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.     

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.     

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.     

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.     

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.     

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.     

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].     

Boundary critical behavior ofd=2 self-avoiding walks on correlated and uncorrelated vacancies

In this paper we present exact results for the critical exponents of interacting self-avoiding walks with ends at a linear boundary. Effective interactions are mediated by vacancies, correlated and uncorrelated, on the dual lattice. By choosing different boundary conditions, several ordinary and special regimes can be described in terms of clusters geometry and of critical and lowtemperature properties of the model. In particular, the problem of boundary exponents at the -point is fully solved, and implications for-point universality are discussed. The surface crossover exponent at the special transition of noninteracting self-avoiding walks is also interpreted in terms of percolation dimensions.     

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.     

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.     

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.     

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.     

Statistics of directed self-avoiding walks on percolation clusters at the percolation threshold

We here study directed self-avoiding walks on site diluted square lattice at the percolation threshold by two parameter real space renormalization group method. We found \(v_\parallel ^{p_c } = 1.00\) and \(v_ \bot ^{p_c } = 0.4348\) from cell-to-cell transformation method. This \(v_ \bot ^{p_c } \) value is then compared with the modified Alexander-Orbach formula that \(v_ \bot ^{p_c } = {{d_S } \mathord{\left/ {\vphantom {{d_S } {2d_L }}} \right. \kern-0em} {2d_L }}\) whered _s is the fracton dimension andd _L is the spreading dimension of the infinite directed percolation cluster.     

Nonlocal Monte Carlo algorithm for self-avoiding walks with fixed endpoints

We study a new Monte Carlo algorithm for generating self-avoiding walks with variable length (controlled by a fugacity) and fixed endpoints. The algorithm is a hybrid of local (BFACF) and nonlocal (cut-and-paste) moves. We find that the critical slowing-down, measured in units of computer time, is reduced compared to the pure BFACF algorithm: _CPU N^2.3 versus N^3.0. We also prove some rigorous bounds on the autocorrelation time for these and related Monte Carlo algorithms.     

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.