Random and deterministic walks on lattices

Physical Review E, 1995

Diffusion of single particles on lattices with random distributions of static barriers (randombarrier model) is investigated by Monte Carlo simulations and the time-dependent effective-medium approximation. The crossover from anomalous to linear diffusive behavior is discussed in terms of a percolation model. For a discrete distribution of barrier heights with a small concentration of "defect" barriers at the percolation threshold, an alternative kind of transition from bulk-controlled to defect-controlled diffusion is observed as the temperature decreases.

2002

Random walks on general graphs play an important role in the understanding of the general theory of stochastic processes. Beyond their fundamental interest in probability theory, they arise also as simple models of physical systems. A brief survey of the physical relevance of the notion of random walk on both undirected and directed graphs is given followed by the exposition of some recent results on random walks on randomly oriented lattices. It is worth noticing that general undirected graphs are associated with (not necessarily Abelian) groups while directed graphs are associated with (not necessarily Abelian) $C^*$-algebras. Since quantum mechanics is naturally formulated in terms of $C^*$-algebras, the study of random walks on directed lattices has been motivated lately by the development of the new field of quantum information and communication.

Journal of Statistical Physics, 1984

Random walks on square lattice percolating clusters were followed for up to 2 • 10 ~ steps. The mean number of distinct sites visited (SN) gives a spectral dimension of d s = 1.30 5:0.03 consistent with superuniversality (d S = 4/3) but closer to the alternative ds= 182/139, based on the low dimensionality correction. Simulations are also given for walkers on an energetically disordered lattice, with a jump probability that depends on the local energy mismatch and the temperature. An apparent fractal behavior is observed for a low enough reduced temperature. Above this temperature, the walker exhibits a "crossover" from fractal-to-Euclidean behavior. Walks on two-and three-dimensional lattices are similar, except that those in three dimensions are more efficient.

Physics Letters A, 1989

The master equation describing the one-dimensional diffusive motion of a particle on an asymmetric random lattice is examined in the two cases ofbond and site disorder. Both models are shown to possess the same long-time properties, thus falling into the same universality class.

Physical Review B, 1980

Random-walk simulations were performed on one-, two-, and three-dimensional, simple and binary lattices with several coordination numbers containing about one million sites. The random walk included a correlation parameter l (Gaussian distribution with given standard deviation) representing a partial directional memory. The walks on the random binary lattices were constrained to sites of one component only (concentration C) with the sites of the second component acting as reflecting microboundaries. All simulations were restricted to the percolating cluster. The simple lattice simulations are compared with the well-known asymptotic analytical expressions for simple random walk (l = 1) and with an expression for correlated walks (I)) 1). The visitation efficiency increases, as expected, with C. It also increases with l for simple and high-C lattices. However, for lower-C lattices the visitation efficiency decreases with l, thus giving rise to "crossover concentrations. " Our results are given in a series of figures of the efficiency or the number of sites visited versus the number of steps, showing the effects of concentration (C), and correlation (l). Applications to exciton percolation and coherence are mentioned.

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

Journal of Physics A: Mathematical and Theoretical, 2008

We report some ideas for constructing lattice models (LMs) as a discrete approach to the reaction-dispersal (RD) or reaction-random walks (RRW) models. The analysis of a rather general class of Markovian and non-Markovian processes, from the point of view of their wavefront solutions, let us show that in some regimes their macroscopic dynamics (front speed) turns out to be different from that by classical reaction-diffusion equations, which are often used as a mean-field approximation to the problem. So, the convenience of a more general framework as that given by the continuous-time random walks (CTRW) is claimed. Here we use LMs as a numerical approach in order to support that idea, while in previous works our discussion was restricted to analytical models. For the two specific cases studied here, we derive and analyze the mean-field expressions for our LMs. As a result, we are able to provide some links between the numerical and analytical approaches studied.

Physical Review E, 2006

We study the persistent random walk of photons on a one-dimensional lattice of random transmittances. Transmittances at different sites are assumed independent, distributed according to a given probability density f͑t͒. Depending on the behavior of f͑t͒ near t = 0, diffusive and subdiffusive transports are predicted by the disorder expansion of the mean square-displacement and the effective medium approximation. Monte Carlo simulations confirm the anomalous diffusion of photons. To observe photon subdiffusion experimentally, we suggest a dielectric film stack for realization of a distribution f͑t͒.

Journal of Statistical Physics, 1989

We present new exact results for a one-dimensional asymmetric disordered hopping model. The lattice is taken infinite from the start and we do not resort to the periodization scheme used by Derrida. An explicit resummation allows for the calculation of the velocity V and the diffusion constant D (which are found to coincide with those given by Derrida) and for demonstrating that V is indeed a self-averaging quantity; the same property is established for D in the limiting case of a directed walk.

Physical Review Letters, 2016

It is recognised now that a variety of real-life phenomena ranging from diffuson of cold atoms to motion of humans exhibit dispersal faster than normal diffusion. Lévy walks is a model that excelled in describing such superdiffusive behaviors albeit in one dimension. Here we show that, in contrast to standard random walks, the microscopic geometry of planar superdiffusive Lévy walks is imprinted in the asymptotic distribution of the walkers. The geometry of the underlying walk can be inferred from trajectories of the walkers by calculating the analogue of the Pearson coefficient.

Journal de Physique, 1985

2014 Nous étudions un modèle de Marches sans retour Dirigées sur un réseau dilué, par différentes approches (arbre de Cayley, développement de faible désordre, calcul Monte-Carlo avec construction de marches jusqu'à 2 000 pas). Ce modèle simple a l'avantage de présenter les traits essentiels du problème controversé des marches sans retour en milieu aléatoire. On montre en particulier que quelle que soit la valeur du désordre, valeur moyenne et valeur plus probable du nombre de marches sont différentes. Abstract. 2014 We consider a model of Directed Self-Avoiding Walks (DSAW) on a dilute lattice, using various approaches (Cayley Tree, weak-disorder expansion, Monte-Carlo generation of walks up to 2 000 steps). This simple model appears to contain the essential features of the controversial problem of self-avoiding walks in a random medium. It is shown in particular that with any amount of disorder the mean value for the number of DSAW is different from its most probable value.

We consider quantum random walks on congested lattices and contrast them to classical random walks. Congestion is modelled with lattices that contain static defects which reverse the walker's direction. We implement a dephasing process after each step which allows us to smoothly interpolate between classical and quantum random walkers as well as study the effect of dephasing on the quantum walk. Our key results show that a quantum walker escapes a finite boundary dramatically faster than a classical walker and that this advantage remains in the presence of heavily congested lattices. Also, we observe that a quantum walker is extremely sensitive to our model of dephasing.

Journal of Physics A: Mathematical and General, 2001

We study the effect of a single excluded site on the diffusion of a particle undergoing random walk in a d-dimensional lattice. The determination of the characteristic function allows to find explicitly the asymptotical behaviour of physical quantities such as the particle average position (drift) x (t) and the mean square deviation x 2 (t) − x 2 (t). Contrarily to the one-dimensional case, where x (t) diverges at infinite times ( x (t) ∼ t 1/2 ) and where the diffusion constant D is changed due to the impurity, the effects of the latter are shown to be much less important in higher dimensions: for d ≥ 2, x (t) is simply shifted by a constant and the diffusion constant remains unaltered although dynamical corrections (logarithmic for d = 2) still occur. Finally, the continuum space version of the model is analyzed; it is shown that d = 1, is the lower dimensionality above which all the effects of the forbidden site are irrelevant. *

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.

Physical review, 2018

The self-avoiding walk on the square site-diluted correlated percolation lattice is considered. The Ising model is employed to realize the spatial correlations of the metric space. As a well-accepted result, the (generalized) Flory's mean field relation is tested to measure the effect of correlation. After exploring a perturbative Fokker-Planck-like equation, we apply an enriched Rosenbluth Monte Carlo method to study the problem. To be more precise, the winding angel analysis is also performed from which the diffusivity parameter of Schramm-Loewner evolution (SLE) theory (κ) is extracted. We find that at the critical Ising (host) system the exponents are in agreement with the Flory's approximation. For the off-critical Ising system we find also a new behavior for the fractal dimension of the walker trace in terms of the correlation length of the Ising system ξ(T), i.e. D SAW F