Directed Percolation Clusters Research Articles

The elastic and directed percolation backbone

We argue that the elastic backbone (EB) (union of shortest paths) on a cylindrical system, studied by Sampaio Filho et al [2018 Phys. Rev. Lett. 120 175701], is in fact the backbone of two-dimensional directed percolation (DP). We simulate the EB on the same system as considered by these authors, and also study the DP backbone directly using an algorithm that allows backbones to be generated in a completely periodic manner. We find that both the EB in the bulk and the DP backbone have a fractal dimension of d b = d B,DP = 1.681 02(15) at the identical critical point p c,DP ≈ 0.705 485 22. We also measure the fractal dimension at the edge of the EB system and for the full DP clusters, and find d e = d DP = 1.840 54(4). We argue that those two fractal dimensions follow from the DP exponents as d B,DP = 2 − 2β/ν ∥ = 1.681 072(12) and d DP = 2 − β/ν ∥ = 1.840 536(6). Our fractal dimensions differ from the value 1.750(3) found by Sampaio Filho et al.

Stochastic Loewner evolution relates anomalous diffusion and anisotropic percolation.

We disclose the origin of anisotropic percolation perimeters in terms of the stochastic Loewner evolution (SLE) process. Precisely, our results from extensive numerical simulations indicate that the perimeters of multilayered and directed percolation clusters at criticality are the scaling limits of the Loewner evolution of an anomalous Brownian motion, being superdiffusive and subdiffusive, respectively. The connection between anomalous diffusion and fractal anisotropy is further tested by using long-range power-law correlated time series (fractional Brownian motion) as the driving functions in the evolution process. The fact that the resulting traces are distinctively anisotropic corroborates our hypothesis. Under the conceptual framework of SLE, our study therefore reveals different perspectives for mathematical and physical interpretations of non-Markovian processes in terms of anisotropic paths at criticality and vice versa.

Microscopic noise, adaptation and survival in hostile environments

The survival of autocatalytic agents in hostile environments depends on their ability to adapt their spatial configuration to local fluctuations. A model of diffusive reactants that extract the advantage of spatio-temporal fluctuations associated with the stochastic wandering of diffusive catalysts is discussed. Two arguments are presented for the basic processes behind this extraordinary behavior. In the first, the local colonies that evolve around any spatially advantageous region overlap in space-time and an infinite directed percolation cluster emerges. The second argument is based on the return probability of a diffusive agent that is shown to yield finite density of active “oases" with an exponentially large contribution to the reactant population. The different range of applicability of these survival lower bounds to small systems is discussed.

Transport properties of directed percolation clusters at the upper critical dimension

We study the transport properties of directed percolation clusters at the upper critical dimensiondc = 4+1, where critical fluctuations induce logarithmic corrections to the leading (mean-field)scaling behaviour. Employing field theory and renormalization group methods we calculatethese logarithmic corrections up to and including the next to leading correction for avariety of observables, viz. the connectivity, i.e., the probability that two given pointsare connected, the average two-point resistance and some of the fractal massesdescribing percolation clusters. Furthermore, we study logarithmic corrections for themultifractal moments of the current distribution on directed percolation clusters.

Logarithmic corrections in directed percolation.

We study directed percolation at the upper critical transverse dimension d=4, where critical fluctuations induce logarithmic corrections to the leading (mean-field) behavior. Viewing directed percolation as a kinetic process, we address the following properties of directed percolation clusters: the mass (the number of active sites or particles), the radius of gyration, and the survival probability. Using renormalized dynamical field theory, we determine the leading and the next to leading logarithmic corrections for these quantities. In addition, we calculate the logarithmic corrections to the equation of state that describes the stationary homogeneous particle density in the presence of a homogeneous particle source.

Directed random walks on directed percolation clusters.

The characteristics of directed random walks on directed percolation clusters are numerically studied. For two-dimensional clusters grown at the critical probability p(c), it is shown that the distance d of the directed random walkers from their most probable end point is determined by a probability distribution p(d) approximately d(-(1+w/nu)), where the values of w and nu are close to the values of the known exponents: w approximately 0.50 and nu approximately 0.63. This probability distribution is independent of the cluster's length t up to d values comparable to the cluster's width approximately t(nu). The results are shown to be consistent with a tree description of the directed percolation clusters.

Directed spiral site percolation on the square lattice

A new site percolation model, directed spiral percolation (DSP), under both directional and rotational (spiral) constraints is studied numerically on the square lattice. The critical percolation threshold $p_c\approx 0.655$ is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension $d_f\approx 1.733$. The clusters generated are anisotropic. Due to the rotational constraint, the cluster growth is deviated from that expected due to the directional constraint. Connectivity lengths, one along the elongation of the cluster and the other perpendicular to it, diverge as $p\to p_c$ with different critical exponents. The clusters are less anisotropic than the directed percolation clusters. Different moments of the cluster size distribution $P_s(p)$ show power law behavior with $|p-p_c|$ in the critical regime with appropriate critical exponents. The values of the critical exponents are estimated and found to be very different from those obtained in other percolation models. The proposed DSP model thus belongs to a new universality class. A scaling theory has been developed for the cluster related quantities. The critical exponents satisfy the scaling relations including the hyperscaling which is violated in directed percolation. A reasonable data collapse is observed in favour of the assumed scaling function form of $P_s(p)$. The results obtained are in good agreement with other model calculations.

Generic sandpile models have directed percolation exponents.

We study sandpile models with stochastic toppling rules and having sticky grains so that with a nonzero probability no toppling occurs, even if the local height of pile exceeds the threshold value. Dissipation is introduced by adding a small probability of particle loss at each toppling. Generically for the models with a preferred direction, the avalanche exponents are those of critical directed percolation clusters. For undirected models, avalanche exponents are those of directed percolation clusters in one higher dimension.

Multifractal current distribution in random-diode networks.

Recently it has been shown analytically that electric currents in a random-diode network are distributed in a multifractal manner [O. Stenull and H. K. Janssen, Europhys. Lett. 55, 691 (2001)]. In the present paper we investigate the multifractal properties of a random diode network at the critical point by numerical simulations. We analyze the currents running on a directed percolation cluster and confirm the field-theoretic predictions for the scaling behavior of moments of the current distribution. It is pointed out that a random diode network is a particularly good candidate for a possible experimental realization of directed percolation.

On directed interacting animals and directed percolation

We study the phase diagram of fully directed lattice animals with nearest-neighbour interactions on the square lattice. This model comprises several interesting ensembles (directed site and bond trees, bond animals, strongly embeddable animals) as special cases and its collapse transition is equivalent to a directed bond percolation threshold. Precise estimates for the animal size exponents in the different phases and for the critical fugacities of these special ensembles are obtained from a phenomenological renormalization group analysis of the correlation lengths for strips of width up to n=17. The crossover region in the vicinity of the collapse transition is analyzed in detail and the crossover exponent $\phi$ is determined directly from the singular part of the free energy. We show using scaling arguments and an exact relation due to Dhar that $\phi$ is equal to the Fisher exponent $\sigma$ governing the size distribution of large directed percolation clusters.

Nonlinear random resistor diode networks and fractal dimensions of directed percolation clusters.

We study nonlinear random resistor diode networks at the transition from the nonpercolating to the directed percolating phase. The resistor-like bonds and the diode-like bonds under forward bias voltage obey a generalized Ohm's law V approximately I(r). Based on general grounds such as symmetries and relevance we develop a field theoretic model. We focus on the average two-port resistance, which is governed at the transition by the resistance exponent straight phi(r). By employing renormalization group methods we calculate straight phi(r) for arbitrary r to one-loop order. Then we address the fractal dimensions characterizing directed percolation clusters. Via considering distinct values of the nonlinearity r, we determine the dimension of the red bonds, the chemical path, and the backbone to two-loop order.

Low-Density Series Expansion for the Domany-Kinzel Model

Domany-Kinzel (DK) model is a family of the 1+1 dimensional stochastic cellular automata with two parameters p 1 and p 2 , which simulate time evolution of interacting active elements in a random medium. By identifying a set of active sites on the spatio-temporal plane with a percolation cluster, we discuss the directed percolation (DP) transitions in the DK model. We parameterize p 1 = p and p 2 = α p with p ∈[0,1] and α∈[0,2] and calculate the mean cluster size and other quantities characterizing the DP cluster as the series of p up to order 51 for several values of α by using a graphical expansion formula recently given by Konno and Katori. We analyze the series by the first- and second-order differential approximations and the Zinn-Justin method and study the dependence on α of the convergence of estimations of critical values and critical exponents. In the mixed site-bond DP region, 1 ≤α≤ 1.3553, the convergence is excellent. As α→2 slowing down of convergence and as α→0 peculiar oscillation of estim...

Transport on directed percolation clusters.

We study random lattice networks consisting of resistorlike and diodelike bonds. For investigating the transport properties of these random resistor diode networks we introduce a field-theoretic Hamiltonian amenable to renormalization group analysis. We focus on the average two-port resistance at the transition from the nonpercolating to the directed percolating phase and calculate the corresponding resistance exponent straight phi to two-loop order. Moreover, we determine the backbone dimension D(B) of directed percolation clusters to two-loop order. We obtain a scaling relation for D(B) that is in agreement with well known scaling arguments.

On the hulls of directed percolation clusters

The properties of the hulls of directed percolation clusters are studied. The scaling and finite-size scaling of many quantities around the percolation threshold pc are derived and a novel Monte Carlo algorithm, which is more than twice as fast as the standard algorithm at pc, has been formulated to study these properties. Simulations have been conducted that enable an estimation of all the exponents involved. In particular, the central exponent, x, relating the average hull length of clusters to their mass, has been estimated to be 0:773.4/ at the percolation threshold. This same exponent is estimated to be 0:905.5/ for p<p c. Thus, this second value should hold for directed animals.

Distributions of self-interactions and voids in (1+1)-dimensional directed percolation.

We investigate the scaling of self-interactions and voids in (1+1)-dimensional directed percolation clusters and backbones. We verify that the meandering of the backbone scales like the directed cluster. A geometric relation between the size distribution and the fractal dimensions of a set of objects is applied to find the scaling properties of self-interactions in directed percolation. Lastly we connect the geometric properties of the back-bone with the avalanche distribution generated by interface dynamics at the depinning transition.

Fractal dimensions of confined clusters in two-dimensional directed percolation

The fractal structure of directed percolation clusters, grown at the percolation threshold inside parabolic-like systems, is studied in two dimensions via Monte Carlo simulations. With a free surface at y=\pm Cx^k and a dynamical exponent z, the surface shape is a relevant perturbation when k<1/z and the fractal dimensions of the anisotropic clusters vary continuously with k. Analytic expressions for these variations are obtained using a blob picture approach.

Fluid-rock interaction in crustal shear zones: A directed percolation approach

The interplay among mechnical, chemical, hydrological, and thermal processes in the evolution of crustal shear zones makes an analytical approach to their study difficult. As an alternative, a stochastic description (using percolation theory) is used to gain insight into reaction softening and volume loss in ductfle deformation zones. Directed percolation is preferred to ordinary percolation as a model because, in common with natural shear zones, directed percolation clusters have high length/width ratios and anisotropic permeability. In addition, transport along the cluster length (for p > p c is linear with time (modeling fluid advection). The process of strain softening is modeled by subjecting a critical cluster to variable amounts of simple shear, resulting in a geometry similar to that of natural shear zones. A stochastic model for the evolution of porosity with time on a directed lattice displays fixed-point behavior for a range of pore-collapse rates. The observation that natural shear zones from a variety of tectonic settings display mean volume losses of 60%-70%; suggests that the system naturally evolves toward a critical state. This can be explained by assuming that pore collapse is regulated by the tendency for fluid pressure to remain close to lithostatic. Volume loss in crustal shear zones appears to be an example of hydromechanical-chemical self-organized critical behavior.

THE STATISTICS OF CRACK BRANCHING DURING FAST CRACK PROPAGATION

We present simple theoretical ideas which allow an understanding of part of the scaling laws recently observed on branched fracture surfaces. Some of these are argued to be common to all critical branched structures, which barely survive when propagating. This is exemplified by numerical results on the directed percolation cluster, which serves as a good toy-model. Finally, we propose a Langevin equation for unbranched cracks in three dimensions, which naturally leads to self-affine structures.

Intermittency studies in directed bond percolation

The self-similar cluster fluctuations of directed bond percolation at the percolation threshold are studied using techniques borrowed from intermittency-related analysis in multi-particle production. Numerical simulations based on the factorial moments for large (1 + 1)-dimensional lattices allow one to handle statistical and boundary effects and show the existence of weak but definite intermittency patterns. The extracted fractal dimensions are in agreement with scaling arguments leading to a new relation linking the intermittency indices to the critical exponents and the fractal dimension of directed percolation clusters.

Self-affine fractal clusters: Conceptual questions and numerical results for directed percolation

In this paper we address the question of the existence of a well defined, non-trivial fractal dimensionD of self-affine clusters. In spite of the obvious relevance of such clusters to a wide range of phenomena, this problem is still open since thedifferent published predictions forD have not been tested yet. An interesting aspect of the problem is that a nontrivial global dimension for clusters is in contrast with the trivial global dimension of self-affine functions. As a much studied example of self-affine structures, we investigate the infinite directed percolation cluster at the threshold. We measuredD ind=2 dimensions by the box counting method. Using a correction to scaling analysis, we obtainedD=1.765(10). This result does not agree with any of the proposed relations, but it favorsD=1+(1-σν∥)/σν⊥, whereν∥ andν⊥ are the correlation length exponents andσ is a Fisher exponent in the cluster scaling.