Structural and dynamical properties of random walk clusters

Journal of Physics A: Mathematical and General, 1984

We study the cluster structure resulting from a nearest-neighbour random walk embedded in a d-dimensional space. Each bond visited by the random walks is regarded as belonging to the cluster. The diffusion exponent an< the fracton dimensional of the fractal cluster in d = 3 is found to be d, = 3.5 f 0.1 and d = 0.57 0.02, using a method of exact enumeration of random walks on these fractals.

Journal of Physics A: Mathematical and General, 1988

Single random walker simulations on energetically disordered percolation clusters (in two dimensions) are presented. Exponential, Gaussian and uniform site energy distributions are investigated. The superposition of spatial and energetic disorder leads to reduced random walk ranges with decreasing temperature. An analogue subordination rule is derived: random walk on an energetically disordered fractal is equivalent to that on a geometrical fractal with a lower spectral dimension. This rule is strictly followed for the exponential distribution but only approximately for the Gaussian and uniform distributions. The last two distributions, and especially the uniform one, show a crossover behaviour analogous to that of random walks on percolation clusters away from criticality.

Journal of Physics A: Mathematical and Theoretical, 2008

We consider random walks (RWs) and self-avoiding walks (SAWs) on disordered lattices directly at the percolation threshold. Applying numerical simulations, we study the scaling behavior of the models on the incipient percolation cluster in space dimensions d = 2, 3, 4. Our analysis yields estimates of universal exponents, governing the scaling laws for configurational properties of RWs and SAWs.

Physical Review B, 1985

Random walks on square-lattice percolation clusters are simulated for interaction ranges spanning one to five nearest-neighbor bonds {R=1,. .. , 5). The relative hopping probability is given by exp(o.r), where r is the number of bonds traversed in one hop and a is a parameter (0&a & 10). The fractal exponent for the random walks is universal. For R=2 (and R=1) we obtain a spectral dimension of d, =1.31+0. 03, in agreement with the Alexander-Orbach conjecture (1.333), and in even better agreement with the Aharony-Stauffer conjecture (1.309). Our results are based on the relation d, ={91/43)f, where S"-N describes the mean number {Sn) of distinct sites visited in N steps for walks originating on all clusters. While the asymptotic limit of f is closely approached after 5000 nominal time steps for a=0, much longer times (& 50000 steps) are required for n »0. We also observe fractal-to-Euclidean crossovers above criticality; again, this crossover takes much longer for a »0.

Physical Review Letters, 2008

We consider self-avoiding walks (SAWs) 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 SAWs, in a good correspondence with an appropriately summed field-theoretical ε = 6 − d-expansion (H.-K. Janssen and O. Stenull, Phys. Rev. E 75, 020801(R) (2007)).

Journal of Physics A: Mathematical and General, 1985

The probability density of the displacement or end-to-end distance of a random walk on the incipient infinite percolation cluster in d = 2 dimensions is studied by an exact enumeration method. Our numerical data suggest specific forms for the probability density both in the chemical distance variable I and the geometric distance r.

Physical Review Letters, 1987

Analysis of Monte Carlo enumerations for diAusion on the fractal structure generated by the random walk on a two-dimensional lattice allows us to predict a behavior &r)n "(1nn)' with v=0. 325~0.01 and a =0.35~0.03. This leads to the conjecture that v=a = -, ' . This value of v, and the presence of logarithmic corrections, are strongly supported by heuristic arguments based on Flory theory and on plausible assumptions.

EPL (Europhysics Letters), 2008

The scaling behavior of self-avoiding walks (SAWs) on the backbone of percolation clusters in two, three and four dimensions is studied by Monte Carlo simulations. We apply the pruned-enriched Rosenbluth chain growth method (PERM). Our numerical results bring about the estimates of critical exponents, governing the scaling laws of disorder averages of the end-to-end distance of SAW configurations. The effects of finite-size scaling are discussed as well.

Physical Review E, 2001

We study the asymptotic shape of self-avoiding random walks (SAW) on the backbone of the incipient percolation cluster in d-dimensional lattices analytically. It is generally accepted that the configurational averaged probability distribution function P B (r, N) for the end-to-end distance r of an N step SAW behaves as a power law for r → 0. In this work, we determine the corresponding exponent using scaling arguments, and show that our suggested 'generalized des Cloizeaux' expression for the exponent is in excellent agreement with exact enumeration results in two and three dimensions.

Physics Procedia, 2010

The scaling behavior of linear polymers in disordered media, modelled by self-avoiding walks (SAWs) on the backbone of percolation clusters in two, three and four dimensions is studied by numerical simulations. We apply the pruned-enriched Rosenbluth chain-growth method (PERM). Our numerical results yield estimates of critical exponents, governing the scaling laws of disorder averages of the configurational properties of SAWs, and clearly indicate a multifractal spectrum which emerges when two fractals meet each other.