The statistics of self-avoiding walks on a disordered lattice

The relevance of lattice disorder on the critical behaviour of self-avoiding walks is discussed. A crossover from nonclassical to classical behaviour seems to take place.Supported by Special Research Area SFB 125Supported by German Academic Exchange Service (DAAD).     

End patterns of self-avoiding walks

Consider a fixed end pattern (a short self-avoiding walk) that can occur as the first few steps of an arbitrarily long self-avoiding walk on ^d. It is a difficult open problem to show that asN , the fraction ofN-step self-avoiding walks beginning with this pattern converges. It is shown that asN , this fraction is bounded away from zero, and that the ratio of the fractions forN andN+2 converges to one. Similar results are obtained when patterns are specified at both ends, and also when the endpoints are fixed.     

From random to self-avoiding walks

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 Π_j=i+2/N(1?ωδ_ij). 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.     

New exponent in self-avoiding walks

A new exponent is reported in the problem of non-intersecting self-avoiding random walks. It is connected with the asymptotic behaviour of the growth of number of such walks. The value of the exponent is found to be nearly 0.90 for all two dimensional and nearly 0.96 for all three dimensional, lattices studied here. It approaches the value 1.0 assymptotically as the dimensionality approaches infinity.     

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.     

Monotonicity of the number of self-avoiding walks

It is shown that the numberc _n of self-avoiding walks of lengthn in ^d is an increasing function ofn.     

Critical behavior of self-avoiding walks on fractals

Using a new graph counting technique suitable for self-similar fractals, exact 18th-order series expansions for SAWs on some Sierpinski carpets are generated. From them, the critical fugacityx _c and critical exponents _SAW and _SAW are obtained. The results show a linear dependence of the critical fugacity with the average number of bonds per site of the lattices studied. We find for some carpets with low lacunarity that _SAW<0.75, thus violating the relation _SAW(fractal) > _SAW (d) for SAWs on the fractals which are embedded in ad-dimensional Euclidean space.     

Multifractality of self-avoiding walks on percolation clusters

We consider self-avoiding walks on the backbone of percolation clusters in space dimensions d=2,3,4. Applying numerical simulations, we show that the whole multifractal spectrum of singularities emerges in exploring the peculiarities of the model. We obtain estimates for the set of critical exponents that govern scaling laws of higher moments of the distribution of percolation cluster sites visited by self-avoiding walks, in a good correspondence with an appropriately summed field-theoretical epsilon=6-d expansion [H.-K. Janssen and O. Stenull, Phys. Rev. E 75, 020801(R) (2007)10.1103/PhysRevE.75.020801].     

Statistics of self-avoiding walks on random lattices

The statistics of directed self-avoiding walks (SAWs) on randomly bond diluted square lattices have been solved exactly and a computer simulation study of the statistics of ordinary SAWs on diluted square lattices has also been performed. The configurational averaging has been performed here over the logarithms of the distribution functions. We find that the critical behaviour remains unchanged below a certain dilution concentrationp ^*, dependent on the length of the chains considered (p ^*=0 forN), and a crossover to a higher order critical behaviour occurs beyond that point.     

Theory of self-avoiding walks on percolation fractals

A phenomenological approach which takes into account the basic geometry and topology of percolation fractal structures and of self-avoiding walks (SAW) is used to derive a new expression for the Flory exponent describing the average radius of gyration of SAWs on fractals. We focus on the radius of gyration and discuss the importance of the intrinsic fractal dimensions of percolation clusters in determining the lower and upper critical dimensions of SAWs. The mean-field version of our new formula corresponds to the Aharony and Harris expression, who used the standard Flory approach for its derivation.On leave from Santipur College, Nadia 741404, India.     

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.     

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.     

Flory approximant for self-avoiding walks on fractals

A Flory approximant for the exponent describing the end-to-end distance of a self-avoiding walk (SAW) on fractals is derived. The approximant involves the fractal dimensionalities of the backbone and of the minimal path, and the exponent describing the resistance of the fractal. The approximant yields values which are very close to those available from exact and numerical calculations.     

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.     

Asymptotic behaviour of self-avoiding walks in continuous space

The asymptotic behaviour of self-avoiding walks that do not lie on a lattice has been investigated by a Monte-Carlo procedure. Its dependence on the fine structure of the walk is discussed.     

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     

On the connective constants of two-dimensional self-avoiding walks

A recent proposal that a new critical exponent characterises the variation of the self-avoiding walk connective constant with lattice co-ordination number is shown to be invalid. Instead, a functional relationship similar to that which holds for the Ising model in two dimensions is found to represent the available data for two-dimensional self-avoiding walks rather well.