on the cohen macaulay property of modular invariant rings abstract if v is a faithful module for a finite group g over a field of characteristic p then the ring of invariants need not be cohen macaulay if p divides the order of g in this article the cohomology of g is used to study the question of cohen macaulayness of the invariant ring one of the results is a classification of all groups for which the invariant ring with respect to the regular representation is cohen macaulay moreover it is proved that if p divides the order of g then the ring of vector invariants of sufficiently many copies of v is not cohen macaulay a further result is that if g is a p group and the invariant ring is cohen macaulay then g is a bireflection group i e it is generated by elements which fix a subspace of v of codimension at most 2

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

the structure of gibbs states and phase coexistence for non symmetric continuum widom rowlinson models we prove the existence of phase transitions in non symmetric r component continuum widom rowlinson models our results are based on an extension of the pirogov sinai theory of phase transitions in general lattice spin systems to continuum systems this generalizes ruelle s extension of the peierls argument for lattices to symmetric continuum widom rowlinson models the pirogov sinai picture of the low temperature phase diagram for spin systems goes over into a phase diagram of the widom rowlinson model at large fugacities z z0 z r 1 there is in z space a point where the system has r pure phases lines with r 1 phases two dimensional surfaces with r 2 phases etc

on the gorenstein property for modular invariants let g gl v be a finite group where v is a finite dimensional vector space over a field f of arbitrary characteristic let s v be the symmetric algebra of v and s v g the ring of g invariants we prove here the following results theorem suppose that g contains no pseudo reflection of any kind 1 if s v g is gorenstein then g sl v 2 if g sl v then the cohen macaulay locus of s v g coincides with its gorenstein locus in particular if s v g is cohen macaulay then it is also gorenstein this extends well known results of k watanabe in case charf g 1 it also confirms a special case of a conjecture due to g kemper e kording g malle b h matzat d vogel and g wiese a similar extension is given to d benson s theorem about the gorenstein property of s v v g the polynomial tensor exterior algebra invariants our proof uses non commutative algebra methods in an essential way

the depth of invariant rings and cohomology abstract let g be a finite group acting linearly on a vector space v over a field k of positive characteristic p and let p g be a sylow p subgroup ellingsrud and skjelbred compositio math 41 1980 233 244 proved the lower bound depth k v g min dim v p 2 dim v for the depth of the invariant ring with equality if g is a cyclic p group let us call the pair g v flat if equality holds in the above in this paper we use cohomological methods to obtain information about the depth of invariant rings and in particular to study the question of flatness for g of order not divisible by p2 it ensues that g v is flat if and only if h1 g k v 0 or dim vp 2 dim v we obtain a formul a for the depth of the invariant ring in the case that g permutes a basis of v and has order not divisible by p2 in this situation g v is usually not flat moreover we introduce the notion of visible flatness of pairs g v and prove that this implies flatness for example the groups sl2 q so3 q su3 q sz q and r q with many interesting representations in defining characteristic are visibly flat in particular if g sl2 q and v is the space of binary forms of degree n or a direct sum of such spaces then g v is flat for all q pr with the exception of a finite number of primes p along the way we obtain results about the buchsbaum property of invariant rings and about the depth of the cohomology modules hi g k v we also determine the support of the positive cohomology h g k v as a module over k v g in the appendix the visibly flat pairs g v of a group with bn pair and an irreducible representation are classified

rings of invariants of finite groups which are hypersurfaces ii let g be a finite subgroup of gl acting naturally on an affine space c of dimension n over the complex number field and denote by v g the quotient variety of c under this action of g an element c of gl is said to be a pseudo reflection if the rank of r 1 is equal to one and g is said to be a reflection group if g is generated by pseudo reflections finite reflection groups were classified by coxeter shephard and todd 8 261 and are characterized by

a remark on different norms and analyticity for many particle interactions we compare a recent result of dobrushin and martirosyan with previous results by gallavotti and miracle sole and by israel and point out that the analytic behavior at high temperatures for many particle interactions is different depending on whether the interactions are weighted with a lattice gas or ising norm or on the other hand with the supremum norm

all automorphisms of the 3 nilpotent free group of countably infinite rank can be lifted abstract it is known from s andreadakis proc london math soc 3 15 1965 239 268 and s bachmuth trans amer math soc 122 1966 1 17 that for a free 3 nilpotent group f of rank n the map aut f aut f is not onto it is proved here that for f countably infinitely generated the map is onto

one dimensional models with 1 r2 interactions the critical behaviour of a general discrete one dimensional model with inverse square interactions is discussed using renormalisation group methods

two dimensional vector invariant rings of abelian p groups abstract for any faithful representation v of a non trivial p group over a field of characteristic p 0 it is known that the ring of vector invariants of m copies of v is not cohen macaulay if m 3 however much less is known about the case m 2 in this paper we show that if m 2 and the group is an abelian p group then the ring of invariants of 2v is a complete intersection in some cases and is not cohen macaulay in most cases as a corollary we obtain that if the field is f p and the ring of invariants of the representation v is a polynomial ring then the ring of invariants of 2v is either a complete intersection or not cohen macaulay

the nonfinite generation of aut g g free metabelian of rank 3 the group of automorphisms of the free metabelian group of rank 3 is not finitely generated let h be a free solvable group of rank n 2 aut h the automorphism group of h and inn h the group of inner automorphisms of h for n 2 it was shown by the authors and e formanek 4 theorem 1 that aut h inn h gl2 z z the ring of integers one consequence of this fact is that aut h is finitely generated f g the purpose of this paper is to prove the following contrasting theorem theorem if g is the free metabelian group of rank three then aut g is not finitely generated this result may also be compared with a theorem of l auslander 1 which states that the automorphism group of a polycyclic group is finitely presented if g denotes the commutator subgroup of g then the kernel of the natural homomorphism aut g 4 aut g g is called the group of ia automorphisms of g and will be denoted by ia g ia g which contains inn g was shown by the authors 3 to be non f g for g as in the theorem to prove that aut g is not f g we need to show that ia g is not f g as an aut g g gl3 z operator group the question arises concerning aut h where h is free metabelian of rank n 3 the immediate feeling that aut h is also non f g may be incorrect for definiteness let n 4 suppose g is free metabelian of rank 3 and consider aut g as embedded in aut h in an obvious manner the nonfinite generatiion of aut g comes from the existence of nontame automorphisms in aut g i e automorphisms of g f f which are not induced by automorphisms of the free group f of rank three and the necessity to include infinitely many nontame automorphisms in any generating set for aut g however the authors have discovered that many of the nontame automorphisms become tame when considered as elements in aut h whether this phenomenon is true for all nontame elements of aut g is unknown as yet but the question of the finite or nonfinite generation of aut h must be considered a difficult open problem with the interesting possibility of finite generation as the answer received by the editors december 31 1980 1980 mathematics subject classification primary 20f28

the cokernel of the johnson homomorphisms of the automorphism group of a free metabelian group in this paper we determine the cokernel of the k th johnson homomorphisms of the automorphism group of a free metabelian group for k 2 and n 4 as a corollary we obtain a lower bound on the rank of the graded quotient of the johnson filtration of the automorphism group of a free group furthermore by using the second johnson homomorphism we determine the image of the cup product map in the rational second cohomology group of the ia automorphism group of a free metabelian group and show that it is isomorphic to that of the ia automorphism group of a free group which is already determined by pettet finally by considering the kernel of the magnus representations of the automorphism group of a free group and a free metabelian group we show that there are non trivial rational second cohomology classes of the ia automorphism group of a free metabelian group which are not in the image of the cup product map

phase diagram of the two dimensional ising antiferromagnet computer assisted proof we study the phase diagram of the ising antiferromagnet on a square lattice in a neighbourhood of ground state critical pointsh 4 t 0 it leads to a question about the valuea c of the critical activity of the hard square lattice gas using a constructive criterion of uniqueness we prove thata c 1 and that the phase diagram of the antiferromagnet does not bulge near mentioned critical points it is a specific feature of this work that the proof was completed with the help of a computer

first order phase transitions in lattice and continuous systems extension of pirogov sinai theory we generalize the notion of ground states in the pirogov sinai theory of first order phase transitions at low temperatures applicable to lattice systems with a finite number of periodic ground states to that of restricted ensembles with equal free energies a restricted ensemble is a gibbs ensemble i e equilibrium probability measure on a restricted set of configurations in the phase space of the system when a restricted ensemble contains only one configuration it coincides with a ground state in the more general case the entropy is also important

description of specifications by means of probability distributions in small volumes under condition of very weak positivity the problem of description of s by means of probability distributions in small volumes with infinite boundary conditions is considered the description of s by means of s consistent systems of probability distributions in volumes of cardinality bounded by n with infinite boundary conditions is established under the condition of very weak positivity particular attention is paid to the most important case n 1 which requires special considerations

on the depth of invariant rings of infinite groups abstract let k be an algebraically closed field for a finitely generated graded commutative k algebra r let cmdef r dim r depth r denote the cohen macaulay defect of r let g be a linear algebraic group over k that is reductive but not linearly reductive we show that there exists a faithful rational representation v of g which we will give explicitly such that cmdef k v k g k 2 for all k n

phase diagrams and cluster expansions for low temperature models i the phase diagram low temperature phase diagrams of two dimensional open image in new window quantum field models are constructed let open image in new window lie in an r 1 dimensional space of perturbations of a polynomial withr degenerate minima perform a scaling open image in new window and assume 1 we constructk distinct states on left begin array 20 c r k end array right hypersurfaces of codimensionk 1 in the space of perturbations an expansion is used to exhibit exponential clustering of the schwinger functions of each of these states at the core of the construction is a general technique for finding the thermodynamically stable phases from a collection of competing minima we draw on ideas of pirogov and sinai 24 for this problem

a non gaussian renormalization group fixed point for hierarchical scalar lattice field theories a rigorous method is developed to handle the large field problems in the wilson kadanoff renormalization group approach to critical lattice systems of unbounded spins we use this method to study in a hierarchical approximation the non gaussian renormalization group fixed point which governs the infrared behaviour of critical lattice field theories in three dimensions the method is an improvement of the analyticity techniques of gawedzki and kupiainen using borel summation techniques we are able to incorporate the large field region into the perturbative region so that the theory is completely described in terms of convergent expansions

moderately ramified actions in positive characteristic in characteristic 2 and dimension 2 wild mathbb z 2 mathbb z actions on k u v ramified precisely at the origin were classified by artin who showed in particular that they induce hypersurface singularities we introduce in this article a new class of wild quotient singularities in any characteristic p 0 and dimension n ge 2 arising from certain non linear actions of mathbb z p mathbb z on the formal power series ring k u 1 dots u n these actions are ramified precisely at the origin and their rings of invariants in dimension 2 are hypersurface singularities with an equation of a form similar to the form found by artin when p 2 in higher dimension the rings of invariants are not local complete intersection in general but remain quasi gorenstein we establish several structure results for such actions and their corresponding rings of invariants

completely analytical interactions constructive description an interactionu is called a completely analytical ca interaction if it satisfies one of 12 given conditions formulated in terms of analyticity properties of the partition functions zv u or correlation decay or truncated correlation bounds or asymptotic behavior of ln zv u v the 12 conditions are presented together with part of the proof of their equivalence the main result of the paper is that each condition is constructive in the following sense instead of checking it in all finite volumesv v it is enough to consider only a finite amount of volumes with restricted size in particular the partition functions z v u for the complex perturbationsu ofu do not vanish for all v v and all with provided this is true only forv with diam v c and but with

on a sharp transition from area law to perimeter law in a system of random surfaces we introduce and study a phase transition which is associated with the spontaneous formation of infinite surface sheets in a bernoulli system of random plaquettes the transition is manifested by a change in the asymptotic behavior of the probability of the formation of a surface spanning a prescribed loop as such this transition offers a generalization of the bond percolation phenomenon at low plaquette densities the probability for large loops is shown to decay exponentially with the loops area whereas for high densities the decay is by a perimeter law furthermore we show that the two phases of the three dimensional plaquette system are in a precise correspondence with the two phases of the dual system of random bonds thus if a natural conjecture about the phase structure of the bond percolation model is true then there is a sharp transition in the asymptotic behavior of the surface events our analysis incorporates block variables in terms of which a non critical system is transformed into one which is close to a trivial high or low density fixed point stochastic geometric effects like those discussed here play an important role in lattice gauge theories

extension of the pirogov sinai theory to a class of quasiperiodic interactions we extend the pirogov sinai theory in such a manner that it applies to a large class of models with small quasiperiodic interactions as perturbations of periodic ones we find general diophantine conditions on the frequency module of the quasiperiodic interactions and derivability conditions on the interaction potentials these conditions allow to prove that the low temperature phase diagram is a homeomorphic deformation of the phase diagram at zero temperature

phase transition in the 3d random field ising model we show that the three dimensional ising model coupled to a small random magnetic field is ordered at low temperatures this means that the lower critical dimension dl for the theory isdl 2 settling a long controversy on the subject our proof is based on an exact renormalization group rg analysis of the system this analysis is carried out in the domain wall representation of the system and it is inspired by the scaling arguments of imry and ma the rg acts in the space of ising models and in the space of random field distributions driving the former to zero temperature and the latter to zero variance

on the absence of spontaneous symmetry breaking and of crystalline ordering in two dimensional systems we develop a unified approach based on araki s relative entropy concept to proving absence of spontaneous breaking of continuous internal symmetries and translation invariance in two dimensional statistical mechanical systems more precisely we show that under rather general assumptions on the interactions all equilibrium states of a two dimensional system have all the symmetries compact internal and spatial of the dynamics except possibly rotation invariance rotation invariance remains unbroken if connected correlations decay more rapidly than the inverse square distance we also prove that two dimensional systems with a non compact internal symmetry group like anharmonic crystals typically do not have gibbs states

an alternate version of pirogov sinai theory a new approach to the pirogov sinai theory of phase transitions is developed not employing the contour models with a parameter the completeness of the phase diagram is proven

microscopic origin of hydrodynamic equations derivation and consequences abstract we describe some recent progress in deriving autonomous hydrodynamic type equations for macroscopic variables from model stochastic microscopic dynamics of particles on a lattice the derivations also yield the microscopic fluctuations about the deterministic macroscopic evolution these grow with time to become infinite when the deterministic solution is unstable a form of microscopic pattern selection is also found

separating invariants and local cohomology the study of separating invariants is a recent trend in invariant theory for a finite group acting linearly on a vector space a separating set is a set of invariants whose elements separate the orbits of g in some ways separating sets often exhibit better behavior than generating sets for the ring of invariants we investigate the least possible cardinality of a separating set for a given g action our main result is a lower bound that generalizes the classical result of serre that if the ring of invariants is polynomial then the group action must be generated by pseudoreflections we find these bounds to be sharp in a wide range of examples

classical lattice gas model with a unique nondegenerate but unstable periodic ground state configuration we construct a classical lattice gas model with a unique periodic ground state configuration such that the peierls condition is not satisfied the ground state configuration is nondegenerate which means that for any fixed energye and any integern the diameter of the support of alln connected local excitations with energy less thane is bounded nevertheless the configuration is not stable it does not give rise to a low temperature phase any translation invariant gibbs state of our model corresponds to quasiperiodic ground state configurations this requires the modification of a recent hypothesis of dobrushin and shlosman

lattice systems with a continuous symmetry we prove that the expansion in powers of the temperature t of the correlation functions and the free energy of the plane rotator model on a d dimensional lattice is asymptotic to all orders in t the leading term in the expansion is the spin wave approximation and the higher powers are obtained by the usual perturbation series we also prove the inverse power decay of the pair correlation at low temperatures for d 3

phase diagrams and cluster expansions for low temperaturep φ 2 models we give a cluster expansion for the schwinger functions of the stable phases found in part i the wightman axioms the mass gap and asymptoticity of perturbation theory follow

on the inverse problem in statistical mechanics for quantum spin systems it is known that for a suitable space of potentials the equilibrium states arew dense in the set of all translation invariant states the problem discussed in this paper is how to recognize such equilibrium states and how to find the corresponding potential a necessary and sufficient condition for a state to be an equilibrium state for some potential is given in sect 3

the interaction potential of the stationary measure of a high noise spinflip process a continuous time spinflip process started from an arbitrary state is proven to yield a gibbs state after any time t 0 if the noise is high enough this is done by approximating the process by discrete time processes the relative energies of those gibbs states are constructed explicitly via a high noise cluster expansion

high temperature differentiability of lattice gibbs states by dobrushin uniqueness techniques we establish conditions for the differentiability to any order of the gibbs states of classical lattice systems with arbitrary compact single spin space and with interactions in the dobrushin uniqueness region the derivatives are expressed as series expansions and are shown to be continuous on the uniqueness region we also provide a procedure for estimating the size of the derivatives these results verify a conjecture of l gross and extend his results in absence of second order phase transitions in the dobrushin uniqueness region journal of statistical physics25 1 57 72 1981 the techniques of this paper are based on those employed by gross

random surfaces with two sided constraints an application of the theory of dominant ground states we consider some models of classical statistical mechanics which admit an investigation by means of the theory of dominant ground states our models are related to the gibbs ensemble for the multidimensional sos model with symmetric constraints x m 2 the main result is that for 0 where 0 does not depend onm the structure of thermodynamic phases in the model is determined by dominant ground states for an evenm a gibbs state is unique and for an oddm the number of space periodic pure gibbs states is two

extension of pirogov sinai theory of phase transitions to infinite range interactions ii phase diagram this paper is the second part of our attempt of an extension of the pirogov sinai theory of phase transitions at low temperatures applicable to the lattice spin systems with finite range interactions to the systems with infinite range interactions employing the cluster expansion method developed in part i and modifying the notion of the truncated contour model introduced by zahradnik we extend the pirogov sinai result on the structure of phase diagrams to our situations as an application we apply our result to potts models with infinite range interactions

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

breaking of periodicity at positive temperatures we discuss a classical lattice gas model without periodic or quasiperiodic ground states the only ground state configurations of our model are nonperiodic thue morse sequences we show that low temperature phases of such models can be ordered in fact we prove the existence of an ordered nonmixing low temperature translation invariant equilibrium state which has nonperiodic gibbs states in its extremal decomposition

an analysis of annni model by peierl s contour method for the three dimensional annni model a converging expression for the curve of the coexistence of the 3 3 phase and ferromagnetic phase is derived for low temperatures using a new extension of the peierls contour method

a unified approach to phase diagrams in field theory and statistical mechanics we construct the phase diagram of any system which admits a low temperature polymer or cluster expansion such an expansion turns the system into a hard core interacting contour model with small but not necessarily positive activities the method uses some of zahradnik s ideas z1 but applies equally well to systems with complex interactions we give two applications first to low temperaturep 2 models with complex couplings and second to a computation of asymptotics of partition functions in periodic volumes if the index of a supersymmetric field theory is known the second application would help determine the number of phases in infinite volume

partially ordered models we provide a formal definition and study the basic properties of partially ordered random fields porf these systems were proposed to model textures in image processing and to represent independence relations between random variables in statistics in the latter case they are known as bayesian networks our random fields are a generalization of probabilistic cellular automata pca and their theory has features intermediate between that of discrete time processes and the theory of statistical mechanical lattice fields its proper definition is based on the notion of partially ordered specification pos in close analogy to the theory of gibbs measures this paper contains two types of results first we present the basic elements of the general theory of porfs basic geometrical issues definition in terms of conditional probability kernels extremal decomposition extremality and triviality reconstruction starting from single site kernels relations between pom and gibbs fields second we prove three uniqueness criteria that correspond to the criteria known as uniform boundedness dobrushin uniqueness and disagreement percolation in the theory of gibbs measures

loci in quotients by finite groups pointwise stabilizers and the buchsbaum property let k v g be the invariant ring of a nite linear group g gl v and let gu be the pointwise stabilizer of a subspace u v we prove that the following numbers associated to the invariant ring do not increase if one passes from k v g to k v gu the minimal number of homogeneous generators the maximal degree of the generators the number of syzygies and other betti numbers the complete intersection defect the dierence between depth and dimension and the type from this theorems of steinberg serre nakajima kac and watanabe and the author follow which say that if k v g is a polynomial ring a hypersurface a complete intersection or cohen macaulay then the same is true for k v gu furthermore k v gu inherits the gorenstein property from k v g we give an algorithm which transforms generators of k v g into generators of k v g u letp be one of the properties mentioned above we consider the locus ofp in v g spec k v g and prove that for x2 spec k v with image x 0 in v g the local ring k v g0 has the property p if and only if p holds for the invariant ring k v g x of the point stabilizer using this we prove that the non cohen macaulay locus in v g is either empty or it has dimension at least one and codimension at least 3 from this we deduce that k v g is buchsbaum if and only if it is cohen macaulay this proves a conjecture of campbell et al

generic frechet differentiability of the pressure in certain lattice systems this note has two goals the first is to give an explicit description theorem 1 of the duals of certain weighted products h of a countable family of banach spaces these products include the usual spaces of interactions which arise in statistical mechanics the second goal is to use this description to prove that if the factor spaces are finite dimensional and the weight functionh satisfies a certain growth condition then the pressure is frechet differentiable wherever it is gateaux differentiable hence is frechet differentiable in a denseg subset

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

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