random walk in a percolation cluster external field dependence random walk in a percolation cluster was studied in the simultaneous application of both dc bias and ac driving fields by monte carlo calculations amplitude and phase shift of the random walk response to the external field of b0 b1 sin omega t were calculated as a function of b1 omega and b0 some new observations are made which cannot be expected simply from the linear superposition of the two separate results for a constant bias field b0 and an ac driving field b1 sin omega t respectively

random walk on fractals numerical studies in two dimensions monte carlo calculations are used to investigate some statistical properties of random walks on fractal structures two kinds of lattices are used the sierpinski gasket and the infinite percolation cluster in two dimensions among other problems the authors study i the range rn of the walker number of distinct visited sites during n steps average value sn variance sigma n and asymptotic distribution ii renewal theory return to the original site probability of return p0 n mean number of returns nu n the probability distribution of the walker position p n r after n steps is discussed the asymptotic behaviour n 1 of these quantities exhibits power laws with associated exponents the numerical values of these exponents are in good agreement with recent theoretical predictions alexander and orbach 1982 rammal and toulouse 1982

random walks on hierarchical lattices at the percolation threshold random walks on hierarchical lattices are considered at the percolation threshold real space renormalisation methods are employed to obtain the exact fractal dimension of the incipient infinite cluster iic df and the fractal dimension of the effective one dimensional resistance length of the iic dr for some hierarchical lattices the recent conjecture of aharony and stauffer does not hold for diamond and wheatstone bridge hierarchical lattices

a diffusion model for transport on loopless aggregates the authors propose a diffusion equation for transport on a loopless aggregate the relevant distance variable is the chemical distance and the spatially dependent coefficients of the diffusion equation are found in terms of b l the number of bonds at chemical distance l from the vertex of the tree and bs l the number of bonds on the skeleton of the tree it is shown that the resulting motion can be modelled in terms of a biased one dimensional random walk with a probability of remaining in place that approaches 1 as l to infinity this is interpreted as being due to motion along dead ends an important consequence of the model is a simple derivation of the expressions relating the diffusion exponent and fracton dimension to the fractal dimension and the intrinsic dimensions of the tree and its skeleton the mean first passage time to go from the vertex to an arbitrary shell is also found

unified continuum description for sub diffusion random walks on multi dimensional comb model abstract the anisotropic sub diffusion random walks on multi dimensional comb structure model have been studied in the continuum approximation the problem is that in the considered model mean square displacements on different directions have different power temporal dependencies so this case essentially differs from usual anisotropic diffusion case and it is not obviously how to describe such a diffusion by a unified way nevertheless the unified continuum description for such random walks has been developed as result the new generalization of fick s law for diffusion current has been obtained for anisotropic anomalous sub diffusion random walks in multi dimensional case namely it has been shown that the effective diffusion coefficient in the generalized fick s law has new operator form components of diffusion tensor instead of constant values have the fractional order temporal derivatives

exact enumeration of random walks with traps a new useful method of exact enumeration of random walks with traps is presented the method is applied to the one dimensional case and is shown to yield excellent agreement with an exact solution the advantage of the method is that it can be applied in higher dimensions numerical results confirm the known asymptotic solution and can be used to estimate the time and concentration needed to approach this limit

topological properties of percolation clusters percolation properties including the total cluster mass s the shell mass b and the linear geometrical size r are studied as a function of the topological chemical distance parameter l all critical exponents are shown to be related to an apparently new exponent nu defined by r approximately lv critical exponents are calculated exactly for percolation clusters on the cayley tree a model for 6d percolation for which s approximately l2 b approximately l and r2 approximately l for diffusion on such clusters one finds that l3 approximately t numerical estimates of the exponents are obtained for other dimensions a conjecture which relates nu to beta and nu is discussed

anomalous acoustic behaviour and backbone structure of percolation clusters a real space renormalisation group method is used to investigate acoustic properties of percolation lattices in the vicinity of the percolation threshold scaling form an expressions for critical exponents x y describing the sound velocity are derived the existence of an anomalous dispersion relation is elucidated dimensionalities dt and dl which represent the geometrical structure of cluster backbones are introduced the exponents x y and the dimensionalities dt dl are expressed in terms of other exponents nu beta t s and beta b for percolation and their explicit values are evaluated from known estimates for nu beta t s and beta b a geometrical picture of the backbone structure is obtained in terms of the dimensionalities

comment on the equivalence between fracton spectral dimensionality and the dimensionality of recurrence the fracton spectral dimensionality d fracton is an important characteristic of fractal processes until now it has not been interpreted as the fractal dimensionality of any well defined fractal set and it has been claimed that d fracton is more intrinsic than the fractal dimensionality in fact d fractol is best understood as an originaldynamical reinterpretation of a well defined previously knownkinetic dimensionality d fracton s twice the fractal codimensionality of the time instants when a fractal process returns to a point it had previously visited

scaling theory for the dynamic density profile at criticality a scaling description for the dynamic density profile of kinetic clusters formed by the sites visited during random walks executed on critical percolation clusters is proposed the analysis is done for both euclidean and fractal spaces various percolative diffusion laws are correctly recovered from this dynamic density profile

probability densities for the displacement of random walks on percolation clusters 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 the numerical data suggest specific forms for the probability density both in the chemical distance variable l and the geometric distance r

fractal dimensionality of polymer chains the authors suggest that the concept of fractal dimensionality provides a useful characterisation of the configurational properties of a single polymer from numerical studies of long polymers traced out by self avoiding walks on a planar lattice they find that the fractal dimensionality is well defined and has a constant value for most scales of length of the chain renormalisation group theory provides a theoretical basis for the concept of fractal dimensionality in polymers

random walks on percolating clusters with energetic disorder 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

random walks on invasion percolation clusters results are presented of numerical simulations of random walks on invasion percolation clusters in two and three dimensions including simulations on continuum percolation clusters at threshold walks of up to 105 steps were taken the results indicate that the fracton dimension of invasion percolation clusters both on regular lattices and in the continuum is the same within numerical errors as the fracton dimension of ordinary percolation clusters at threshold

diffusion on percolation clusters at criticality the concept of fractal dimensionality is used to study the problem of diffusion on percolation clusters the authors find from monte carlo simulations that the fractal dimensionality of a random walk on a critical percolation cluster in three dimensional space is d 3 3 or 0 1 where the size of the cluster is restricted to be larger than the span of the walk and is d 3 9 or 0 1 for a walk on clusters not subject to this restriction for two dimensional space they find d approximately d approximately 2 7 or p0 1 the exponent d and d is related to the scaling of the average length r of n steps via rd varies as n the fracton dimensionality which is related to the density of states was found to be 1 26 or 0 1 these results are in good agreement with the predictions of alexander and orbach 1982

image prediction based on neighbor embedding methods this paper describes two new intraimage prediction methods based on two data dimensionality reduction methods nonnegative matrix factorization nmf and locally linear embedding these two methods aim at approximating a block to be predicted in the image as a linear combination of k nearest neighbors determined on the known pixels in a causal neighborhood of the input block variable k can be seen as a parameter controlling some sort of sparsity constraints of the approximation vector the impact of this parameter as well as of the nonnegativity and sum to one constraints for the addressed prediction problem has been analyzed the prediction and rd performances of these two new image prediction methods have then been evaluated in a complete image coding and decoding algorithm simulation results show gains up to 2 db in terms of the psnr of the reconstructed signal after coding and decoding of the prediction residue when compared with h 264 avc intraprediction modes up to 3 db when compared with template matching and up to 1 db when compared with a sparse prediction method

diffusion on random systems above below and at their percolation threshold in two and three dimensions a detailed monte carlo study is presented for classical diffusion random walks on random l l triangular and l l l simple cubic lattices with l up to 4096 and 256 respectively the speed of a cyber 205 vector computer is found to be about one order of magnitude larger than that of a usual cdc cyber 76 computer to reach the asymptotic scaling regime walks with up to 10 million steps were simulated with about 10 jl steps in total for l 256 at the percolation threshold we review and extend the dynamical scaling description for the distance traveled as function of time the diffusivity above the threshold and the cluster radius below earlier discrepancies between scaling theory and computer experiment are shown to be due to insufficient monte carlo data the conductivity exponent is found to be 2 0 0 2 in three and 1 28 0 02 in two dimensions our data in three dimensions follow well the finite size scaling theory below the threshold the approach of the distance traveled to its asymptotic value is consistent with theoretical speculations and an exponent 2 5 independent of dimensionality the correction to scaling exponent at pc seems to be larger in two than in three dimensions

transport in a two component randomly composite material scaling theory and computer simulations of termite diffusion near the superconducting limit the question of transport in a random two component mixture is addressed to this end two models are precisely formulated that effectively extend to two component mixtures the de gennes ant model of one component systems we develop and test a scaling theory and we solve some of the problems associated with the limit where one of the components is superconducting our work provides the first practical realisation of the termite model of a random superconducting network which performs a normal walk in the normal regions of the material but performs a new and unique form of random walk in the superconducting regions we find that the divergence of the electrical conductivity at the percolation threshold can be described by this random walk and that the critical exponent s is given by s 1 3 f 0 1 in d 2 if we perturb away from the pure superconduct ing limit we find that the electrical properties can still be described by the random walk model with a crossover exponent i 0 4 d 2 moreover we find that the diffusion constant in this region is described by a scaling form so that data can be made to collapse upon a single curve whose form is governed by the exponents s and i

diffusion and fracton dimensionality on fractals and on percolation clusters the authors present simulations of diffusion on an exact fractal and on percolation clusters at criticality for two and three dimensions the results for the fractal support the rammal and toulouse proposition 1983 that ds n dn varies as b n s n the results for percolation are in excellent agreement with the alexander and orbach conjecture 1982 that the fracton dimensionality d 4 3

self avoiding walks on percolation clusters the author studies self avoiding walks saws on percolation clusters a scaling function representation for r the mean end to end distance is proposed which describes a crossover from ordinary saws to saws on fractals he distinguishes between saws on a single cluster for which r approximately nx where x nu 1 and saws on all clusters for which r approximately nz where z nu 2 and n is the number of monomers in the walk he estimates nu 1 d 2 approximately 1 285 and the correction to scaling exponent omega d 2 approximately 1 3 and nu 1 d 3 approximately 1 38 two plausible generalisations of the flory 1953 approximation for nu 1 are investigated and it is argued that none of them provides a satisfactory approximation for nu 1 at all dimensions

percolation on infinitely ramified fractals we present a family of exact fractals with a wide range of fractal and fracton dimensionalities this includes the case of the fracton dimensionality of 2 which is critical for diffusion this is achieved by adjusting the scaling factor as well as an internal geometrical parameter of the fractal these fractals include the cases of finite and infinite ramification characterized by a ramification exponentp the infinite ramification makes the problem of percolation on these lattices a nontrivial one we give numerical evidence for a percolation transition on these fractals this transition is tudied by a real space renormalization group technique on lattices with fractal dimensionality d between 1 and 2 the critical exponents for percolation depend strongly on the geometry of the fractals

diffusion on fractal lattices and the fractal einstein relation finitely ramified fractal lattices show anomalous diffusion with r2 varies as t2h there is a hierarchy of transit times which are shown by monte carlo simulation to satisfy tau n varies as alpha n for large n where alpha b1h and b is the base of the lattice the lattice resistivity scales with size as rho varies as rn where r is characteristic of the lattice exact renormalisation group methods yield alpha and r analytically and verify the einstein relation for fractals alpha rn