On Continuous-Time Self-Avoiding Random Walk in Dimension Four

We study the distribution of the end-to-end distance of continuous-time self-avoiding random walks (CTRW) in dimension four from two viewpoints. From a real-space renormalization-group map on probabilities, we conjecture the asymptotic behavior of the end-to-end distance of a weakly self-avoiding random walk (SARW) that penalizes two-body interactions of random walks in dimension four on a hierarchical lattice. Then we perform the Monte Carlo computer simulations of CTRW on the four-dimensional integer lattice, paying special attention to the difference in statistical behavior of the CTRW compared with the discrete-time random walks. In this framework, we verify the result already predicted by the renormalization-group method and provide new results related to enumeration of self-avoiding random walks and calculation of the mean square end-to-end distance and gyration radius of continous-time self-avoiding random walks.     

On the non-universality of a critical exponent for self-avoiding walks

We have extended the enumeration of self-avoiding walks on the Manhattan lattice from 28 to 53 steps and for self-avoiding polygons from 48 to 84 steps. Analysis of this data suggests that the walk generating function exponent γ = 1.3385 ± 0.003, which is different from the corresponding exponent on the square, triangular and honeycomb lattices. This provides numerical support for an argument recently advanced by Cardy, to the effect that excluding walks with parallel nearest-neighbour steps should cause a change in the exponent γ. The lattice topology of the Manhattan lattice precludes such parallel steps.     

Efficient Implementation of the Pivot Algorithm for Self-avoiding Walks

The pivot algorithm for self-avoiding walks has been implemented in a manner which is dramatically faster than previous implementations, enabling extremely long walks to be efficiently simulated. We explicitly describe the data structures and algorithms used, and provide a heuristic argument that the mean time per attempted pivot for N-step self-avoiding walks is O(1) for the square and simple cubic lattices. Numerical experiments conducted for self-avoiding walks with up to 268 million steps are consistent with o(log N) behavior for the square lattice and O(log N) behavior for the simple cubic lattice. Our method can be adapted to other models of polymers with short-range interactions, on the lattice or in the continuum, and hence promises to be widely useful.     

Simulation study of random sequential adsorption of mixtures on a triangular lattice

Random sequential adsorption of binary mixtures of extended objects on a two-dimensional triangular lattice is studied numerically by means of Monte Carlo simulations. The depositing objects are formed by self-avoiding random walks on the lattice. We concentrate here on the influence of the symmetry properties of the shapes on the kinetics of the deposition processes in two-component mixtures. Approach to the jamming limit in the case of mixtures is found to be exponential, of the form: θ(t) ∼ θ_jam - Δθ exp(- t/σ), and the values of the parameter σ are determined by the order of symmetry of the less symmetric object in the mixture. Depending on the local geometry of the objects making the mixture, jamming coverage of a mixture can be either greater than both single-component jamming coverages or it can be in between these values. Results of the simulations for various fractional concentrations of the objects in the mixture are also presented.     

Monte Carlo Study of Four-Dimensional Self-avoiding Walks of up to One Billion Steps

We study self-avoiding walks on the four-dimensional hypercubic lattice via Monte Carlo simulations of walks with up to one billion steps. We study the expected logarithmic corrections to scaling, and find convincing evidence in support the scaling form predicted by the renormalization group, with an estimate for the power of the logarithmic factor of 0.2516(14), which is consistent with the predicted value of 1/4. We also characterize the behaviour of the pivot algorithm for sampling four dimensional self-avoiding walks, and conjecture that the probability of a pivot move being successful for an N-step walk is \(O([ \log N ]^{-1/4})\).     

The critical behaviour of two-dimensional self-avoiding random walks

We present exact results for the mean end-to-end distance of self-avoiding random walks on several planar lattices. For the square lattice, we extend the known results from walks with 20 steps to walks with 22 steps, and for the triagular lattice from 14 to 16 steps. For the honeycomb lattice we went up to 34 steps, for the two-choice square lattice up to 44 steps, and for the 4-choice triagular lattice up to 19 steps. The extrapolated valuev=0.747±0.001 (provided the correction-to-scalng exponent is not appreaciably smaller than unity) is in disagreement with both Flory's value and the recent estimate of Derrida. We claim that a different analysis of Derrida's data supports this value.Address from 1st April–30th September 1982: Department of Chemical Physics, Weizmann Institute of Science, Rehovot 76100, IsraelAddress from 1st April-30th September 1982, Department of Chemical Physics, Weizmann Institute of Science, Rehovot, 76100, Israel     

Critical exponent for the loop erased self-avoiding walk by Monte Carlo methods

A Monte Carlo simulation was performed for loop-erased self-avoiding walks (LESAW) to ascertain the exponentv for the Z² and Z³ lattices. The estimated values were 2v=1.600±0.006 in two dimensions and 2v=1.232±0.008 in three dimensions, leading to the conjecturev=4/5 for the two-dimensional LESAW. These results add to existing evidence that the loop-erased self-avoiding walks are not in the same universality class as self-avoiding walks.     

Step-step correlations in restricted and self-avoiding random walks

Static step-step correlations for restricted and self-avoiding random walks (SARW) on quadratic and simple cubic lattices are studied with the help of Monte Carlo simulation technique. For the SARW ofN steps our results, for largeN tend to the theoretically predicted values. An analysis of the correlations for SARW in terms of the individual restrictions indicate that its value between steps at a distancer have a dominant contribution coming from the restriction prohibiting polygon closures of sides2r (r2). Our results also show that the contribution of successive restrictions to the mean-square end-to-end distance of a SARW decay with the same exponent as that of the correlations for the SARW.     

Renormalization of self-avoiding walks by non-linear transformations

Renormalizations of self-avoiding lattice walks by non-linear transformations are discussed. A procedure to obtain approximate renormalization group equations, together with the value of the critical index ν, is carried through for the triangular and the square lattice.     

Two-dimensional oriented self-avoiding walks with parallel contacts

Oriented self-avoiding walks (OSAWs) on a square lattice are studied, with binding energies between steps that are oriented parallel across a face of the lattice. By means of exact enumeration and Monte Carlo simulation, we reconstruct the shape of the partition function and show that this system features of first-order phase transition from a free phase to a tight-spiral phase at _s =log(), where -2.638 is the growth constant for SAWs. With Monte Carlo simulations we show that parallel contacts happen predominantly between a step close to the end of the OSAW and another step nearby; this appears to cause the expected number of parallel contacts to saturate at large lengths of the OSAW.     

Unbiased sampling of globular lattice proteins in three dimensions

We present a Monte Carlo method that allows efficient and unbiased sampling of Hamiltonian walks on a cubic lattice. Such walks are self-avoiding and visit each lattice site exactly once. They are often used as simple models of globular proteins, upon adding suitable local interactions. Our algorithm can easily be equipped with such interactions, but we study here mainly the flexible homopolymer case where each conformation is generated with uniform probability. We argue that the algorithm is ergodic and has dynamical exponent z=0. We then use it to study polymers of size up to 64(3)=262 144 monomers. Results are presented for the effective interaction between end points, and the interaction with the boundaries of the system.     

Transforming Fixed-Length Self-avoiding Walks into Radial SLE<Subscript>8/3</Subscript>

We conjecture a relationship between the scaling limit of the fixed-length ensemble of self-avoiding walks in the upper half plane and radial SLE_8/3 in this half plane from 0 to i. The relationship is that if we take a curve from the fixed-length scaling limit of the SAW, weight it by a suitable power of the distance to the endpoint of the curve and apply the conformal map of the half plane that takes the endpoint to i, then we get the same probability measure on curves as radial SLE_8/3. In addition to a non-rigorous derivation of this conjecture, we support it with Monte Carlo simulations of the SAW. Using the conjectured relationship between the SAW and radial SLE_8/3, our simulations give estimates for both the interior and boundary scaling exponents. The values we obtain are within a few hundredths of a percent of the conjectured values.     

Group theory,Markov chains,and excluded volume effect in polymers

A restricted walk of orderr on a lattice is defined as a random walk in which polygons withr vertices or less are excluded. A study of restricted walks for increasingr provides an understanding of how the transition in properties is effected from random to self-avoiding walks which is important in our understanding of the excluded volume effect in polymers and in the study of many other problems. Here the properties of restricted walks are studied by the transition matrix method based on the theory of Markov chains. A group theoretical method is used to reduce the transition matrix governing the walk in a systematic manner and to classify the eigenvalues of the transition matrix according to the various representations of the appropriate group. It is shown that only those eigenvalues corresponding to two particular representations of the group contribute to the correlations among the steps of the walk. The distributions of eigenvalues for walks of various ordersr on the two-dimensional triangular lattice and the three-dimensional face-centered cubic lattice are presented, and they are shown to have some remarkable features.     

Monte carlo study of the interacting self-avoiding walk model in three dimensions

We consider self-avoiding walks on the simple cubic lattice in which neighboring pairs of vertices of the walk (not connected by an edge) have an associated pair-wise additive energy. If the associated force is attractive, then the walk can collapse from a coil to a compact ball. We describe two Monte Carlo algorithms which we used to investigate this collapse process, and the properties of the walk as a function of the energy or temperature. We report results about the thermodynamic and configurational properties of the walks and estimate the location of the collapse transition.     

Random walks with short-range interaction and mean-field behavior

We introduce a model of self-repelling random walks where the short-range interaction between two elements of the chain decreases as a power of the difference in proper time. The model interpolates between the lattice Edwards model and the ordinary random walk. We show by means of Monte Carlo simulations in two dimensions that the exponentv _MF obtained through a mean-field approximation correctly describes the numerical data and is probably exact as long as it is smaller than the corresponding exponent for self-avoiding walks. We also compute the exponent and present a numerical study of the scaling functions.     

Self-avoiding walks on randomly diluted lattices

We discuss how the introduction of quenched impurities changes the exponents of a self-avoiding walk on a lattice. We find that , the exponent for the number of walks, does not change. On the other hand the exponent for the mean square end to end distance does change. This is caused by a singular normalization atp=p _c , which is necessary to compensate for the allowed number of walks on the diluted lattice.     

Kinetic-growth self-avoiding walks on small-world networks

Kinetically-grown self-avoiding walks have been studied on Watts-Strogatz small-world networks, rewired from a two-dimensional square lattice. The maximum length L of this kind of walks is limited in regular lattices by an attrition effect, which gives finite values for its mean value 〈L 〉. For random networks, this mean attrition length 〈L 〉 scales as a power of the network size, and diverges in the thermodynamic limit (system size N ↦∞). For small-world networks, we find a behavior that interpolates between those corresponding to regular lattices and randon networks, for rewiring probability p ranging from 0 to 1. For p < 1, the mean self-intersection and attrition length of kinetically-grown walks are finite. For p = 1, 〈L 〉 grows with system size as N^1/2, diverging in the thermodynamic limit. In this limit and close to p = 1, the mean attrition length diverges as (1-p)^-4. Results of approximate probabilistic calculations agree well with those derived from numerical simulations.     

Study of Monte Carlo methods for generating self-avoiding walks

The concept of fractal dimensionality is used to study different statistical methods for generating self-avoiding walks (SAWs). The reliability of SAWs traced by the enrichment technique and the dynamic Monte Carlo technique is verified. The number of dynamic cycles which represent a single independent SAW ofN ₀ steps is found to be about 0.1N ₀ ³ . We show that the enrichment process for generating SAWs may be presented as a critical phenomenon.     

Exact Sampling of Self-avoiding Paths via Discrete Schramm-Loewner Evolution

We present an algorithm, based on the iteration of conformal maps, that produces independent samples of self-avoiding paths in the plane. It is a discrete process approximating radial Schramm-Loewner evolution growing to infinity. We focus on the problem of reproducing the parametrization corresponding to that of lattice models, namely self-avoiding walks on the lattice, and we propose a strategy that gives rise to discrete paths where consecutive points lie an approximately constant distance apart from each other. This new method allows us to tackle two non-trivial features of self-avoiding walks that critically depend on the parametrization: the asphericity of a portion of chain and the correction-to-scaling exponent.