a monte carlo method for series expansions the authors describe a simple and general algorithm to calculate series expansions in enumeration problems to large orders approximately by a monte carlo method it can be used to generate unbiased samples in cluster studies e g linear or branched polymers random surfaces etc in any dimension they calculate the number of site animals of size n on the square lattice for n or 50 and their average size to better than 1 accuracy

finite size scaling and universality in the spectrum of the quantum ising chain i periodic and antiperiodic boundary condition the spectrum of the quantum ising chain is studied in the finite size scaling limit for periodic and antiperiodic boundary conditions the finite size corrections are computed for all energy gaps the results of the model are used to test the privman fisher universality hypothesis and conformal theory at the critical point it is shown how to build the spectrum from the irreducible representations of the virasoro algebra in addition the author finds evidence for an unsuspected symmetry for systems both non critical and of finite size

critical exponents of self avoiding walks on a family of truncated n simplex lattices the authors have studied self avoiding walks saws on a family of truncated n simplex lattices which provide a family of fractals in which the fractal dimension d can be varied to a wide range while the spectral dimension d is held almost fixed by means of exact renormalisation group transformations they have calculated the critical exponents nu alpha and gamma and the connectivity constant mu of the saws for n 5 and 6 they propose an approximate theory for calculating the critical exponents of the saws on fractals which is expected to be accurate at large values of n they show that the theory gives results which are in good agreement with exact values even for small values of n and leads to simple closed relations for the exponents

a numerical study of diffusion and conduction in a 2d random medium investigates numerically the diffusion and conduction properties of a two dimensional random lattice hopping model the results are compared with a recent field theoretical renormalisation group approach showing that two is the upper critical dimension for such models the predicted scaling law with a universal logarithmic behaviour of the velocity is clearly observed the finiteness of the diffusion constant are well as its regular dependence versus the strength of randomness are evenly in good agreement with analytical results the author also determines the frequency dependence of the ac conductivity tensor

random walks with feedback on fractal lattices we study numerically a random walk under the competitive processes of a self organized feedback coupling characterized by a strength lambda and an underlying fractal lattice whereas a fractal structure favors a subdiffusive behavior a dynamical feedback leads either to localization in case of an attractive feedback lambda 0 or to superdiffusion for a repulsive memory strength lambda 0 under the influence of both processes the dynamical exponent z is changed for a sierpinski gasket or a sierpinski carpet with repulsive feedback coupling we get 2 z 1 04 or 2 z 1 08 respectively when an attractive feedback is dominant the system offers localization as in the case of a random walk in regular lattices the numerical results are strongly supported by analytical studies based on scaling arguments

diffusive behaviour of self attractive walks we study an attractive random walk called the satw in one dimension and show that the walk is diffusive we show rigorously that asymptotically where is the average of the square of the number of distinct points visited by the walk in n steps since the average square of the distance to the origin is of the same order as we get the result that asymptototically

monte carlo series analysis of irreversible self avoiding walks i the indefinitely growing self avoiding walk igsaw high precision monte carlo data are used to estimate the exponents which govern the asymptotic behaviour of the recently introduced indefinitely growing self avoiding walk in two dimensions for this walk the exponent gamma is by definition equal to one applying the same methods which are used to extract the exponents from exact series enumeration the authors give an estimate for the exponent nu of 0 567 or 0 003 the leading corrections to this asymptotic behaviour are also calculated

low lying excitations in the square triangle random tiling model we consider the sub dominant eigenstates of the transfer matrix for the square triangle random tiling model on an infinite strip of width l a numerical algorithm for generation of the corresponding solution of the bethe ansatz is developed numerical finite size scaling analysis of the associated eigenvalues reveals the presence of both integer and non integer critical exponents the analytical value of one of the non integer exponents is found it is also shown numerically that along with the leading correction to the free energy density for some excitations there is a term proportional to

the ashkin teller quantum chain and conformal invariance the authors study the finite size effects for the fourth state one dimensional quantum chain introduced by kohmoto et al 1981 using different boundary conditions from the corrections to the ground state energy for periodic boundary conditions they verify that the central charge of the virasoro algebra is equal to one in the whole domain of criticality their numerical study suggests a new relation between the corrections to the ground state energy with free boundary conditions and with periodic boundary conditions various critical exponents including the surface ones and those corresponding to the para fermions are determined

crossover in the one dimensional self directed walk the self directed walk is studied in one dimension in this walk with memory the jump probability is given by wn or i 1 exp or g delta n i 1 where delta n is the difference between the number of times the sites in the forward and backward directions have been visited after n steps when g o there is a crossover between a gaussian random walk and an intermediate regime where the radius of gyration grows like n2 followed by a crossover to the asymptotic regime where the walk is directed when g 0 a single crossover is obtained between the gaussian random walk and a saturation regime at large n when the walk is self attracting

field theoretic formulation of an epidemic process with immunisation the author describes a model of the spread of an epidemic on a lattice in which sites may be infected by their neighbours after which they recover and are subsequently immune it corresponds to a modification of the directed percolation problem and also to a growth model of clusters the model is analysed using a continuum field theory both within a self consistent approximation and using an epsilon expansion below six transverse dimensions

statistical mechanics of non intersecting line systems statistical assemblies of non intersecting closed lines loops with a variable loop multiplicity are considered which may have relevance for defect models of certain phase transitions a loop gas model on a lattice is formulated and shown to be formally equivalent to an m anisotropic n vector model in the limit n 0 in two dimensions the equivalence with the eight vertex model in a field is discussed in a few special cases the novel model is equivalent to known ones

scaling and crossover in the one dimensional true self avoiding walk the authors apply scaling and crossover arguments to the one dimensional true self avoiding walk which avoids itself with strength g the problem is formulated in terms of a grand canonical ensemble a real space renormalisation group analysis shows that for large repulsion x e g is a relevant variable causing crossover from the self repelling chain src limit with crossover exponent phi 1 a physical interpretation of this result is given in terms of competition between the correlation length and the average distance before the walk turns back on itself the resulting flow suggests the existence of an intermediate attractive fixed point which makes the exponent v different from src and random walk values for all x between 0 and 1 in agreement with recent monte carlo results

non universal behaviour of self attracting walks a recently proposed model for self attracting walks is studied using exact enumeration techniques the probability of a step is proportional to exp nu where n 1 for sites already visited by the walker and n 0 for the others with u 0 series for the mean square displacement rn2 approximately n2 nu and the mean number of visited sites sn approximately ns of n step walks are calculated in one to four dimensions in all dimensions anomalous diffusion is observed and exponents nu and s vary continuously with the strength parameter u the results are compared with simulations and with previous results for static and dynamic models of generalized random walks in one dimension the behaviour in two and three dimensions may describe anomalous diffusion in real systems

statistics of the true self avoiding walk in one dimension numerical results of a monte carlo study for the true self avoiding walk in one dimension are presented for any positive value of the strength parameter 0 g infinity the root mean square displacement rn and the range sn of the walk are characterised by two universal exponents nu 2 3 or 0 003 and s 2 3 or 0 01 respectively for negative g self attracting walk both rn and sn exhibit a saturation effect limn to infinity rn rinfinity g and limn to infinity sn sinfinity g with rinfinity g approximately g 1 and sinfinity g approximately g 1 at g or approximately 0 a simple scaling analysis in n and g is proposed and found to be consistent with the monte carlo results

self avoiding random walk a brownian motion model with local time drift a natural model for a self avoiding brownian motion inr d when specialised and simplified tod 1 becomes the stochastic differential equation x t b t int limits 0 t g x s l s x s ds where l t x t 0 x r is the local time process ofx thoughx is not markovian an analogue of the ray knight theorem holds for l x x r which allows one to prove in many cases of interest that mathop lim limits t to infty x t t exists almost surely and to identify the limit

irreducible tensor description i a classical gas a classical moderately rarefied simple monatomic gas is considered and it is supposed that its behavior is governed by boltzmann s equation then after a brief review of the fundamental properties of the symmetric traceless tensors the irreducible equations of transfer for the relative symmetric traceless moments of the one point distribution function f are systematically derived and related to those of johnston subsequently grad s expansion of the distribution function in terms of reducible three dimensional hermite polynomials which does not fit in together and with the equations of transfer just mentioned is rigorously transformed into its irreducible counterpart fashioned by mathematical apparatus such as one dimensional laguerre polynomials and ikenberry s tensorial harmonics finally some useful conversion formulas between the relative symmetric traceless moments and the tensorial laguerre ikenberry expansion coefficients of f are deduced and the irreducible variant of grad s moment t

operator dimensions and surface exponents for the non linear schrodinger model at t 0 using the euler maclaurin formula the authors compute finite size corrections to the ground and excited state energies and momenta this enables them to obtain all possible operator scaling dimensions at the critical point t 0 and surface exponents for a variety of boundary conditions they extend the predictions of conformal invariance to include green functions with oscillating terms

renormalisation of random walks with memory the renormalisation of field theoretic models of the true self avoiding random walk is analysed for short range interactions the field theory is shown to be renormalisable only in a special case which corresponds to the problem of a random walk in a random environment and non renormalisable otherwise the long range version of the field theory is shown to be renormalisable in a larger region of the parameter space in the case which corresponds to the original long range true self avoiding random walk the renormalisation group analysis is carried out to two loop order anomalous dimensions of various composite operators are calculated and the crossover between scaling regimes described by short range and long range models is discussed

spiral growing self avoiding walk a new universality class the author introduces the concept of a spiral growing self avoiding walk sgsaw by imposing privman s spiralling constraint ibid vol 16 p l571 1983 on the growing self avoiding walk introduced by majid et al phys rev lett vol 52 p 1257 1984 and lyklema and kremer ibid vol 17 p l691 1984 he studies the critical properties of the sgsaw using monte carlo simulation the exponent nu associated with the average radius of gyration sn is 0 8 on the other hand the critical exponent nu r associated with the end to end distance rn is much less than nu the latter is in sharp contrast to the result nu nu r for the usual saw directed gsaws are shown to behave like directed usual saws

the boltzmann equation for persistent scattering models abstract the nonlinear boltzmann equation n l b e for the persistent scattering model of futcher and hoare can be completely solved by a straightforward application of the fourier transform method for general maxwell models it then follows i that the n l b e for this model possesses the bobylev krook wu mode as a special similarity solution ii that all solutions lying inside a certain hilbert space can be given in the form of an expansion in eigenfunctions of the linearized boltzmann equation the coefficients of which can be determined recursively iii that the cornille gervois solutions lying outside the above hilbert space can be constructed not only for the persistent scattering model but also for a general class of maxwell models the above results valid for models with a continuous energy variable are generalized to include the discrete persistent scattering model

critical behaviour of two interacting linear polymer chains exact results for a state of interpenetration of chains on a fractal lattice a model of two interacting chemically different linear polymer chains on a fractal lattice is proposed to study segregation and interpenetration of chains using the real space renormalization group transformation the value of the contact exponent at the point of the interpenetration transition is calculated exactly for the 2d sierpinski gasket

on the classical solutions of the boltzmann equation on etudie le probleme aux valeurs initiales pour l equation de boltzmann on montre que des solutions au sens classique existent pour tout temps si la donnee initiale est assez proche de l equilibre on se restreint a un domaine cubique et on considere seulement la condition aux limites periodique

approximation of the boltzmann equation by discrete velocity models two convergence results related to the approximation of the boltzmann equation by discrete velocity models are presented first we construct a sequence of deterministic discrete velocity models and prove convergence as the number of discrete velocities tends to infinity of their solutions to the solution of a spatially homogeneous boltzmann equation second we introduce a sequence of markov jump processes interpreted as random discrete velocity models and prove convergence as the intensity of jumps tends to infinity of these processes to the solution of a deterministic discrete velocity model

monte carlo renormalisation group for the true self avoiding walk following the general ideas by swendsen 1982 a monte carlo renormalisation group mcrg estimation of the end to end exponent nu for the one dimensional true self avoiding walk tsaw has been carried out because the tsaw is known to reach its asymptotic behaviour only very slowly an approximate expression for the fixed point has been determined by monte carlo techniques this approximate fixed point has been used as a starting point for the mcrg the final result for the exponent nu is 0 665 or 0 004

the generalised true self avoiding walk a model with continuously variable exponent ν the true self avoiding walk is generalised by admitting that the self avoidance parameter depends on the past history of the walk for the generalised model the end to end distance exponent nu is calculated by a method that gives the same result as a flory type theory for the usual true self avoiding walk the exponent nu is predicted to be continuously variable with one of the parameters of the generalised model the resulting expression for the exponent nu is confirmed by monte carlo simulations

the nontriviality of the one dimensional problem of a true self avoiding walk it is shown that the root mean square distance r2 12 for the self avoiding walk problem is related to the number of steps n as r2 approximately n2 nu with nu 2 3 rather than the self obvious nu 1 this coincides with the mean field theory result obtained by pietronero 1983 which erroneously was not extended to the case d 1

renormalisation of the true self avoiding walk it is shown that the renormalisation of the true self avoiding walk of amit et al 1982 is its upper critical dimensionality two involves at least two possibly three coupling constants fixed points to first order in epsilon 2 d are identified and the asymptotic behaviour at d 2 is discussed

self attracting walk on lattices we have studied a model of self attracting walk proposed by sapozhnikov using the monte carlo method the mean square displacement and the mean number of visited sites are calculated for one two and three dimensional lattices in one dimension the walk shows diffusive behaviour with however in two and three dimensions we observed a non universal behaviour i e the exponent varies continuously with the strength of the attracting interaction

renormalisation of the long range true self avoiding walk the authors show that the true self avoiding walk with long range interactions leads to a renormalisable field theory they compute the characteristic exponent of the gyration radius up to first order in the difference between the upper critical dimension and space dimensionality and compare it with the estimates based on flory like arguments

diffusive limits for true or myopic self avoiding random walks and self repellent brownian polymers in d 3 the problems considered in the present paper have their roots in two different cultures the true or myopic self avoiding walk model tsaw was introduced in the physics literature by amit et al phys rev b 27 1635 1645 1983 this is a nearest neighbor non markovian random walk in mathbb z d which prefers to jump to those neighbors which were less visited in the past the self repelling brownian polymer model srbp initiated in the probabilistic literature by durrett and rogers probab theory relat fields 92 337 349 1992 independently of the physics community is the continuous space time counterpart a diffusion in mathbb r d pushed by the negative gradient of the mollified occupation time measure of the process in both cases similar long memory effects are caused by a path wise self repellency of the trajectories due to a push by the negative gradient of softened local time we investigate the asymptotic behaviour of tsaw and srbp in the non recurrent dimensions first we identify a natural stationary in time and ergodic distribution of the environment the local time profile as seen from the moving particle the main results are diffusive limits in the case of tsaw for a wide class of self interaction functions we establish diffusive lower and upper bounds for the displacement and for a particular more restricted class of interactions we prove full clt for the finite dimensional distributions of the displacement in the case of srbp we prove full clt without restrictions on the interaction functions these results settle part of the conjectures based on non rigorous renormalization group arguments equally valid for the tsaw and srbp cases in amit et al 1983 the proof of the clt follows the non reversible version of kipnis varadhan theory on the way to the proof we slightly weaken the so called graded sector condition

spiral self avoiding walks self avoiding walks with a spiral constraint on the square lattice are enumerated up to 40 steps numerical evidence suggests that they belong to a new universality class

on the boltzmann equation in unbounded space far from equilibrium and the limit of zero mean free path this paper studies loeb solutions of the boltzmann equation in unbounded space under natural initial conditions of finite mass energy and entropy an existence theory for large initial data is presented maxwellian behaviour is obtained in the limits of zero mean free path and of infinite time in the standard space homogeneous hard potential case the infinite time limit is of strongl 1 type

iterative solution of the boltzmann equation we define an iterative scheme to solve the nonlinear boltzmann equation conservation rules are maintained at each iterative step we apply this method to a spatially uniform and isotropic velocity distribution function on the maxwell and very hard particle models a particular example is evaluated and results are compared with the exact solution it shows to be a very fast convergent approach

singularities in the nonisotropic boltzmann equation abstract we consider solutions of the nonlinear boltzmann equation nlbe with nonisotropic singular initial conditions the nlbe is transformed into an integral equation which is solved iteratively the evolution of delta and step singularities in the distribution function during the initial layer is discussed and compared with the isotropic case

information theoretic extended entropy for steady heat condition a phase space distribution for a far from equilibrium steady state in which thermodynamic temperature t and heat flux j are specified is calculated by maximising the information theoretic entropy when this distribution is used to calculate free energy f the latter is obtained as a function of the distribution moduli which in turn are calculated from a consistency condition as functions of j and t the coefficient of the o j2 term in f can be expressed in terms of the radial distribution function g2 and three particle distribution g3 a numerical estimate is made for a hard sphere model of liquid ar at 87 k and high density using the percus yevick solution for g2 and the kirkwood superposition for g3 the estimate confirms the conclusion of an earlier theory that non linear effects at liquid density are negligible except possibly for unrealistically large j fluctuation theory is invoked to make an estimate of the negligibly small o j4 term in f

cell renormalisation of growth processes true self avoiding walks and growing animals the authors present a generating function formalism for growth processes in general and introduce a new cell renormalisation scheme as examples they treat the true self avoiding walk and the eden process or growing animals in two and three dimensions using small cells the results in both cases indicate a substantial increase in the fractal dimension d compared with their equilibrium counterparts the case of the eden process may however suggest a peculiar convergence behaviour as the cell size b tends to infinity

self directed walk a monte carlo study in two dimensions the self directed walk is studied in three dimensions monte carlo simulations on the simple cubic lattice provide an estimate of the radius of gyration exponent nu 0 67 or 0 01 in agreement with the authors recent flory like theory leading to nu 2 d when 2 or d or 4

solution to the nonlinear boltzmann equation for maxwell models for nonisotropic initial conditions the known solution to the spatially homogeneous nonlinear boltzmann equation for maxwell models in a series of laguerre polynomials is extended to include nonisotropic initial conditions existence proofs for a class of solutions are supplied the equations for the generalized nonisotropic laguerre moments are derived in explicit form for two and three dimensional models further it is shown that the ordinary moments satisfy the same set of equations as the hermite polynomial moments

a short note on the true self avoiding walk the scaling laws for the true self avoiding walk and a fokker planck equation for the end to end probability distribution are combined in order to calculate the scaling functions and the end to end distance in one dimension explicitly

exact solutions of the nonlinear boltzmann equation a review is given of research activities since 1976 on the nonlinear boltzmann equation and related equations of boltzmann type in which several rediscoveries have been made and several conjectures have been disproved subjects are i the bkw solution of the boltzmann equation for maxwell molecules first discovered by krupp in 1967 and the krook wu conjecture concerning the universal significance of the bkw solution for the large v t behavior of the velocity distribution functionf v t ii moment equations and polynomial expansions off v t iii model boltzmann equation for a spatially uniform system of very hard particles that can be solved in closed form for general initial conditions iv for maxwell and non maxwell type molecules there exist solutions of the nonlinear boltzmann equation with algebraic decrease at connections with nonuniqueness and violation of conservation laws v conjectured super h theorem and the bkw solution vi exactly soluble one dimensional boltzmann equation with spatial dependence

superdiffusive bounds on self repellent brownian polymers and diffusion in the curl of the gaussian free field in d 2 we consider two models of random diffusion in random environment in two dimensions the first example is the self repelling brownian polymer this describes a diffusion pushed by the negative gradient of its own occupation time measure local time the second example is a diffusion in a fixed random environment given by the curl of massless gaussian free field

on self repelling walks the authors investigate the properties of self repelling walks otherwise known as true self avoiding walks in both one and two dimensions for a range of values of the repulsion parameter g 0 2 or g or 10 0 in one dimension they have obtained 24 terms of the generating function of the mean square end to end distance rn2 while on the two dimensional square lattice they have obtained 12 15 terms in one dimension they find the data to be well fitted by rn2 n43 a b n13 c n o 1 n and in two dimensions by rn2 dn mod lnn mod alpha 1 0 ln mod lnn mod lnn with alpha approximately 0 5 estimates of the amplitudes a and d are also obtained

series study of the one dimensional true self avoiding walk the true self avoiding walk problem is formulated using a grand canonical approach and exact enumeration methods are used to calculate the average end to end distance for one dimensional true self avoiding walks with up to 21 steps the results are in agreement with a universality picture obtained both from monte carlo simulations and from scaling and crossover arguments the extrapolated value of the end to end distance exponent nu is nu 0 67 or 0 04

powerlike decreasing solutions of the boltzmann equation for a maxwell gas we study the homogeneous isotropic nonlinear boltzmann equation for a maxwellian interaction we show that solutions decreasing like inverse powers of the energy are physically acceptable both in the linearized and the quadratic problem because all moments may not exist we introduce a generalized generating function and a finite differential system for generalized sonine moments is derived these new solutions may lead to small relaxation rates and justify in most cases the linear approximation

a comparative study of interacting random walk models the authors analyse four interacting random walk models from a comparative point of view concentrating on one dimensional versions they consider an interacting random walk model recently introduced by stanley et al 1983 the domb joyce model 1972 and the true self avoiding walk of amit et al 1983 in addition they introduce a model based on the weighting of turning points whose properties are related to those of the ising spin chain fixed points and their stability are identified for both attractive and repulsive correlations in all four cases they compare the mechanisms involved in each of these models and comment on the role played in each by its particular form of correlation between steps the different universal behaviours are shown to arise from competition between short range and long range and particularly cumulative memory effects and between local and global normalisation conditions

true self avoiding walks on a percolation cluster the authors investigate by monte carlo simulations the critical behaviour of true self avoiding walks tsaws on a percolation cluster performed very close to the percolation threshold specifically they generate tsaws on a site percolated incipient infinite cluster for various values of the self avoidance parameter g 0 they found that such walks exhibit critical behaviour different from that of ordinary self avoiding walks and also from that of random walks of no constraint the flory exponent obtained was v 0432 or 0 005 for all g 0 which agrees well with the flory type formula suggested by rammal 1984

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

applications of reciprocity to non linear extended thermodynamics kinetic equations abstract reciprocity relations and integrability conditions can be used in non linear extended irreversible thermodynamics to calculate forces and phenomenological coefficients uniquely in terms of parameters in kinetic equations derived from a model one assumes a set of n even variables i and corresponding i with an equation from the model for i independent thermodynamic forces terms are assumed known comparison of powers of in the model equation with corresponding powers in the canonical onsager casimir formalism yields equations to determine all quantities in the latter save the higher dependence of entropy which is affected by uncertainties in measurement by introducing a phenomenological ansatz one can relate these higher entropy terms to corresponding terms in the model equation a numerical illustration is made for a relaxing scalar structural parameter appropriate to a simple liquid

wild s solution of the nonlinear boltzmann equation the relaxation to equilibrium of a spatially uniform pseudo maxwellian gas is considered a modified wild expansion is defined for solving the nonlinear boltzmann equation the positivity and asymptotic conditions as well as the conservation rules are maintained at each truncation order some particular examples are evaluated the comparison with exact solutions illustrates the very fast convergence of this method

nonlinear reciprocity and the maximum entropy formalism abstract the maximum entropy formalism calculates a phase space distribution p x which maximizes the information theoretic entropy subject to specified values of internal energy and state variables i and i i 1 n the latter are averages calculated from p of dynamical functions ai and ai ilai respectively where l is the liouville operator from p one extracts a fluctuation distribution g a v for the values of the a and a a kinetic equation for the fluctuation distribution has been derived by grabert using projection operators appropriate to a system in thermal contact with its surroundings and g is proposed as an ansatz for its solution from the first moments of the kinetic equation after introduction therein of g one extracts phenomenological equations for ai and hi whose terms can be grouped to exhibit onsager symmetry even when nonlinear terms are included the groupings can be affected exactly but not uniquely

onsager reciprocity in the nonlinear regime a system is assumed in the framework of extended non equilibrium thermodynamics to be described by state variables i which are ensemble averages of dynamical functions ai and their time derivatives i gai a fokker planck type equation is derived by the zwanzig projection operator technique for g a v the amplitude of the probability that a a and a v the first moments of this equation give the phenomenological equations which can be cast in onsager casimir canonical form with forces and coefficients exhibiting reciprocity and non linear in both and variables it is necessary to assume that g is sharply peaked in equilibrium and that a markovian approximation holds for the particular sets of variables and time scale chosen the arguments are illustrated by applying them to heat conduction in a liquid or solid

renormalisability and renormalisation of the true self avoiding random walks renormalisation of the recently proposed field theoretic model for the true self avoiding random walk is analysed apart from a special case which corresponds to the problem of a random walk in a random environment the model is shown to be non renormalisable in the case of short range interactions contrary to earlier conjectures the long range version of this field theoretic model is shown to be renormalisable in certain regions of the parameter space and renormalisation group analysis of the model is carried out to two loop order in the region corresponding to the true self avoiding random walk with long range repulsion in a different region of the parameter space the anomalous dimension of the diffusion coefficient is determined exactly in perturbation theory

true self avoiding walk on fractals the authors investigate the true self avoiding walk on fractal spaces the number sn of sites visited during an n step walk and the root mean square displacement rn are calculated via extensive monte carlo simulations on the 2d sierpinski gasket for all positive values of the repulsion parameter g they found the same exponents s 0 815 or 0 001 and nu 0 510 or 0 005 respectively both rn and sn are shown to exhibit a simple scaling behaviour in n and g the authors compare the results with a simple flory like prediction for the exponent nu