interpretation of gwmfe the geometrical mechanical interpretation simplified regularization terms and the treatment of systems a catalog of inner products which occur in gwmfe is given with particular attention paid to those involving second order operators after presenting an example of the 2d phenomenon of grid collapse and discussing the need for long time regularization the paper reports on the application of the 2d code to several nontrivial problems nonlinear arsenic diffusion in the manufacture of semiconductors the drift diffusion equations for semiconductor device simulation the buckley leverett black oil equations for reservoir simulation and the motion of surfaces by mean curvature

moving mesh partial differential equations mmpdes based on the equidistribution principle this paper considers several moving mesh partial differential equations that are related to the equidistribution principle several of these are new and some correspond to discrete moving mesh equations that have been used by others their stability is analyzed and it is seen that a key term for most of these moving mesh pdes is a source like term that measures the level of equidistribution it is shown that under weak assumptions mesh crossing cannot occur for most of them finally numerical experiments for these various moving mesh pdes are performed to study their relative properties

construction of the incipient infinite cluster for spread out oriented percolation above 4 1 dimensions we construct the incipient infinite cluster measure iic for sufficiently spread out oriented percolation on d for d 1 4 1 we consider two different constructions for the first construction we define n e by taking the probability of the intersection of an event e with the event that the origin is connected to x n d summing this probability over x d and normalising the sum to get a probability measure we let n and prove existence of a limiting measure the iic for the second construction we condition the connected cluster of the origin in critical oriented percolation to survive to time n and let n under the assumption that the critical survival probability is asymptotic to a multiple of n 1 we prove existence of a limiting measure with in addition we study the asymptotic behaviour of the size of the level set of the cluster of the origin and the dimension of the cluster of the origin under our methods involve minor extensions of the lace expansion methods used in a previous paper to relate critical oriented percolation to super brownian motion for d 1 4 1

new monte carlo method for the self avoiding walk we introduce a new monte carlo algorithm for the self avoiding walk saw and show that it is particularly efficient in the critical region long chains we also introduce new and more efficient statistical techniques we employ these methods to extract numerical estimates for the critical parameters of the saw on the square lattice we find 2 63820 0 00004 0 00030 1 352 0 006 0 025 v 0 7590 0 0062 0 0042 where the first error bar represents systematic error due to corrections to scaling subjective 95 confidence limits and the second bar represents statistical error classical 95 confidence limits these results are based on saws of average length 166 using 340 hours cpu time on a cdc cyber 170 730 we compare our results to previous work and indicate some directions for future research

self avoiding walk enumeration via the lace expansion we introduce a new method for the enumeration of self avoiding walks based on the lace expansion we also introduce an algorithmic improvement called the two step method for self avoiding walk enumeration problems we obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3 we enumerate 32 step self avoiding polygons in d 3 26 step self avoiding polygons in d 4 30 step self avoiding walks in d 3 and 24 step self avoiding walks and polygons in all dimensions d 4 we analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8 we also provide major extensions of 1 d expansions for the connective constant and for two critical amplitudes

a new moving mesh algorithm for the finite element solution of variational problems the paper is devoted to the description and application of a new iterative mesh optimization algorithm for the finite element solution of variational problems set in infinite dimensional spaces the optimality criterion is that the mesh should be such that the variational energy functional evaluated at the finite element approximation be minimized such a criterion has a relatively long history in the finite element literature the chief merit of the procedure presented in this paper is that each node of the mesh and the corresponding nodal value of the discrete approximation are updated by solving sequentially local minimization problems with very few degrees of freedom it is shown that this procedure reduces the energy functional monotonically without the need to solve the global discrete problem at intermediate stages applications to partial differential equations are considered numerical results in two dimensions are obtained by incorporating the algorithm into bank s well known pltmg elliptic solver

polymers in a weak random potential in dimension four rigorous renormalization group analysis correlation functions of the edwards model of polymers at weak coupling are defined and studied at the critical point in dimension four by a rigorous renormalization group method which validates at any order perturbative renormalization group results on their behaviour at large distances remainders are controlled by a new argument which enlarges the use of methods of constructive field theory to models of statistical physics

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

analysis of an algorithm for generating locally optimal meshes for l 2 approximation by discontinuous piecewise polynomials this paper discusses the problem of constructing a locally optimal mesh for the best l 2 approximation of a given function by discontinuous piecewise polynomials in the one dimensional case it is shown that under certain assumptions on the approximated function baines algorithm m j baines math comp 62 1994 pp 645 669 for piecewise linear or piecewise constant polynomials produces a mesh sequence which converges to an optimal mesh the rate of convergence is investigated a two dimensional modification of this algorithm is proposed in which both the nodes and the connection between the nodes are self adjusting numerical results in one and two dimensions are presented

decay of correlations in nearest neighbor self avoiding walk percolation lattice trees and animals we consider nearest neighbor self avoiding walk bond percolation lattice trees and bond lattice animals on z d the two point functions of these models are respectively the generating function for self avoiding walks from the origin to x z d the probability of a connection from the origin to x and the generating functions for lattice trees or lattice animals containing the origin and x using the lace expansion we prove that the two point function at the critical point is asymptotic to const x 2 d as x for d 5 for self avoiding walk for d 19 for percolation and for sufficiently large d for lattice trees and animals these results are complementary to those of ann probab 31 2003 349 408 where spread out models were considered in the course of the proof we also provide a sufficient and rather sharp if d 4 condition under which the two point function of a random walk on z d is asymptotic to const x 2 d as x

geometric analysis of φ4 fields and ising models parts i and ii we provide here the details of the proof announced in 1 that ind 4 dimensions the even 4 euclidean field theory with a lattice cut off is inevitably free in the continuum limit in the single phase regime the analysis is nonperturbative and is based on a representation of the field variables or spins in ising systems as source sink creation operators in a system of random currents which may be viewed as the mediators of correlations in this dual representation the onset of long range order is attributed to percolation in an ensemble of sourceless currents and the physical interaction in the 4 field and other aspects of the critical behavior in ising models are directly related to the intersection properties of long current clusters an insight into the criticality of the dimensiond 4 is derived from an analogy foreseen by k symanzik with the intersection properties of paths of brownian motion other results include the proof that in certain respect the critical behavior in ising models is in exact agreement with the mean field approximation in high dimensionsd 4 but not in the low dimensiond 2 for which we establish the universality of hyperscaling

thermodynamics for the zero level set of the brownian bridge the random set of instants where the brownian bridge vanishes is constructed in terms of a random branching process the hausdorff measure supported by this set is shown to be equivalent to the partition function of a special class of disordered systems this similarity is used to show rigorously the existence of a phase transition for this particular class of disordered systems moreover it is shown that at high temperature the specific free energy has the strong self averaging property and that at low temperature it has no self averaging property the unicity at high temperature and the existence of many limits at low temperature are established almost surely in the disorder

lace expansion for the ising model the lace expansion has been a powerful tool for investigating mean field behavior for various stochastic geometrical models such as self avoiding walk and percolation above their respective upper critical dimension in this paper we prove the lace expansion for the ising model that is valid for any spin spin coupling for the ferromagnetic case we also prove that the expansion coefficients obey certain diagrammatic bounds that are similar to the diagrammatic bounds on the lace expansion coefficients for self avoiding walk as a result we obtain gaussian asymptotics of the critical two point function for the nearest neighbor model with d gg 4 and for the spread out model with d 4 and l gg 1 without assuming reflection positivity

regularity properties and pathologies of position space renormalization group transformations scope and limitations of gibbsian theory we reconsider the conceptual foundations of the renormalization group rg formalism and prove some rigorous theorems on the regularity properties and possible pathologies of the rg map our main results apply to local in position space rg maps acting on systems of bounded spins compact single spin space regarding regularity we show that the rg map defined on a suitable space of interactions formal hamiltonians is always single valued and lipschitz continuous on its domain of definition this rules out a recently proposed scenario for the rg description of first order phase transitions on the pathological side we make rigorous some arguments of griffiths pearce and israel and prove in several cases that the renormalized measure is not a gibbs measure for any reasonable interaction this means that the rg map is ill defined and that the conventional rg description of first order phase transitions is not universally valid for decimation or kadanoff transformations applied to the ising model in dimension d 3 these pathologies occur in a full neighborhood 0 h e of the low temperature part of the first order phase transition surface for block averaging transformations applied to the ising model in dimension d 2 the pathologies occur at low temperatures for arbitrary magnetic field strength pathologies may also occur in the critical region for ising models in dimension d 4 we discuss the heuristic and numerical evidence on rg pathologies in the light of our rigorous theorems in addition we discuss critically the concept of gibbs measure which is at the heart of present day classical statistical mechanics we provide a careful and we hope pedagogical overview of the theory of gibbsian measures as well as the less familiar non gibbsian measures emphasizing the distinction between these two objects and the possible occurrence of the latter in different physical situations we give a rather complete catalogue of the known examples of such occurrences the main message of this paper is that despite a well established tradition gibbsiannessshould not be taken for granted

a central limit theorem for a one dimensional polymer measure let s n n n 0 be a random walk on the integers having bounded steps the self repellent resp self avoiding walk is a sequence of transformed path measures which discourage resp forbid self intersections this is used as a model for polymers previously we proved a law of large numbers that is we showed the convergence of s n n toward a positive number under the polymer measure the present paper proves a classical central limit theorem for the self repellent and self avoiding walks that is we prove the asymptotic normality of s n n n for large n the proof refines and continues results and techniques developed previously

borel type bounds for the self avoiding walk connective constant let be the self avoiding walk connective constant on we show that the asymptotic expansion for c 1 in powers of 1 2d satisfies borel type bounds this supports the conjecture that the expansion is borel summable

the triangle condition for percolation aizenman and newman introduced an unverified condition the triangle condition which has been shown to imply that a number of percolation critical exponents take their mean field values and which is expected to hold above six dimensions for nearest neighbour percolation we prove that the triangle condition is satisfied in sufficiently high dimensions for the nearest neighbour model and above six dimensions for a class of spread out models the proof uses an expansion which is related to the lace expansion for self avoiding walk percolation is a simple probabilistic model for which many interesting problems remain unsolved the basic objects in percolation are random graphs on an infinite lattice typically there is a critical density for the graph edges known as bonds below which there is zero probability that a fixed point in the lattice is part of an infinite connected graph but above which this probability is strictly positive this abrupt change in behaviour has been used in chemistry and statistical physics to model phase transitions in a variety of physical systems such as fluid flow through a porous medium hence the name percolation random resistor networks and the gelation of branched polymers for an introduction see 15 20 to define the models we are interested in we consider the of dimensional integer lattice z whose elements are referred to as sites and the set of pairs b x y of distinct sites the bonds to each bond b is associated an independent bernoulli random variable nb9 with nb 1 with probability p jb and nb 0 with probability p jh the jb are fixed and z invariant and p is a parameter the nearest neighbour model is defined by taking jb 1 for those b x y such that x y i 1 and jb 0 otherwise expectation with respect to the joint distribution of the nb is denoted by if nb 1 we say b is occupied and otherwise b is vacant given a bond configuration nb two sites x and y are said to be connected if there is a path from x to y which consists of occupied bonds the set of sites which are connected to x is called the connected cluster of x and is denoted c x the probability that x is connected to y is written tp x y 1 ip x y i x and y are connected i y e c x p where i denotes the indicator function denoting by c x the number of sites in c x the expected cluster size or susceptibility is given by

long time asymptotics of solutions for the coupled dispersive ab system with initial value problems abstract in this paper we study the basic riemann hilbert rh problem of the coupled dispersive ab system with the schwartz class of initial value problems starting from the lax pair we use the deift zhou nonlinear steepest descent method to analyze the obtained rh problem such that the long time asymptotics of the solutions of this coupled ab system is found as t

the hausdorff dimension of the two dimensional edwards random walk the authors prove that the hausdorff dimension of the two dimensional edwards random walk 1965 is two with probability one this result is in contradiction with the widely accepted conjecture that the hausdorff dimension of a random walk is the inverse of the critical exponent v actually v is 3 4 for the two dimensional edwards random walk

the number and size of branched polymers in high dimensions we consider two models of branched polymers lattice trees on thed dimensional hypercubic lattice i the nearest neighbor model in sufficiently high dimensions and ii a spread out or long range model ford 8 in which trees are constructed from bonds of length less than or equal to a large parameterl we prove that for either model the critical exponent for the number of branched polymers exists and equals 5 2 and that the critical exponentv for the radius of gyration exists and equals 1 4 this improves our earlier results for the corresponding generating functions the proof uses the lace expansion together with an analysis involving fractional derivatives which has been applied previously to the self avoiding walk in a similar context

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out

periodic solutions and whitham equations for the ab system effective formulae for the single phase periodic solution of the ab system are obtained the solution depends on a complete set of four complex parameters riemann invariants the derivation of whitham equations which describe a slow evolution of the modulated periodic solution is presented the general theory is illustrated by application to the problem of soliton creation on the front of a sharp pulse

self avoiding walk in five or more dimensions i the critical behaviour we use the lace expansion to study the standard self avoiding walk in thed dimensional hypercubic lattice ford 5 we prove that the numbercn ofn step self avoiding walks satisfiescn a n where is the connective constant i e 1 and that the mean square displacement is asymptotically linear in the number of steps i e v 1 2 a bound is obtained forcn x the number ofn step self avoiding walks ending atx the correlation length is shown to diverge asymptotically like z 1 2 the critical two point function is shown to decay at least as fast as x 2 and its fourier transform is shown to be asymptotic to a multiple ofk 2 ask 0 i e 0 we also prove that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion the infinite self avoiding walk is constructed in this paper we prove these results assuming convergence of the lace expansion the convergence of the lace expansion is proved in a companion paper

critical two point functions and the lace expansion for spread out high dimensional percolation and related models we consider spread out models of self avoiding walk bond percolation lattice trees and bond lattice animals on mathbb z d having long finite range connections above their upper critical dimensions d 4 self avoiding walk d 6 percolation and d 8 trees and animals the two point functions for these models are respectively the generating function for self avoiding walks from the origin to x in mathbb z d the probability of a connection from 0 to x and the generating function for lattice trees or lattice animals containing 0 and x we use the lace expansion to prove that for sufficiently spread out models above the upper critical dimension the two point function of each model decays at the critical point as a multiple of x 2 d as x to infty we use a new unified method to prove convergence of the lace expansion the method is based on x space methods rather than the fourier transform our results also yield unified and simplified proofs of the bubble condition for self avoiding walk the triangle condition for percolation and the square condition for lattice trees and lattice animals for sufficiently spread out models above the upper critical dimension

logarithmic correction for the susceptibility of the 4 dimensional weakly self avoiding walk a renormalisation group analysis we prove that the susceptibility of the continuous time weakly self avoiding walk on mathbb z d in the critical dimension d 4 has a logarithmic correction to mean field scaling behaviour as the critical point is approached with exponent frac 1 4 for the logarithm the susceptibility has been well understood previously for dimensions d 5 using the lace expansion but the lace expansion does not apply when d 4 the proof begins by rewriting the walk two point function as the two point function of a supersymmetric field theory the field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers by providing a setting where the methods of the companion papers are applied together the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner