Random walk on self-avoiding walk: a model for conductivity of linear polymers

An exhaustive analysis of the shape properties of discrete self-avoiding random walks is presented. The dependence of the main parameters of the probability distributions from the length of the walk and the differences between these distributions for conventional and self-avoiding walks are studied. By means of high-precision Monte Carlo simulations it is shown that the characteristics of the shape of self-avoiding random walks, when regarded as a function of the walk length, present the expected asymptotic behaviour. The differences with conventional random walks depend upon the observable considered: the probability distributions of the principal inertia eigenvalues of self-avoiding walks spread around the most probable value more widely than the corresponding distributions for unrestricted walks, while in the cases of the asphericity or ratios of inertia eigenvalues, the distributions for self-avoiding walks are somewhat more peaked than their counterparts for conventional random walks. Analytical expressions for the probability distributions of inertia moment ratios and the two-dimensional asphericity are given. For common random walks there is an excellent agreement between these analytical distributions and the Monte Carlo data. In this work it is established that this concordance is maintained if the same analytical distributions are applied to the self-avoiding case. This means that within the precision of the simulations the functional form of the mentioned distributions does not vary when passing from conventional to self-avoiding walks.

We show that the ‘erasing-larger-loops-first’ (ELLF) method, which was first introduced for erasing loops from the simple random walk on the Sierpiński gasket, does work also for non-Markov random walks, in particular, self-repelling walks to construct a new family of self-avoiding walks on the Sierpiński gasket. The one-parameter family constructed in this method continuously connects the loop-erased random walk and a self-avoiding walk which has the same asymptotic behavior as the ‘standard’ self-avoiding walk. We prove the existence of the scaling limit and study some path properties: The exponent $ν$ governing the short-time behavior of the scaling limit varies continuously in the parameter. The limit process is almost surely self-avoiding, while it has path Hausdorff dimension $1/ν $, which is strictly greater than $1$.

In this paper, we study non-trapped self-avoiding walks (SAWs), which are SAWs that can be extended to an arbitrary length. We apply the flatPERM Monte Carlo method for this purpose, generating non-trapped SAWs of length up to 1024 on the square, triangular and hexagonal lattices, and calculating their number and mean squared end-to-end distance. We find strong numerical evidence that the growth constant μ and entropic and metric exponents γ and ν are identical for non-trapped and all SAWs, and in particular that the exponents are also universal for non-trapped SAWs. We also calculate the limiting ratio of non-trapped to all SAWs. We see some evidence of an n−1/2 correction-to-scaling term which is not present for all SAWs.

We use the real-space renormalization group method to study the critical behavior of self-avoiding walks (SAWs) on both site percolation clusters at percolation threshold and site lattice animals in a square lattice. The correlation length exponents of SAWs are found to be\(\nu _{SAW}^{p^c } = 0.767\) on the percolation clusters atpc andνSAWLA=0.804 on lattice animals. These results are compared with Kremer's suggestion of modified Flory formula\(\nu = 3/(\bar d + 2)\) where\(\bar d\) is the fractal dimension of the fractal object.

IRREVERSIBLE SELF AVOIDING WALKS

Polymers in random media: An introduction

We consider the phenomena of entanglement of the two interacting self‐avoiding walks (SAW) situated in a member of the three‐dimensional Sierpinski Gasket (SG) fractal family. We focus our attention to determine number of point contacts between the two SAW paths M, which turns out to be a set of power laws whose characteristics depend predominantly on the interactions between SAW steps. The phase diagrams have been establised and corresponding values of the contact critical exponents φ, associated with the two‐path mutual contacts, have been found.

The self-organized criticality on the random fractal networks has many motivations, like the movement pattern of fluid in the porous media. In addition to the randomness, introducing correlation between the neighboring portions of the porous media has some nontrivial effects. In this paper, we consider the Ising-like interactions between the active sites as the simplest method to bring correlations in the porous media, and we investigate the statistics of the BTW model in it. These correlations are controlled by the artificial "temperature" T and the sign of the Ising coupling. Based on our numerical results, we propose that at the Ising critical temperature T_{c} the model is compatible with the universality class of two-dimensional (2D) self-avoiding walk (SAW). Especially the fractal dimension of the loops, which are defined as the external frontier of the avalanches, is very close to D_{f}^{SAW}=4/3. Also, the corresponding open curves has conformal invariance with the root-mean-square distance R_{rms}∼t^{3/4} (t being the parametrization of the curve) in accordance with the 2D SAW. In the finite-size study, we observe that at T=T_{c} the model has some aspects compatible with the 2D BTW model (e.g., the 1/log(L)-dependence of the exponents of the distribution functions) and some in accordance with the Ising model (e.g., the 1/L-dependence of the fractal dimensions). The finite-size scaling theory is tested and shown to be fulfilled for all statistical observables in T=T_{c}. In the off-critical temperatures in the close vicinity of T_{c} the exponents show some additional power-law behaviors in terms of T-T_{c} with some exponents that are reported in the text. The spanning cluster probability at the critical temperature also scales with L^{1/2}, which is different from the regular 2D BTW model.

Renormalization group approaches to polymers in disordered media

Distribution functions of properties of critical percolation clusters are calculated using the ``$H$-cell'' real space renormalization group (RG). We consider structural properties which span two terminals on percolation clusters. These include the lengths of (minimal, average edge-to-edge, longest) self-avoiding walks, the number of the singly connected bonds, and the masses of percolation clusters, as well as of the backbone. We show that the RG corresponds to a (Galton-Watson) branching process, and apply theorems developed in the mathematical literature. We derive recursion relations for the distribution functions, and find exact functional forms for their asymptotic tail behavior at both small and large arguments. The results for the minimal paths have implications on the (multifractal) distribution of wave functions, while the singly connected bonds determine the moments of Ising correlations on these clusters. Our results compare well with existing simulation data.

A two-dimensional lattice-gas model with square symmetry is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. It has been shown that the precision of the method depends strongly not only on the number of sites in the block but also on its symmetry. In general, the accuracy of the method increases with the number of sites in the block. The minimal relative error in determining the critical values of the interaction parameters is equal to \(\). Using the RSRG method, we have explored phase diagrams of both a two-dimensional Ising spin model and of a square lattice gas with lateral interactions between adparticles. We also have investigated the influence of the attractive and repulsive interactions on both the thermodynamic properties of the lattice gas and the diffusion of adsorbed particles over surface. We have calculated adsorption isotherms and coverage dependences of the pair correlation function, isothermal susceptibility and the chemical diffusion coefficient. In addition, we have included in our analysis the interaction of the activated particle in the saddle point with its nearest neighbors. We have also used Monte Carlo (MC) technique to calculate these dependences. Despite the fact that both methods constitute very different approaches, the correspondence of the numerical data is surprisingly good. Therefore, we conclude that the RSRG approach can be applied to characterize the thermodynamic and kinetic properties of systems of particles with strong lateral interactions.

The t dependence of the average of the end-to-end distance (Rt2) of t-step random walks on self-avoiding walks is studied by small cell real-space renormalisation. The results are in qualitative agreement with earlier Monte Carlo data.

Monte Carlo simulation of a confined random-walk chain

The prediction of conformal invariance for the statistics of self-avoiding random walks restricted to both semi-infinite and wedge-shaped geometries are tested by extrapolating exact enumerations. Close agreement is found, both for the angular distribution of the end-to-end vector, and for the dependence of the critical exponent gamma 2 on the opening angle of the wedge.

A brief review will be given of the current situation in the theory of self-avoiding walks (SAWs). The Domb-Joyce model first introduced in 1972 consists of a random walk on a lattice in which eachN step configuration has a weighting factor Π i=0 N−2 Πj=i+2/N(1−ωδij). Herei andj are the lattice sites occupied by the ith and jth points of the walk. When ω=0 the model reduces to a standard random walk, and when ω=1 it is a self-avoiding walk. The universality hypothesis of critical phenomena will be used to conjecture the behavior of the model as a function ofω for largeN. The implications for the theory of dilute polymer solutions will be indicated.