Single random walker on disordered lattices

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.

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.

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 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 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. B, Condensed matter, 1994

results are obtained for random walks of excitation on a one-dimensional lattice with a Gaussian energy distribution of site energies. The distribution 4(t) of waiting times is studied for different degrees of energetic disorder. It is shown that at T=0, %'(t) is described by a biexponential dependence and at T@0 the distribution %(t) broadens due to the power-law "tail" t '~t hat corresponds to the description of %(t) in the framework of the continuous-time random walk model. The parameter y depends linearly on T for strong (T~O) and moderate disorder. For the case of T=O the number of new sites S(t) visited by a walker is calculated at t~~. The results are in accordance with Monte Carlo data. The survival probability 4(t) for strong disorder in the long-time limit is characterized by the power-law dependence 4(t)-t s with p=cy, where c is the trap concentration and for moderate disorder the decay 4(t) is faster than t

Physical Review E, 1998

We utilize our previously reported model of energetically disordered lattices to study diffusion properties, where we now add the effect of a directional bias in the motion. We show how this leads to ballistic motion at low temperatures, but crosses over to normal diffusion with increasing temperature. This effect is in addition to the previously observed subdiffusional motion at early times, which is also observed here, and also crosses over to normal diffusion at long times. The interplay between these factors of the two crossover points is examined here in detail. The pertinent scaling laws are given for the crossover times. Finally, we deal with the case of the frequency dependent bias, which alternates ͑switches͒ its direction with a given frequency, resulting in a different type of scaling. ͓S1063-651X͑98͒11008-5͔

The average number SN (t) of distinct sites visited up to time t by N noninteracting random walkers all starting from the same origin in a disordered fractal is considered. This quantity SN (t) is the result of a double average: an average over random walks on a given lattice followed by an average over different realizations of the lattice. We show for two-dimensional percolation clusters at criticality (and conjecture for other stochastic fractals) that the distribution of the survival probability over these realizations is very broad in Euclidean space but very narrow in the chemical or topological space. This allows us to adapt the formalism developed for Euclidean and deterministic fractal lattices to the chemical language, and an asymptotic series for SN (t) analogous to that found for the non-disordered media is proposed here. The main term is equal to the number of sites (volume) inside a "hypersphere" in the chemical space of radius L[ln(N )/c] 1/v where L is the root-mean-square chemical displacement of a single random walker, and v and c determine how fast 1 − Γt(ℓ) (the probability that a given site at chemical distance ℓ from the origin is visited by a single random walker by time t) decays for large values of ℓ/L:

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.

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.