end to end distance from the green s function for a hierarchical self avoiding walk in four dimensions in bei92 we introduced a levy process on a hierarchical lattice which is four dimensional in the sense that the green s function for the process equals frac 1 x 2 if the process is modified so as to be weakly self repelling it was shown that at the critical killing rate mass squared c the green s function behaves like the free one now we analyze the end to end distance of the model and show that its expected value grows as a constant times sqrt t log frac 1 8 t left 1 o left frac log log t log t right right which is the same law as has been conjectured for self avoiding walks on the simple cubic lattice 4 the proof uses inverse laplace transforms to obtain the end to end distance from the green s function and requires detailed properties of the green s function throughout a sector of the complex plane these estimates are derived in a companion paper bi02

heat kernel coefficients of the laplace operator on the d dimensional ball we present a very quick and powerful method for the calculation of heat kernel coefficients it makes use of rather common ideas as integral representations of the spectral sum mellin transforms non trivial commutation of series and integrals and skillful analytic continuation of zeta functions on the complex plane we apply our method to the case of the heat kernel expansion of the laplace operator on a d dimensional ball with either dirichlet neumann or in general robin boundary conditions the final formulas are quite simple using this case as an example we illustrate in detail our scheme which serves for the calculation of an in principle arbitrary number of heat kernel coefficients in any situation when the basis functions are known we provide a complete list of new results for the coefficients b3 b10 corresponding to the d dimensional ball with all the mentioned boundary conditions and d 3 4 5

the broken supersymmetry phase of a self avoiding random walk we consider a weakly self avoiding random walk on a hierarchical lattice ind 4 dimensions we show that for choices of the killing ratea less than the critical valueacthe dominant walks fill space which corresponds to a spontaneously broken supersymmetry phase we identify the asymptotic density to which walks fill space a to be a supersymmetric order parameter for this transition we prove that a ac a log ac a 1 2 asa ac which is mean field behavior with logarithmic corrections as expected for a system in its upper critical dimension

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

the diffusive phase of a model of self interacting walks we consider simple random walk onzd perturbed by a factor exp t pjt wheret is the length of the walk and j t sum nolimits 0 leqslant i 0 ford 2 the diffusion constant is equal to 1 but ford 2 it is renormalized ford 1 andp 3 2 we prove diffusion for all real positive or negative ford 2 the scaling limit is brownian motion but ford 2 it is the edwards model with the wrong sign of the coupling when 0 which governs the limiting behaviour the latter arises since for p frac 4 d 2 t p j t is the discrete self intersection local time this establishes existence of a diffusive phase for this model existence of a collapsed phase for a very closely related model has been proven in work of bolthausen and schmock

green s function for a hierarchical self avoiding walk in four dimensions this is the second of two papers on the end to end distance of a weakly self repelling walk on a four dimensional hierarchical lattice it completes the proof that the expected value grows as a constant times sqrt t log frac 1 8 t left 1 o left frac log log t log t right right which is the same law as has been conjectured for self avoiding walks on the simple cubic lattice 4 apart from completing the program in the first paper the main result is that the green s function is almost equal to the green s function for the markov process with no self repulsion but at a different value of the killing rate which can be accurately calculated when the interaction is small furthermore the green s function is analytic in in a sector in the complex plane with opening angle greater than

a renormalisation group method i gaussian integration and normed algebras this paper is the first in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields fermion fields or both our immediate motivation is a specific model involving both boson and fermion fields which arises as a representation of the continuous time weakly self avoiding walk in this paper we define normed algebras suitable for a renormalisation group analysis and develop methods for performing analysis on these algebras we also develop the theory of gaussian integration on these normed algebras and prove estimates for gaussian integrals the concepts and results developed here provide a foundation for the continuation of the method presented in subsequent papers in the series

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

systematics of the relationship between vacuum energy calculations and heat kernel coefficients casimir energy is a nonlocal effect its magnitude cannot be deduced from heat kernel expansions even those including the integrated boundary terms on the other hand it is known that the divergent terms in the regularized but not yet renormalized total vacuum energy are associated with the heat kernel coefficients here a recent study of the relations among the eigenvalue density the heat kernel and the integral kernel of the operator e t h is exploited to characterize this association completely various previously isolated observations about the structure of the regularized energy emerge naturally for over 20 years controversies have persisted stemming from the fact that certain presumably physically meaningful terms in the renormalized vacuum energy density in the interior of a cavity become singular at the boundary and correlate to certain divergent terms in the regularized total energy the point of view of the present paper promises to help resolve these issues

improving explicit bounds for the solutions of second order linear differential equations this paper improves the algorithm for the construction of explicit bounds for the solutions of second order linear differential equations of the type p x y x q x y x 0 p x q x 0 x x 0 described in a recent paper of the authors the main improvements come from the introduction of new functionals devised by trevisan and from the application of an enhanced version of the mean value theorem for integrals when some functions of the integrand are monotonic

accurate continuous numerical solutions of time dependent mixed partial differential problems in this paper a separation of variables method for the solution of time dependent problems of the type u t t c t u x x 0 0 subject to u 0 t u p t 0 and u x 0 x u t x 0 g x is developed first an exact series solution of the problem is given given an admissible error e 0 and a bounded domain d t x t 0 x p 0 t t a continuous numerical solution is constructed so that the approximation error is uniformly upper bounded by e in d t

new bounds for the solutions of second order linear differential equations this paper presents a new method to construct explicit bounds for the solutions of second order linear differential equations of the type p x y x q x y x 0 p x q x 0 x x 0 which complements the method described in recent papers of the authors the method lies in the use of beesack s version of opial s inequality some results on asymptotics of the solution for the case p x 1 are also included

a renormalisation group method v a single renormalisation group step this paper is the fifth in a series devoted to the development of a rigorous renormalisation group method applicable to lattice field theories containing boson and or fermion fields and comprises the core of the method in the renormalisation group method increasingly large scales are studied in a progressive manner with an interaction parametrised by a field polynomial which evolves with the scale under the renormalisation group map in our context the progressive analysis is performed via a finite range covariance decomposition perturbative calculations are used to track the flow of the coupling constants of the evolving polynomial but on their own perturbative calculations are insufficient to control error terms and to obtain mathematically rigorous results in this paper we define an additional non perturbative coordinate which together with the flow of coupling constants defines the complete evolution of the renormalisation group map we specify conditions under which the non perturbative coordinate is contractive under a single renormalisation group step our framework is essentially combinatorial but its implementation relies on analytic results developed earlier in the series of papers the results of this paper are applied elsewhere to analyse the critical behaviour of the 4 dimensional continuous time weakly self avoiding walk and of the 4 dimensional n component varphi 4 model in particular the existence of a logarithmic correction to mean field scaling for the susceptibility can be proved for both models together with other facts about critical exponents and critical behaviour

casimir energy of a ball and cylinder in the zeta function technique a simple method is proposed to construct the spectral zeta functions required for calculating the electromagnetic vacuum energy with boundary conditions given on a sphere or on an infinite cylinder when calculating the casimir energy in this approach no exact divergencies appear and no renormalization is needed the starting point of the consideration is the representation of the zeta functions in terms of contour integral further the uniform asymptotic expansion of the bessel function is essentially used after the analytic continuation needed for calculating the casimir energy the zeta functions are presented as infinite series containing the riemann zeta function with rapidly falling down terms the spectral zeta functions are constructed exactly for a material ball and infinite cylinder placed in a uniform endless medium under the condition that the velocity of light does not change when crossing the interface as a special case perfectly conducting spherical and cylindrical shells are also consi

casimir forces for moving boundaries with robin conditions we consider a massless scalar field in 1 1 dimensions that satisfies a robin boundary condition at a non relativistic moving boundary using the perturbative approach introduced by ford and vilenkin we compute the total force on the moving boundary in contrast to what happens for the dirichlet and neumann boundary conditions in addition to a dissipative part the force acquires also a dispersive one further we also show that with an appropriate choice for the mechanical frequency of the moving boundary it is possible to turn off the vacuum dissipation almost completely

next order asymptotic expansion for n marginal optimal transport with coulomb and riesz costs abstract motivated by a problem arising from density functional theory we provide the sharp next order asymptotics for a class of multimarginal optimal transport problems with cost given by singular long range pairwise interaction potentials more precisely we consider an n marginal optimal transport problem with n equal marginals supported on r d and with cost of the form i j x i x j s in this setting we determine the second order term in the n asymptotic expansion of the minimum energy for the long range interactions corresponding to all exponents 0 s d we also prove a small oscillations property for this second order energy term our results can be extended to a larger class of models than power law type radial costs such as non rotationally invariant costs the key ingredient and main novelty in our proofs is a robust extension and simplification of the fefferman gregg decomposition 20 26 extended here to our class of kernels and which provides a unified method valid across our full range of exponents our first result generalizes a recent work of lewin lieb and seiringer 36 who dealt with the second order term for the coulomb case s 1 d 3

the dirichlet to robin transform a simple transformation converts a function satisfying a dirichlet boundary condition to a function satisfying a robin generalized neumann condition under sufficiently restrictive conditions one can arrange that the old function is the solution of a solvable differential equation and the new one solves an interesting differential equation in the simplest cases this observation enables the exact construction of the green functions for the wave heat and schrodinger problems with a robin boundary condition the resulting physical picture is that the field can exchange energy with the boundary and a delayed reflection from the boundary results in more general situations the method allows at least approximate and local construction of the appropriate reflected solutions and hence a classical path analysis of the green functions and the associated spectral information by this method we solve the wave equation on an interval with one robin and one dirichlet endpoint and thence derive several variants of a gutzwiller type expansion for the density of eigenvalues the variants are consistent except for an interesting subtlety of distributional convergence that affects only the neighbourhood of zero in the frequency variable

the vacuum energy for two cylinders with one increasing in size we consider the vacuum energy for the configuration of two cylinders and obtain its asymptotic expansion if the radius of one of these cylinders becomes large while the radius of the other one and their separation are kept fixed we calculate explicitly the next to leading order correction to the vacuum energy for the radius of the other cylinder becoming large or small

a simple method for finite range decomposition of quadratic forms and gaussian fields we present a simple method to decompose the green forms corresponding to a large class of interesting symmetric dirichlet forms into integrals over symmetric positive semi definite and finite range properly supported forms that are smoother than the original green form this result gives rise to multiscale decompositions of the associated gaussian free fields into sums of independent smoother gaussian fields with spatially localized correlations our method makes use of the finite propagation speed of the wave equation and chebyshev polynomials it improves several existing results and also gives simpler proofs

finite range decomposition for gaussian measures with improved regularity abstract we consider a family of gradient gaussian vector fields on the torus z l n z d adams koteck muller and independently bauerschmidt established the existence of a uniform finite range decomposition of the corresponding covariance operators i e the covariance can be written as a sum of covariance operators supported on increasing cubes with diameter l k we improve this result and show that the decay behaviour of the kernels in fourier space can be controlled then we show the regularity of the integration map that convolves functionals with the partial measures of the finite range decomposition in particular the new finite range decomposition avoids the loss of regularity which arises in the renormalisation group approach to anisotropic problems in statistical mechanics

four dimensional weakly self avoiding walk with contact self attraction we consider the critical behaviour of the continuous time weakly self avoiding walk with contact self attraction on mathbb z 4 for sufficiently small attraction we prove that the susceptibility and correlation length of order p for any p 0 have logarithmic corrections to mean field scaling and that the critical two point function is asymptotic to a multiple of x 2 this shows that small contact self attraction results in the same critical behaviour as no contact self attraction a collapse transition is predicted for larger self attraction the proof uses a supersymmetric representation of the two point function and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self avoiding walk without self attraction

explicit bounds for the solutions of second order linear differential equations this paper deals with the construction of explicit bounds for solutions of second order linear differential equations of the type p x y x q x y x 0 p x q x 0 x x 0 the construction is based on the study of the evolution of two complementary functionals involving y x in the sequence of zeroes of y x and y x based on that both a theoretical bound and an algorithm to explicitly calculate that bound are presented an illustrative example shows that the bounds proposed here improve previous results

monte carlo study of four dimensional self avoiding walks of up to one billion steps we study self avoiding walks on the four dimensional hypercubic lattice via monte carlo simulations of walks with up to one billion steps we study the expected logarithmic corrections to scaling and find convincing evidence in support the scaling form predicted by the renormalization group with an estimate for the power of the logarithmic factor of 0 2516 14 which is consistent with the predicted value of 1 4 we also characterize the behaviour of the pivot algorithm for sampling four dimensional self avoiding walks and conjecture that the probability of a pivot move being successful for an n step walk is o log n 1 4

finite range decomposition of gaussian processes let be the finite difference laplacian associated to the lattice z d for dimension d 3 a 0 and l a sufficiently large positive dyadic integer we prove that the integral kernel of the resolvent g a a 1 can be decomposed as an infinite sum of positive semi definite functions v n of finite range v n x y 0 for x y o l n equivalently the gaussian process on the lattice with covariance g a admits a decomposition into independent gaussian processes with finite range covariances for a 0 v n has a limiting scaling form l n left d 2 right gamma c left tfrac x y l n right as n as a corollary such decompositions also exist for fractional powers 2 0 2 the results of this paper give an alternative to the block spin renormalization group on the lattice

on summability of distributions and spectral geometry modulo the moment asymptotic expansion the cesaro and parametric behaviours of distributions at infinity are equivalent on the strength of this result we construct the asymptotic analysis for spectral densities arising from elliptic pseudodifferential operators we show how cesaro developments lead to efficient calculations of the expansion coefficients of counting number functionals and green functions the bosonic action functional proposed by chamseddine and connes can more generally be validated as a cesaro asymptotic development

compressed self avoiding walks bridges and polygons we study various self avoiding walks saws which are constrained to lie in the upper half plane and are subjected to a compressive force this force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half space in the case of bridges this is the unique end point in the case of saws or self avoiding polygons this corresponds to all vertices of maximal height we first use the conjectured relation with the schramm loewner evolution to predict the form of the partition function including the values of the exponents and then we use series analysis to test these predictions

spectral zeta functions for a cylinder and a circle spectral zeta functions s for the massless scalar fields obeying the dirichlet and neumann boundary conditions on a surface of an infinite cylinder are constructed these functions are defined explicitly in a finite domain of the complex plane s containing the closed interval of real axis 1 re s 0 proceeding from this the spectral zeta functions for the boundary conditions given on a circle boundary value problem on a plane are obtained without any additional calculations the casimir energy for the relevant field configurations is deduced

mixed problems for separated variable coefficient wave equations analytic numerical solutions with a priori error bounds this paper deals with the construction of accurate analytic numerical solutions of mixed problems related to the separated variable dependent wave equation utt b t a x uxx 0 0 based on the study of the growth of eigenfunctions of the underlying sturm liouville problems an exact theoretical series solution is firstly obtained explicit bounds allow truncation of the series solution so that the error of the truncated approximation is less than e1 in a bounded domain d x t 0 x l 0 t d since the approximation involves only a finite number of exact eigenvalues i2 1 i n0 the admissible error for the approximated eigenvalues i2 1 i n0 is determined in order to construct an analytical numerical solution of the mixed problem involving only approximated eigenvalues i2 so that the total error is less than e uniformly in d uniqueness of solutions is also treated

the hybrid spectral problem and robin boundary conditions the hybrid spectral problem where the field satisfies dirichlet conditions d on part of the boundary of the relevant domain and neumann n on the remainder is discussed in simple terms a conjecture for the c1 coefficient is presented and the conformal determinant on a 2 disc where the d and n regions are semi circles is derived comments on higher coefficients are made a separable second order hemisphere hybrid problem is introduced that involves robin boundary conditions and leads to logarithmic terms in the heat kernel expansion which are evaluated explicitly

finite temperature casimir energy in closed rectangular cavities a rigorous derivation based on a zeta function technique we derive rigorously explicit formulae of the casimir free energy at finite temperature for massless scalar field and electromagnetic field confined in a closed rectangular cavity with different boundary conditions by a zeta regularization method we study both the low and high temperature expansions of the free energy in each case we write the free energy as a sum of a polynomial in temperature plus exponentially decay terms we show that the free energy is always a decreasing function of temperature in the cases of massless scalar field with the dirichlet boundary condition and electromagnetic field the zero temperature casimir free energy might be positive in each of these cases there is a unique transition temperature as a function of the side lengths of the cavity where the casimir energy changes from positive to negative when the space dimension is equal to two and three we show graphically the dependence of this transition temperature on the side lengths of the cavity finally we also show that we can obtain the results for a non closed rectangular cavity by letting the size of some directions of a closed cavity go to infinity and we find that these results agree with the usual integration prescription adopted by other authors

the cesaro behaviour of distributions a theory of summability for generalized functions of one real variable is presented this theory generalizes the classical theory of summability of series and integrals the relationship between the average behaviour of generalized functions according to this summability theory and the distributional behaviour is studied several applications are given

scalar casimir densities for cylindrically symmetric robin boundaries the wightman function the vacuum expectation values of the field square and the energy momentum tensor are investigated for a massive scalar field with a general curvature coupling parameter in the region between two coaxial cylindrical boundaries it is assumed that the field obeys general robin boundary conditions on bounding surfaces the application of a variant of the generalized abel plana formula allows us to extract from the expectation values the contribution from single shells and to present the interference part in terms of exponentially convergent integrals the vacuum forces acting on the boundaries are presented as the sum of self action and interaction terms the first one contains well known surface divergences the interaction forces between the cylindrical boundaries are finite and are attractive for special cases of dirichlet and neumann scalars for the general robin case the interaction forces can be both attractive or repulsive depending on the coefficients in the boundary conditions the total casimir energy is evaluated by using the zeta function regularization technique it is shown that it contains a part which is located on bounding surfaces the formula for the interference part of the surface energy is derived and the energy balance is discussed

construction and borel summability of infrared φ 4 4 by a phase space expansion we construct the thermodynamic limit of the critical massless 4 model in 4 dimensions with an ultraviolet cutoff by means of a partly renormalized phase space expansion this expansion requires in a natural way the introduction of effective or running constants and the infrared asymptotic freedom of the model i e the decay of the running coupling constant plays a crucial role we prove also that the correlation functions of the model are the borel sums of their perturbation expansion

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

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

asymptotic behaviour of the solutions of second order functional differential equations abstract this paper presents integral criteria to determine the asymptotic behaviour of the solutions of second order nonlinear differential equations of the type y x q x f y x 0 with q x 0 and f y odd and positive for y 0 as x tends to it also compares them with the results obtained by chanturia 1975 in 11 for the same problem

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

local and global casimir energies for a semitransparent cylindrical shell the local casimir energy density and the global casimir energy for a massless scalar field associated with a function potential in a 3 1 dimensional circular cylindrical geometry are considered the global energy is examined for both weak and strong coupling the latter being the well studied dirichlet cylinder case for weak coupling through the total energy is shown to vanish by both analytic and numerical arguments based both on green s function and zeta function techniques divergences occurring in the calculation are shown to be absorbable by renormalization of physical parameters of the model the global energy may be obtained by integrating the local energy density only when the latter is supplemented by an energy term residing precisely on the surface of the cylinder the latter is identified as the integrated local energy density of the cylindrical shell when the latter is physically expanded to have finite thickness inside and outside the function shell the local energy density diverges as the surface of the shell is approached the divergence is weakest when the conformal stress tensor is used to define the energy density a real global divergence first occurs in as anticipated but the proof is supplied here for the first time this divergence is entirely associated with the surface energy and does not reflect divergences in the local energy density as the surface is approached

automatic solution of sturm liouville problems using the pruess method abstract we develop algorithms based on coefficient approximation for the automatic solution of regular and singular sturm liouville problems

an explicit formula for the casimir energy of an arbitrary compact smooth surface without boundary an explicit formula for the finite part of the casimir energy of a scalar field x t defined on a smooth compact two dimensional riemannian manifold x without boundary is derived the formula involves the metric tensor on x and its derivatives

a renormalisation group method ii approximation by local polynomials this paper is the second in a series devoted to the development of a rigorous renormalisation group method for lattice field theories involving boson fields fermion fields or both the method is set within a normed algebra mathcal n of functionals of the fields in this paper we develop a general method localisation to approximate an element of mathcal n by a local polynomial in the fields from the point of view of the renormalisation group the construction of the local polynomial corresponding to f in mathcal n amounts to the extraction of the relevant and marginal parts of f we prove estimates relating f and its corresponding local polynomial in terms of the t phi semi norm introduced in part i of the series

oscillation criteria for odd order neutral equations abstract in this paper four main results are obtained for oscillation of all solutions of the odd order neutral differential equation formula where p i t 0 i 0 i r r i 1 2 m and p 0 theorem 1 shows that if in addition to the above 0 p i x x n 1 and for some 0 1 all solutions of the first order delay equation formula are oscillatory then all solutions of are oscillatory in particular when m 1 and p 1 t p 1 0 then p 1 n 1 1 p n e n n 1 n 1 implies that all solutions of are oscillatory in such case theorem 4 1 due to gopalsamy et al czech math j 42 1992 313 323 reduces to p 1 n 1 1 p n which in view of the known inequality 1 e n n 1 n 1 j math anal appl 139 1989 311 318 graef et al j math anal appl 155 1991 562 571 and ladas et al proc amer math soc 113 1991 123 133 have been generalized in theorems 2 3 and 4 respectively

the asymptotic evaluation of a class of path integrals ii the asymptotic behavior of a class of wiener like path integrals functions of the local time is determined these integrals are of interest in themselves and also arise very naturally in the theory of disordered systems we show that by making use of a grassman algebra i e a set of anticommuting variables the earlier treatment of this problem can be greatly simplified in particular the previous use of replica trick which involves a difficult to justify analytic continuation in the number of field components is thus avoided

the analytic regularization zeta function method and the cut off method in the casimir effect the zero point energy associated with a hermitian massless scalar field in the presence of perfectly reflecting plates in a 3d flat space time is discussed a new technique to unify two different methods the zeta function and a variant of the cut off method used to obtain the so called casimir energy is presented and the proof of the analytic equivalence between both methods is given

an upper bound for the distance between a zero and a critical point of a solution of a second order linear differential equation this paper presents an upper bound for the distance between a zero and a critical point of a solution of the second order linear differential equation p x y q x y x 0 with p x q x 0 it also compares it with previous results

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

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