Long-range random walk on percolation clusters

The Journal of Chemical Physics, 1985

We perform random walk simulations on binary three-dimensional simple cubic lattices covering the entire ratio of open/closed sites (fractionp) from the critical percolation threshold to the perfect crystal. We observe fractal behavior at the critical point and derive the value of the number-of-sites-visited exponent, in excellent agreement with previous work or conjectures, but with a new and imprOVed computational algorithm that extends the calculation to the long time limit. We show the crossover to the classical Euclidean behavior in these lattices and discuss its onset as a function of the fractionp. We compare the observed trends with the two-dimensional case.

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 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.

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 Letters, 1999

2D Percolation path exponents x P ℓ describe probabilities for traversals of annuli by ℓ non-overlapping paths, each on either occupied or vacant clusters, with at least one of each type. We relate the probabilities rigorously to amplitudes of O(N = 1) models whose exponents, believed to be exact, yield x P ℓ = (ℓ 2 − 1)/12. This extends to half-integers the Saleur-Duplantier exponents for k = ℓ/2 clusters, yields the exact fractal dimension of the external cluster perimeter, DEP = 2 − x P 3 = 4/3, and also explains the absence of narrow gate fjords, as originally found by Grossman and Aharony. 05.50.+q, 64.60.Fr, 05.45.Df, 64.60.Ak The fractal geometry of critical percolation clusters has been of interest both for intrinsic reasons and as a window on a range of phenomema. It is characterized by fractal dimensions of various sets [1,2], e.g., of the connected clusters, their backbones, the sets of pivotal (singly-connecting) bonds, the clusters' boundaries (hulls), and their external (accessible) perimeters. A set S is said here to be of fractal dimension D S if the density of points in S within a box of linear size R decays as

Physical Review E, 2004

The scaling properties of self-avoiding walks on a d-dimensional diluted lattice at the percolation threshold are analyzed by a field-theoretical renormalization group approach. To this end we reconsider the model of Y. Meir and A. B. Harris (Phys. Rev. Lett., 63, 2819 (1989)) and argue that via renormalization its multifractal properties are directly accessible. While the former first order perturbation did not agree with the results of other methods, we find that the asymptotic behavior of a self-avoiding walk on the percolation cluster is governed by the exponent νp = 1/2+ε/42+110ε 2 /21 3 , ε = 6 − d. This analytic result gives an accurate numeric description of the available MC and exact enumeration data in a wide range of dimensions 2 ≤ d ≤ 6.

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.

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.

The incipient infinite cluster appearing at the bond percolation threshold can be decomposed into singly connected ''links'' and multiply connected ''blobs.'' Here we decompose blobs into objects known in graph theory as 3-blocks. A 3-block is a graph that cannot be separated into disconnected subgraphs by cutting the graph at two or fewer vertices. Clusters, blobs, and 3-blocks are special cases of k-blocks with kϭ1, 2, and 3, respectively. We study bond percolation clusters at the percolation threshold on two-dimensional ͑2D͒ square lattices and three-dimensional cubic lattices and, using Monte Carlo simulations, determine the distribution of the sizes of the 3-blocks into which the blobs are decomposed. We find that the 3-blocks have fractal dimension d 3 ϭ1.2Ϯ0.1 in 2D and 1.15Ϯ0.1 in 3D. These fractal dimensions are significantly smaller than the fractal dimensions of the blobs, making possible more efficient calculation of percolation properties. Additionally, the closeness of the estimated values for d 3 in 2D and 3D is consistent with the possibility that d 3 is dimension independent. Generalizing the concept of the backbone, we introduce the concept of a ''k-bone,'' which is the set of all points in a percolation system connected to k disjoint terminal points ͑or sets of disjoint terminal points͒ by k disjoint paths. We argue that the fractal dimension of a k-bone is equal to the fractal dimension of k-blocks, allowing us to discuss the relation between the fractal dimension of k-blocks and recent work on path crossing probabilities.