deterministic brownian motion generated from differential delay equations this paper addresses the question of how brownian like motion can arise from the solution of a deterministic differential delay equation to study this we analytically study the bifurcation properties of an apparently simple differential delay equation and then numerically investigate the probabilistic properties of chaotic solutions of the same equation our results show that solutions of the deterministic equation with randomly selected initial conditions display a gaussian like density for long time but the densities are supported on an interval of finite measure using these chaotic solutions as velocities we are able to produce brownian like motions which show statistical properties akin to those of a classical brownian motion over both short and long time scales several conjectures are formulated for the probabilistic properties of the solution of the differential delay equation numerical studies suggest that these conjectures could be universal for similar types of chaotic dynamics but we have been unable to prove this

stationary distribution of a nonlinear system driven by a chaotic force a simple nonlinear system driven by a chaotic force is discussed x t 1 t x t x t 2 the chaotic force t is defined by t kg yn 1 for n t n 1 n 0 1 2 where yn 1 is a chaotic sequence of a map f y yn 1 0 5 yn 0 5 as g y two cases are considered a g yn 1 yn 1 y0 and b g yn 1 yn 1 yn 1 the relaxation process of this system is investigated theoretically the and k dependence of the stationary distribution of x is discussed it is shown that for small the stationary distribution exhibits a drastic change according to k and the correlation of yn the fractal structure of the stationary distribution is found the theoretical results are shown to be in a good agreement with numerical ones which have been done for the logistic map as f y

a class of superrigid group von neumann algebras we prove that for any group g in a fairly large class of generalized wreath product groups the associated von neumann algebra lg completely remembers the group g more precisely if lg is isomorphic to the von neumann algebra l of an arbitrary countable group then must be isomorphic to g this represents the rst superrigidity result pertaining to group von neumann algebras

tricriticality and persistency of trails and silhouettes trails and silhouettes polymers with loops are generated using exact enumeration on a face centred cubic lattice by assigning a tunable fugacity factor with each intersection it is observed that as the fugacity for intersections is increased the trail and silhouette configurations change from swollen to compact ones indicating the existence of tricritical points strong divergence of the specific heats for different path lengths is used to locate these tricritical points the dlog pade scheme is used to compute the corresponding tricritical exponents in the limit of large path lengths the tricritical exponents thus obtained show that these tricritical points are distinct from the usual theta point and that trails and silhouettes do not belong to the saw universality class the tricritical exponents further show that trails and silhouettes do not belong to the same universality class persistency arising from fixing the initial step is also investigated and it is found that odd moments of the persistency obey a power law that can be expressed in terms of the critical exponent nu and the path length l

nonperturbative tricritical exponents of trails i exact enumeration on a triangular lattice as the fugacity of intersections in trails intersecting but non overlapping lattice walks is increased the configurations change from swollen ones with the scaling exponents of self avoiding walks to compact ones separating the two regimes is a potentially new tricritical point with no perturbative renormalisation fixed point associated with it supporting evidence for the existence of a tricritical point its likely location and exponents are computed for the first time from exact enumeration of all trails up to length of fifteen lattice constants on the triangular lattice the divergence of the specific heat indicates the location of the tricritical point generalised ratio and pade methods are used to extract the scaling exponents for the number of configurations and their end to end distance

microscopic chaos and gaussian diffusion processes abstract in this paper we construct and analyze a prototypical model of microscopic chaos in particular we extend the results of beck and shimizu to the case where the microscopic time scale is no longer small the upshot is that a non ornstein uhlenbeck deterministic process can generate a gaussian diffusion process

higher order correlations of tchebyscheff maps we determine the higher order correlation functions of iterates of tchebyscheff polynomials of arbitrary order n the non vanishing correlations can be described by suitable sets of graphs these graphs allow for a perturbative evaluation of invariant densities for langevin like systems that are driven by deterministic chaotic forces generated by tchebyscheff maps rather than gaussian white noise

specific heat and crossover exponents at the tricritical collapse of trails from the exact enumeration on a triangular lattice of trail configurations at the tricritical regime as the fugacity for intersections is increased the authors find phi t 0 68 8 for the crossover exponent and hence alpha t 0 5 2 for the specific heat exponent it indicates the collapse transition of trails to differ from that of polymer chains at the theta point in two dimensions

generalized langevin equation with chaotic force the generalized langevin equation with chaotic force is investigated x t 0tdt t t x t t where t t t t x2 the chaotic force t is defined by t yn 1 y for n t n 1 n 0 1 2 where yn 1 is a chaotic sequence yn 1 f yn the time evolution of x t which is generated by the chaotic force is discussed the approach of the distribution function of x to a stationary distribution is studied it is shown that the distribution function satisfies the fokker planck type equation with the memory effect in the small limit the relation between the invariant density of f y and the stationary distribution of x is discussed

study of the θ point by enumeration of self avoiding walks on the triangular lattice the author reports series expansion analyses of self avoiding walks with nearest neighbour bond interactions the estimates 2 nu t 1 07 or 0 05 and phi 0 64 or 0 05 for the correlation and crossover exponents at the theta point were obtained by examining the number of walks and the end to end distance data up to 16 steps on the triangular lattice

chaotic force in brownian motion brownian motion driven by a chaotic force is dicussed v t v t t the force t takes the value y 1 t for tn t tn 1 n 0 1 2 where tn t n 1j 0 xj the effect of the chaotic time interval xn and the chaotic magnitude yn on the stationary distribution of v is investigate in the following cases i yn is chaotic and xn is constant ii yn is regular and xn is chaotic iii yn and xn are chaotic and iv xn is strongly correlated with yn xn yn 0 5 the t dependence of the stationary distribution of v is discussed for t 1 the stationary distribution in each case is shown to have a gaussian form the diffusion constant is calculated the theoretical results are shown to be in a good agreement with the numerical ones

approach to gaussian stochastic behavior for systems driven by deterministic chaotic forces we consider skew product dynamical systems that describe the stroboscopic dynamics of a damped particle subjected to a chaotic kick force in a suitable scaling limit the dynamics converges to the ornstein uhlenbeck process we investigate the deterministic chaotic corrections in the vicinity of this gaussian limit case for various examples of chaotic forces we present numerical evidence that for certain classes of chaotic forces the deterministic chaotic corrections of the invariant density are universal we provide analytical results for forces generated by tchebyscheff maps and sketch a renormalization group theory in the space of probability densities

perturbation theory analysis of chaos i transition from quasiperiodicity to chaos abstract a three dimensional differential system is presented to discuss the transition from quasiperiodicity to chaos in dissipative systems the system includes the lorenz model and a time dependent hamiltonian system as special cases an asymptotic solution is calculated by using the nonlinear scale method the transition is discussed theoretically with the help of the solution the time evolution of the system is decomposed into a fastly varying part and a slowly varying part the slowly varying part is shown to be described in terms of a langevin type equation the origin of the stochastic nature and the reconciliation between the stochastic nature and the deterministic one of chaos are elucidated clearly the theoretical result is shown to be in a good agreement with that of its numerical simulation

lattice trails i exact results considers the problem of lattice trails introduced by malakis 1975 1976 the author proves the existence of a finite connective constant and establishes a result for the growth of n step trails tn analogous to the best known result for self avoiding walks tn lambda n expt o n for the honeycomb d 2 and lave s lattice d 3 the author establishes a counting theorem from which one can deduce the exact value of the connective constant lambda for the honeycomb lattice lambda 2 2 square root 2 further it follows that the trail problem is in the same universality class as the self avoiding walk problem for those lattices an exact amplitude relation between trails and self avoiding walks and between dumb bells and trails and self avoiding walks is established the non existence of a counting theorem for arbitrary lattices is established an inequality for the triangular lattice connective constant is proved a high density expansion for lambda for the d dimensional hybercubic lattice is also obtained the author argues that contrary to recent suggestions the model is in the same universality class as the self avoiding walk model

four dimensional polymer collapse ii interacting self avoiding trails we have simulated four dimensional interacting self avoiding trails isat on the hypercubic lattice with standard interactions at a wide range of temperatures up to length 4096 and at some temperatures up to length 16384 the results con3rm the earlier prediction using data from a non standard model at a single temperature of a collapse phase transition occurring at 3nite temperature moreover they are in accordwith the phenomenological theory originally proposed by lifshitz grosberg andkhokhlov in three d andrecently given new impetus by its use in the description of simulational results for four dimensional interacting self avoiding walks isaw in fact we argue that the available data is consistent with the conclusion that the collapse transitions of isat andisaw lie in the same universality class in contrad iction with long standing predictions we deduce that there exists a pseudo 3rst order transition for isat in four dimensions at 3nite lengths while the thermodynamic limit is described by the standard polymer mean 3eld theory giving a second order transition in contradiction to the prediction that the upper critical dimension for isat is du 4 c 2001 elsevier science b v all rights reserved

persistency studies of trails and silhouettes on square and triangular lattices the authors study the persistencies of trails and silhouettes on two dimensional square and triangular lattices these persistencies are induces by fixing the first step of the walk in the positive x direction trails are walks which are not allowed to overlap on themselves but may self intersect whereas silhouettes are just the shadows of trails associated with each self intersection is an attractive interaction energy epsilon mod epsilon mod thus a fugacity factor f i theta theta eitkbt ei theta where i is the number of intersections may be defined this has the advantage of providing a handle on the temperature and such dependence of persistencies on temperature is studied for the first time in this paper they are found to obey a law that may be expressed as a function of the critical exponents x2k 1l theta approximately lpk nu theta f l where l is the chain length nu is the correlation exponent f l approximately loge l or some weak function of l e g lw where w 1 0 p is a parameter and k 0 1 2

persistency properties of models of polymers on simple cubic and face centred cubic lattices the first four odd moments of the persistence lengths of trails and silhouettes are studied on two different 3d lattices the simple cubic lattice and the face centred cubic lattice variations of the averaged persistence lengths with chain lengths l and inverse temperatures theta are systematically examined it is found that the averaged persistence lengths scale with a scaling law of the form xl2k 1 theta approximately lpk nu theta f l where nu is the correlation exponent p is a parameter k 0 1 2 and f l approximately constant in contrast to the results in 2d where f l approximately loge l

relaxation and bifurcation in brownian motion driven by a chaotic force brownian motion driven by a chaotic sequence of iterates of a map f y which may depend on a bifurcation parameter is discussed n t t t where t kyn 1 for n t n 1 n 0 1 2 and yn 1 f yn the time evolution equation for the distribution function of the velocity is derived if 1 the distribution tends to the stationary one which has a similar form as the invariant density of f y in the limit of 0 the distribution function satisfies a fokker planck type equation with memory effects the dependence of relaxation processes on the bifurcation parameter is also investigated the fluctuation dissipation theorem is discussed the theoretical results are shown to be in a good agreement with numerical ones which have been done for the tent map and the logistic map

c simplicity of free products with amalgamation and radical classes of groups abstract we give new characterizations to ensure that a free product of groups with amalgamation has a simple reduced group c algebra and provide a concrete example of an amalgam with trivial kernel such that its reduced group c algebra has a unique tracial state but is not simple moreover we show that there is a radical class of groups for which the reduced group c algebra of any group is simple precisely when the group has a trivial radical corresponding to this class

chaotic brownian network abstract a system of many brownian particles interacting via the chaotic force is proposed as a new model of neural network which is called the chaotic brownian network the network is applied to the associative memory problem the coupling constants are determined by the hebb rule it is shown that some neuron pairs which are specified by the stored patterns exhibit the synchronized motions with the same sign or with the opposite sign the relation between the synchronized motion and the chaotic wandering motion is discussed the network is shown to retrieve the stored patterns or the reverse ones systematically and very quickly it is shown that the network can store and retrieve the number of patterns comparable to the number of neurons

dynamical systems of langevin type abstract dynamical systems of langevin type are deterministic mappings that arise if the gaussian white noise of a langevin equation is replaced by a deterministic chaotic dynamics we describe the various types of mappings that can be obtained in this way and investigate typical transition scenarios from complicated non gaussian to langevin like behavior that occur if a time scale parameter is changed we define a new characteristic quantify fir ergodic mappings called effective langevin radius which is related to the critical time scale where the marginal invariant density of the dynamical system of langevin type loses its differentiability the invariant density is shown to contain useful information on the entire characteristic functional of the chaotic driving force some general symmetry properties of the invariant densities are discussed

lattice trails ii numerical results for pt i see ibid vol 18 p 567 1985 a numerical study of the properties of lattice trails on the honeycomb square triangular and simple cubic lattices is made critical points are estimated for all lattices and upper and lower bounds established extensive series have been obtained and series analysis of both trail generating functions and mean square end to end distance series are not inconsistent with the conclusion that the problem is in the same universality class as the self avoiding walk problem a pseudo star triangle transformation is defined and the analyticity properties of that function coupled with previous exact results clearly supports that conclusion for the triangular lattice as well as providing excellent unbiased critical point estimates the author also shows that the connective constant for d or 2 dimensional hypercubic trails is strictly greater than the corresponding quantity for saws

fractal structure and gaussian distribution in chaotic brownian motion in brownian motion driven by a chaotic sequence of iterates of a map f y x t x t f t where f t yn 1 for n t n 1 n 1 2 and y n 1 f yn the fractal structure and the dependence of the recurrence relation xn 1 xn where xn x t n are studied the recurrence relation is shown to change from a shape similar to f y into a diagonal strip which does not depend on the details of f y if is decreased consequently it is shown that the stationary distribution of xn changes from a similar shape as the invariant density of f y into a gaussian shape

from the perron frobenius equation to the fokker planck equation we show that for certain classes of deterministic dynamical systems the perron frobenius equation reduces to the fokker planck equation in an appropriate scaling limit by perturbative expansion in a small time scale parameter we also derive the equations that are obeyed by the first and second order correction terms to the fokker planck limit case in general these equations describe non gaussian corrections to a langevin dynamics due to an underlying deterministic chaotic dynamics for double symmetric maps the first order correction term turns out to satisfy a kind of inhomogeneous fokker planck equation with a source term for a special example we are able solve the first and second order equations explicitly

dimension and stable rank in the k theory of c algebras in topological k theory which can be viewed as the algebraic side of the theory of vector bundles some of the interesting properties which one investigates are for example the conditions under which bundles must possess trivial direct summands or the extent to which the cancellation property for direct whitney sums of bundles holds such properties turn out to be controlled in part by the dimension of the base space and results describing the nature of this control are among what are frequently called stability results 17 during the past few years there has been a sudden flowering of a theory for c algebras see surveys 35 36 37 since c algebras are profitably thought of as non commutative locally compact spaces with the finitely generated projective modules being the appropriate generalization of vector bundles according to swan s theorem 40 it would be natural to look for stability results for c algebras but until now the theory has been focussed on the c groups of c algebras and there has been little discussion of stability properties presumably in part for lack of an appropriate concept of dimension for c algebras one of the main objectives of this paper is to introduce a concept of dimension for c algebras which directly generalizes the classical concept of dimension for compact spaces 16 20 22 and to develop some techniques for calculating it notably for c algebras which are obtained as crossed product algebras for an action of the group of integers in algebraic k theory which is the generalization of topological c theory to rings a substantial number of stability properties have been discovered 2 39 42 48 within that theory the concept which has played a role most analogous to that of dimension is the bass stable rank but despite its many successes the bass stable rank has often been difficult to calculate even for relatively uncomplicated rings another main objective of the present paper is to make accessible for application to c algebras some of the stability results from algebraic c theory this comes about from the fact that the concept of dimension which we introduce dominates the bass stable rank i was led to seek stability results for c algebras by my desire to determine whether the cancellation property for projective modules holds for the irrational rotation c algebras discussed in 30 which as elliott 11 has suggested can appropriately be called 2 dimensional non commutative tori in more concrete terms the question is whether two projections in such an algebra which have the same trace must be unitarily equivalent the present paper provides the general theory which is needed for the affirmative answer which i obtained in a subsequent paper i will derive the

tricritical points of trails their euler digraphs and graphs exact results on the sierpinski gasket the collapse transitions of trails their oriented graphs and silhouettes eulerian digraphs and graphs respectively as the fugacity for crossings is increased are investigated by exact decimation on the 2d sierpinski gasket recursion relations between the generating functions for the three basic configurations on consecutive levels are derived for all models the authors find tricritical points which move along a line in a four dimensional parameter space as the fugacity is varied and terminate at a decoupled first or second order multicritical point it suggests these models belong to distinct universality classes which differ from that of self attracting polymer chains which do not undergo a collapse theta transition on this fractal lattice

driven chaotic motion in single and double well potentials abstract the stochastic motions in stable and bistable potentials are compared on the basis of a langevin like equation with a deterministic chaotic process replacing gaussian white noise ergodic and mixing properties are obtained under rather general circumstances in the case of the symmetric double well potential the system exhibits a behavior quite different from what one observes for stable potentials as the control parameters are varied the system undergoes a phase transition from an ergodic to a nonergodic phase lyapunov exponents and fractal dimensions are studied for more detailed insight

identification of a polymer growth process with an equilibrium multicritical collapse phase transition the meeting point of swollen collapsed and crystalline polymers we have investigated a polymer growth process on the triangular lattice where the configurations produced are self avoiding trails we show that the scaling behavior of this process is similar to the analogous process on the square lattice however while the square lattice process maps to the collapse transition of the canonical interacting self avoiding trail isat model on that lattice the process on the triangular lattice model does not map to the canonical equilibrium model on the other hand we show that the collapse transition of the canonical isat model on the triangular lattice behaves in a way reminiscent of the point of the interacting self avoiding walk isaw model which is the standard model of polymer collapse this implies an unusual lattice dependency of the isat collapse transition in two dimensions by studying an extended isat model we demonstrate that the growth process maps to a multicritical point in a larger parameter space in this extended parameter space the collapse phase transition may be either point like second order or first order and these two are separated by a multicritical point it is this multicritical point to which the growth process maps furthermore we provide evidence that in addition to the high temperature gaslike swollen polymer phase coil and the low temperature liquid drop like collapse phase globule there is also a maximally dense crystal like phase crystal at low temperatures dependent on the parameter values the multicritical point is the meeting point of these three phases our hypothesized phase diagram resolves the mystery of the seemingly differing behaviors of the isaw and isat models in two dimensions as well as the behavior of the trail growth process

free products of c algebras small spatial free products of c algebras are constructed under certain conditions they have properties similar to those proved by paschke and salinas for the algebras c gi g2 where g1 g2 are discrete groups the freeproduct analogs of noncommutative bernoulli shifts are discussed 0 introduction let k be a field consider the category of unital algebras over k it is well known that this category admits coproducts free products of algebras 2 heuristically the free product of algebras is the algebra generated by them with no relations except for the identification of unit elements if k c the complex numbers and we consider unital algebras we can easily define a operation on the free products let a b be unital c algebras and a b their free product which is a unital algebra the question arises in what ways may one define a pre c norm on a b that extends the norms on a and b guided by analogy with tensor products we expect to have a choice among many pre c norms giving rise to many c free products of a and b one natural norm is 1c i i sup 1it c ih st representation of a b the representations of a b are in 1 1 correspondence with pairs of representations of a and b which act on the same hilbert space let a b be the completion of a b in this norm it is easy to see that this construction defines a coproduct in the category of c algebras and that a b is the biggest free product of a and b analogous to the biggest tensor product a 0 b if g 1 g2 are discrete groups we obtain c g1 c g2 c gi g2 where g1 g2 is the free product group and this is analogous to the relation c g1 0 c g2 c g1 x g2 this paper is motivated by the question is there a smallest c product a b in analogy to the smallest tensor product a b satisfying a relation cr gi g2 cr gi cr g2 received by the editors january 28 1981 and in revised form may 19 1981 1980 mathematics subject classification primary 46l05 secondary 46l55 1 982 american mathematical society 0002 9947 82 00001022 04 25

universal amplitudes ratio at the collapse transition of polymers in two dimensions first values of the universal ratio of the mean square radius of gyration and the mean square end to end distance for 2d linear polymers at their collapse transition are calculated by the scanning simulation method two models are considered i self avoiding walks saws with nearest neighbour attractions for which this ratio is found to be 0 179 or 0 003 at the theta point ii self intersecting trails at their tricritical point for which the value 0 166 or 0 002 is obtained this provides yet another indication that these models may belong to different tricritical universality classes the authors have also computed this ratio for trails at the high temperature limit with non interacting intersections and found 0 141 or 0 001 in excellent agreement with earlier estimates of this ratio for saws in their swollen phase

non perturbative tricritical exponents of trails ii exact enumerations on square and simple cubic lattices for pt i see ibid vol 21 p 773 1988 the authors present exact enumerations of trails self intersecting but non overlapping lattice walks tabulated according to their length l number of intersection i and end to end distance for the square lattice up to l 21 i 7 and for the simple cubic lattice up to l 15 i 5 they introduce a fugacity for intersection to explore the transition to a collapsed phase through a novel tricritical point which is unaccessible by the renormalisation group approach the existence of tricritical points in both lattices is manifested by the divergence of the specific heat the values of the tricritical couplings and exponents are extracted by a dlogpade analysis they find for the three dimensional lattice nu t approximately 0 48 and gamma t approximately 0 43 nu t approximately 0 52 and gamma t approximately 1 25 for the square lattice

hopf algebras and the logarithm of the s transform in free probability let k be a positive integer and let g k denote the set of all joint distributions of k tuples a 1 a k in a noncommutative probability space a such that a 1 a k 1 g k is a group under the operation of the free multiplicative convolution we identify g k as the group of characters of a certain hopf algebra y k then by using the log map from characters to infinitesimal characters of y k we introduce a transform ls for distributions g k ls is a power series in k noncommuting indeterminates z 1 z k its coefficients can be computed from the coefficients of the r transform of by using summations over chains in the lattices nc n of noncrossing partitions the ls transform has the linearizing property that ls ls ls g k such that in the particular case k 1 one has that y 1 is naturally isomorphic to the hopf algebra sym of symmetric functions and that the ls transform is very closely related to the logarithm of the s transform of voiculescu by the formula ls z z log s z g 1 in this case the group g 1 can be identified as the group of characters of sym in such a way that the s transform its reciprocal 1 s and its logarithm log s relate in a natural sense to the sequences of complete elementary and respectively power sum symmetric functions

brownian motion from deterministic dynamics abstract certain deterministic dynamical systems exhibit a transition from non gaussian chaotic to brownian motion behavior when a suitable scaling limit is applied for a simple hyperbolic model system we analyze the structure of the corresponding strange attractor determine the invariant measure and elucidate the transition scenario leading to a gaussian stochastic process

perturbation theory analysis of chaos iii period doubling bifurcations in a simple model abstract by using the nonlinear scale method the asymptotic solution of a three dimensional differential system is calculated to elucidate the period doubling bifurcation route to chaos as an stochastic description of chaos a langevin type equation with a vacillation force is derived the stochastic and deterministic nature of the vacillation force is studied

the statistics of collapsing square lattice trails with a fixed number of vertices of degree 4 a trail on the square lattice with a fixed number k of vertices of degree 4 is called a k trail we model polymer collapse using k trails by incorporating an interaction energy which is proportional to the number of nearest neighbour contact edges of the trail it is known that the number of square lattice n edge closed open k trails can be bounded above and below to o nk by the number of n step self avoiding circuits walks this along with pattern theorems for self interacting self avoiding circuits and walks are used herein to establish upper and lower bounds to o nk for the collapsing free energy of k trails in terms of self avoiding circuits or walks as appropriate we also use pattern theorems to obtain bounds on the limiting nearest neighbour contact density for collapsing k trails finally we investigate k trails with a fixed density of nearest neighbour contacts and show that their limiting entropy per monomer is independent of k

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

classification of inductive limits of 1 dimensional nccw complexes abstract a classification result is obtained for the c algebras that are stably isomorphic to inductive limits of 1 dimensional noncommutative cw complexes with trivial k 1 group the classifying functor cu is defined in terms of the cuntz semigroup of the unitization of the algebra for the simple c algebras covered by the classification cu reduces to the ordered k 0 group the cone of traces and the pairing between them as an application of the classification it is shown that the crossed products by a quasi free action o 2 r are all isomorphic for a dense set of positive irrational numbers

existence of four dimensional polymer collapse i kinetic growth trails abstract we present the results of simulations of kinetic growth trails kgt bond avoiding walks in four dimensions we use a mapping from a kinetic growth model to a static model of self interacting trails isat at a particular temperature to argue that this temperature is precisely the collapse temperature of four dimensional interacting trails to do this we show that the kinetic growth trails behave neither like static non interacting trails which should behave as excluded volume dominated four dimensional polymers that is self avoiding walks or collapsed four dimensional polymers but rather show an intermediate behaviour this is the first indication of collapse in any four dimensional lattice polymer model and so may be helpful in deciding which of the competing models of polymers is a good model in lower dimensions we have calculated various exponents of the kgt model and identified them with certain critical exponents of the static isat problem