Random walks on fractals: higher moments

Physical Review E, 2003

We analyze random walk through fractal environments, embedded in three-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments ͑i.e., of the displacements between two consecutive hittings͒ is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite well a power-law in the case where the dimension D F of the fractal is less than 2, there is though, always a finite rate of unaffected escape. Random walks through fractal sets with D F р2 can thus be considered as defective Levy walks. The distribution of jump increments for D F Ͼ2 is decaying exponentially. The diffusive behavior of the random walk is analyzed in the frame of continuous time random walk, which we generalize to include the case of defective distributions of walk increments. It is shown that the particles undergo anomalous, enhanced diffusion for D F Ͻ2, the diffusion is dominated by the finite escape rate. Diffusion for D F Ͼ2 is normal for large times, enhanced though for small and intermediate times. In particular, it follows that fractals generated by a particular class of self-organized criticality models give rise to enhanced diffusion. The analytical results are illustrated by Monte Carlo simulations.

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.

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.

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.

Physica A: Statistical Mechanics and its Applications, 1989

The study of random fractals on d-dimensional hypercubic space enables us to define (d -1) classes of percolating structures in these fractals. Such percolating structures are shown here when d = 2 and d = 3. The fractal dimensions of these sets at the percolation thresholds are: 312 for d = 2 corresponding to a threaded structure, and 2 and 8/3 for d = 3, corresponding respectively to a threaded structure and a twisted structure. The study of random walk on these percolating structures reveals the harmonic analysis of electrical and transport properties.

Physical Review A, 1990

Anisotropic diffusion on a Sierpi'nski gasket is studied using renormalization equations for the probability densities of waiting times. The effect of an external field on a random walk is described in terms of the dependence of hopping probabilities, mean waiting times, and standard deviations on the hopping distance and on the intensity of the field. Our procedure can also be applied to other deterministic fractals.

Physical Review E, 1995

The distributions of the number of distinct sites S visited by random walks of n steps in the infinite cluster of two-dimensional lattices at the percolation threshold are studied. Different lattice sizes, different origins of the walks, and different realizations of the disorder are investigated by Monte Carlo simulations. The distribution of the mean values of (S") appears to have selfaveraging features. The probability distribution of the normalized values of (S") is investigated with respect to its multifractal behavior. The distributions of the probabilities p($) for fixed S" are presented and analyzed. These distributions are wide and their moments show behavior that cannot be characterized by multifractal scaling exponents.

Journal of Physics A: Mathematical and General, 1989

A renormalisation theory is developed to study the critical behaviour of selfavoiding random walks on multifractals. Critical exponents and connectivity constants are calculated for walks on a class of square multifractal lattices using a finite lattice renormalisation. The effect of the multifractal disorder is considered for both annealed and quenched disorder .

Physical Review E, 2015

We investigate site percolation on a weighted planar stochastic lattice (WPSL), which is a multifractal and whose dual is a scale-free network. Percolation is typically characterized by a threshold value p c at which a transition occurs and by a set of critical exponents β, γ , ν which describe the critical behavior of the percolation probability P (p), mean cluster size S(p), and the correlation length ξ . Besides, the exponent τ characterizes the cluster size distribution function n s (p c ) and the fractal dimension d f characterizes the spanning cluster. We numerically obtain the value of p c and of all the exponents. These results suggest that the percolation on WPSL belong to a separate universality class than on all other planar lattices.

Physical review. E, Statistical, nonlinear, and soft matter physics, 2001

Stretched exponential relaxation [exp-(t/tau)(beta(K))] is observed in a large variety of systems but has not been explained so far. Studying random walks on percolation clusters in curved spaces whose dimensions range from 2 to 7, we show that the relaxation is accurately a stretched exponential and is directly connected to the fractal nature of these clusters. Thus we find that in each dimension the decay exponent beta(K) is related to well-known exponents of the percolation theory in the corresponding flat space. We suggest that the stretched exponential behavior observed in many complex systems (polymers, colloids, glasses...) is due to the fractal character of their configuration space.

Physical review. A, 1987

Static and dynamic properties of the fractal sets generated by free and k-tolerant walks are analyzed in detail. A rather complete picture is obtained for the set of the intersections of free random walks, using correlation functions like the probability of visiting one or more sites m times. Monte Carlo enumerations, jointly with rather sophisticated numerical analysis, are used to determine the fractal dimension of the set of the self-intersections of k-tolerant walks. The results are used to throw new light on the Flory argument for polymer chains with excluded-volume effects; the universal behavior of k-tolerant walks is explained in a coarse-grained reinterpretation of the Flory approximation. Diffusion on the same class of walks allows us to discuss also their universality with respect to dynamical properties. In particular, a spectral dimension equal to -, is obtained for the free random walk in d =-2 dimensions.

Physical Review Letters, 1984

Chaos, Solitons & Fractals, 2019

In this paper we analyse random walk on a fractal structure, specifically fractal curves, using the recently develped calculus for fractal curves. We consider only unbiased random walk on the fractal stucture and find out the corresponding probability distribution which is gaussian like in nature, but shows deviation from the standard behaviour. Moments are calculated in terms of Euclidean distance for a von Koch curve. We also analyse Levy distribution on the same fractal structure, where the dimension of the fractal curve shows significant contribution to the distrubution law by modyfying the nature of moments. The appendix gives a short note on Fourier transform on fractal curves.

Journal of Luminescence, 1984

Luminescence from naphthalene alloys is quenched by long-range exciton hops. These are modeled by long-range random walks on long-range percolation clusters with a range-dependent hopping time. Both linear and exponential range dependencies are simulated, over nearest to fifth nearest neighbor hops. At critical percolation thresholds the random walk properties obey the superuniversality hypothesis (spectral or fracton dimension of about 4/3). However this asymptotic limit is approached at different rates for different functional forms of the hopping time (constant, r, er, 10r).

Physical Review E, 2013

We present exact, analytic results for the mean time to trapping of a random walker on the class of deterministic Sierpinski graphs embedded in d 2 Euclidean dimensions, when both nearest-neighbor (NN) and next-nearestneighbor (NNN) jumps are included. Mean first-passage times are shown to be modified significantly as a consequence of the fact that NNN transitions connect fractals of two consecutive generations.