Narrow Results

The social percolation model proposed by Solomon et al. as well as its modification are studied in two to four dimensions for the phenomena of self-organized criticality. Stability in the models is obtained and the systems are shown to automatically drift towards the percolation threshold.

The Social Percolation model recently proposed by Solomon et al. is studied on the Ising correlated inhomogeneous network. The dynamics in this is studied so as to understand the role of correlations in the social structure. Thus, the possible role of the structural social connectivity is examined.

There is a discrepancy between the standard view of equilibrium through price adjustment in economics and the observation of large fluctuations in stock markets. We study here a simple model where agents decisions not only depend upon their individual preferences but also upon information obtained from their neighbors in a social network. The model shows that information diffusion coupled to the adjustment process drives the system to criticality with large fluctuations rather than converging smoothly to equilibrium.

Diffusion on 2D site percolation clusters at p = 0.7, 0.8, and 0.9 above p_c on the square lattice in the presence of two crossed bias fields, a local bias B and a global bias E, has been investigated. The global bias E is applied in a fixed global direction whereas the local bias B imposes a rotational constraint on the motion of the diffusing particle. The rms displacement R_t ~ t^k in the presence of both biases is studied. Depending on the strength of E and B, the behavior of the random walker changes from diffusion to drift to no-drift or trapping. There is always diffusion for finite B with no global bias. A crossover from drift to no-drift at a critical global bias E_c is observed in the presence of local bias B for all disordered lattices. At the crossover, value of the rms exponent changes from k = 1 to k < 1, the drift velocity v_t changes from constant in time t to decreasing power law nature, and the "relaxation" time τ has a maximum rate of change with respect to the global bias E. The value of critical bias E_c depends on the disorder p as well as on the strength of local bias B. Phase diagrams for diffusion, drift, and no-drift are obtained as a function of bias fields E and B for these systems.

We study the ratio of the number of sites in the largest and second largest clusters in random percolation. Using the scaling hypothesis that the ratio <M₁>/<M₂> of the mean cluster sizes M₁ and M₂ scales as f ((p - p_c) L^1/ν), we employ finite-size scaling analysis to find that <M₁>/<M₂> is nonuniversal with respect to the boundary conditions imposed. The mean <M₁/M₂> of the ratios behaves similarly although with a distinct critical value reflecting the relevance of mass fluctuations at the percolation threshold. These zero exponent ratios also allow for reliable estimates of the critical parameters at percolation from relatively small lattices.

We compare the results of a very efficient algorithm that we have proposed to study the time evolution of percolation clusters when the occupation probability swept through the critical value in the same sample and in a single run with another algorithm proposed by Newman and Ziff to allow fast calculations of the standard percolation model. Both have a complexity per site that is roughly independent of the size of the system. Our results show that for the derivative threshold distribution, the results (exponent = 1.8 ± 0.2) are closer to Wester while for the cumulative distribution (exponent = 1.5 ± 0.1), they are closer to Newman and Ziff.

The Sznajd model for the spread of opinions is applied to the incipient infinite percolation cluster on the square lattice, with and without power-law correlations for the occupied sites. We trace the phase transition between the two fixed points of all spins up and all spins down.

We study the Gierer–Meinhardt model of reaction-diffusion on a site-disordered square lattice. Let p be the site occupation probability of the square lattice. For p greater than a critical value p_c, the steady state consists of stripe-like patterns with long-range connectivity. For p < p_c, the connectivity is lost. The value of p_c is found to be much greater than that of the site percolation threshold for the square lattice. In the vicinity of p_c, the cluster-related quantities exhibit power-law scaling behavior. The method of finite-size scaling is used to determine the values of the fractal dimension d_f, the ratio, γ/ν, of the average cluster size exponent γ and the correlation length exponent ν. The values appear to indicate that the disordered GM model belongs to the universality class of ordinary percolation.

A novel approach to parallelize the well-known Hoshen–Kopelman algorithm has been chosen, suitable for simulating huge lattices in high dimensions on massively-parallel computers with distributed memory and message passing. This method consists of domain decomposition of the simulated lattice into strips perpendicular to the hyperplane of investigation that is used in the Hoshen–Kopelman algorithm. Systems of world record sizes, up to L = 4 000 256 in two dimensions, L = 20 224 in three, and L = 1036 in four, gave precise estimates for the Fisher exponent τ, the corrections to scaling Δ₁, and for the critical number density n_c.

The restructuring process of diagenesis in the sedimentary rocks is studied using a percolation type model. The cementation and dissolution processes are modeled by the culling of occupied sites in rarefied and growth of vacant sites in dense environments. Starting from sub-critical states of ordinary percolation the system evolves under the diagenetic rules to critical percolation configurations. Our numerical simulation results in two dimensions indicate that the stable configuration has the same critical behavior as the ordinary percolation.

In order to study fluctuations in percolating systems, lattices for sizes up to L = 100 000 have been simulated several thousand times using the Hoshen–Kopelman algorithm. Distributions of cluster numbers are Gaussians for small clusters and half-sided quasi-Gaussians for large clusters. The variance of cluster numbers is proportional to the mean, with power-law deviations for small clusters. Higher moments like skewness and kurtosis were also studied.

Dynamic mean field theory is applied to the problem of forest fires. The starting point is the Monte Carlo simulation in a lattice of a million cells. The statistics of the clusters is obtained by means of the Hoshen–Kopelman algorithm. We get the map p_n → p_{n + 1}, where p_n is the probability of finding a tree in a cell, and n is the discrete time. We demonstrate that the time evolution of p is chaotic. The arguments are provided by the calculation of the bifurcation diagram and the Lyapunov exponent. The bifurcation diagram reveals several windows of stability, including periodic orbits of length three, five and seven. For smaller lattices, the results of the iteration are in qualitative agreement with the statistics of the forest fires in Canada in the years 1970–2000.

A model of the dynamics of appearance of a new collective feeling, in addition and opposite to an existing one, is presented. Using percolation theory, the collective feeling of insecurity is shown to be able to coexist with the opposite collective feeling of safety. Indeed this coexistence of contradictory social feelings result from the simultaneous percolation of two infinite clusters of people who are respectively experiencing a safe and unsafe local environment. Therefore opposing claims on national debates over insecurity are shown to be possibly both valid.

A generalized magnetically controlled ballistic rain-like deposition (MBD) model of granular piles has been numerically investigated in 2D. The grains are taken to be elongated disks whence characterized by a two-state scalar degree of freedom, called "nip", their interaction being described through a Hamiltonian. The results are discussed in order to search for the effect of nip flip (or grain rotation from vertical to horizontal and conversely) probability in building a granular pile. The characteristics of creation of + (or -) nip's clusters and clusters of holes (missing nips) are analyzed. Two different cluster-mass regimes have been identified, through the cluster-mass distribution function which can be exponential or have a power law form depending on whether the nip flip (or grain rotation) probability is large or small. Analytical forms of the exponent are empirically found in terms of the Hamiltonian parameters.

We study percolation as a critical phenomenon on a random multifractal support. The scaling exponent β related to the mass of the infinite cluster and the fractal dimension of the percolating cluster d_f are quantities that have the same value as the ones from the standard two-dimensional regular lattice percolation. The scaling exponent ν related to the correlation length is sensitive to the local anisotropy and assumes a value different from standard percolation. We compare our results with those obtained from the percolation on a deterministic multifractal support. The analysis of ν indicates that the deterministic multifractal is more anisotropic than the random multifractal. We also analyze connections with correlated percolation problems and discuss some possible applications.

This article describes the investigation of morphological variations among two sets of neuronal cells, namely a control group of wild type mouse cells and a group of cells of a transgenic line. Special attention is given to singular points in the neuronal structure, namely the branching points and extremities of the dendritic processes. The characterization of the spatial distribution of such points is obtained by using a recently reported morphological technique based on forced percolation and window-size compensation, which is particularly suited to the analysis of scattered points, presenting several coexisting densities. Different dispersions were identified in our statistical analysis, suggesting that the transgenic line of neurons is characterized by a more pronounced morphological variation. A classification scheme based on a canonical discriminant function was also considered in order to identify the morphological differences.

We analyze a random resistor–inductor–capacitor (RLC) lattice model of two-dimensional metal–insulator composites. The results are compared with Bruggeman's and Landauer's Effective Medium Approximations where a discrepancy was observed for some frequency zones. Such a discrepancy is mainly caused by the strong conductivity fluctuations. Indeed, a two-branches distribution is observed for low frequencies. We show also by increasing the system size that at p_c the so-called Drude peak vanishes; it increases for vanishing losses.

Directed spiral percolation (DSP) is a new percolation model with crossed external bias fields. Since percolation is a model of disorder, the effect of external bias fields on the properties of disordered systems can be studied numerically using DSP. In DSP, the bias fields are an in-plane directional field (E) and a field of rotational nature (B) applied perpendicular to the plane of the lattice. The critical properties of DSP clusters are studied here varying the direction of E field and intensities of both E and B fields in two-dimensions. The system shows interesting and unusual critical behavior at the percolation threshold. Not only the DSP model is found to belong in a new universality class compared to that of other percolation models but also the universality class remains invariant under the variation of E field direction. Varying the intensities of the E and B fields, a crossover from DSP to other percolation models has been studied. A phase diagram of the percolation models is obtained as a function of intensities of the bias fields E and B.

In order to investigate the dependence on lattice size of several observables in percolation, the Hoshen–Kopelman algorithm was modified so that growing lattices could be simulated. By this way, when simulating a lattice of size L, lattices of smaller sizes can be simulated in the same run for free, saving computing time.

Here, site percolation in three dimensions was studied. Lattices of up to L = 5000, with many L-steps in between, have been simulated, for occupation probabilities of p = 0.25, p = 0.3, p = p_c = 0.311608, and p = 0.35.

This paper presents an algorithm, which computes the number of anti-red bonds in a simple cubic lattice (site percolation) for different sizes and densities. Our interest was the fractal dimension of anti-red bonds at the percolation threshold. The value is found to be 1.18 ± 0.01. Two different theories proposed by Conigilio resp. Gouyet suggested a fractal dimension of 1.25 resp. 0.9. Thus, we can exclude the theory of Gouyet and are consistent with the one by Coniglio.