Critical behavior of a random diode network

We study the percolation properties of a random diode network (RDN) which contains two kinds of directed bonds on a square lattice. This network is a special case of the random insulation-resistor-diode network. Both Monte Carlo simulations and series expansion for the percolation probability show that an estimated critical exponent, beta=0.1794+/-0.008, is different from known values for a conventional insulation-resistor-diode network. RDN belongs to neither the isotropic percolation universality class nor to the directed percolation universality, which we attribute to a difference of symmetry breakdown around the critical point.     

Non-equilibrium critical phenomena and phase transitions into absorbing states

This review addresses recent developments in non-equilibrium statistical physics. Focusing on phase transitions from fluctuating phases into absorbing states, the universality class of directed percolation is investigated in detail. The survey gives a general introduction to various lattice models of directed percolation and studies their scaling properties, field-theoretic aspects, numerical techniques, as well as possible experimental realizations. In addition, several examples of absorbing-state transitions which do not belong to the directed percolation universality class will be discussed. As a closely related technique, we investigate the concept of damage spreading. It is shown that this technique is ambiguous to some extent, making it impossible to define chaotic and regular phases in stochastic non-equilibrium systems. Finally, we discuss various classes of depinning transitions in models for interface growth which are related to phase transitions into absorbing states.     

Directed Percolation with Colors and Flavors

A model of directed percolation processes with colors and flavors that is equivalent to a population model with many species near their extinction thresholds is presented. We use renormalized field theory and demonstrate that all renormalizations needed for the calculation of the universal scaling behavior near the multicritical point can be gained from the one-species Gribov process (Reggeon field theory). In addition this universal model shows an instability that generically leads to a total asymmetry between each pair of species of a cooperative society, and finally to unidirectionality of the interspecies couplings. It is shown that in general the universal multicritical properties of unidirectionally coupled directed percolation processes with linear coupling can also be described by the model. Consequently the crossover exponent describing the scaling of the linear coupling parameters is given by =1 to all orders of the perturbation expansion. As an example of unidirectionally coupled directed percolation, we discuss the population dynamics of the tournaments of three species with colors of equal flavor.     

Flowing Sand—A Possible Physical Realization of Directed Percolation

A simple model for flowing sand on an inclined plane is introduced. The model is related to recent experiments by Douady and Daerr and reproduces some of the experimentally observed features. Avalanches of intermediate size appear to be compact, placing the critical behavior of the model into the universality class of compact directed percolation. On very large scales, however, the avalanches break up into several branches, leading to a crossover from compact to ordinary directed percolation. Thus, systems of flowing granular matter on an inclined plane could serve as a first physical realization of directed percolation.     

Diffusion on the Scaling Limit of the Critical Percolation Cluster in the Diamond Hierarchical Lattice

We construct critical percolation clusters on the diamond hierarchical lattice and show that the scaling limit is a graph directed random recursive fractal. A Dirichlet form can be constructed on the limit set and we consider the properties of the associated Laplace operator and diffusion process. In particular we contrast and compare the behaviour of the high frequency asymptotics of the spectrum and the short time behaviour of the on-diagonal heat kernel for the percolation clusters and for the underlying lattice. In this setting a number of features of the lattice are inherited by the critical cluster.     

Renormalized Field Theory of Resistor Diode Percolation

We study resistor diode percolation at the transition from the non-percolating to the directed percolating phase. We derive a field theoretic Hamiltonian which describes not only geometric aspects of directed percolation clusters but also their electric transport properties. By employing renormalization group methods we determine the average two-port resistance of critical clusters, which is governed by a resistance exponent . We calculate to two-loop order.     

Phase diagram of optimal paths

We show that by choosing appropriate distributions of the randomness the search for optimal paths links diverse problems of disordered media, such as directed percolation, invasion percolation, and directed and nondirected spanning polymers. We also introduce a simple and efficient algorithm, which solves the d-dimensional model numerically in O(N(1+df/d)) steps, where df is the fractal dimension of the path. Using extensive simulations in two dimensions, we identify the phase boundaries of the directed polymer universality class. A new strong-disorder phase occurs where the optimum paths are self-affine with parameter-dependent scaling exponents. Furthermore, the phase diagram contains directed and nondirected percolation as well as the directed random walk models at specific points and lines.     

Active-to-absorbing phase transition subjected to the velocity fluctuations in the frozen limit case

The directed bond percolation process is studied in the presence of compressible velocity fluctuations with long-range correlations. We discuss a construction of a field theoretic action and a way of obtaining its large scale properties using the perturbative renormalization group. The most interesting results for the frozen velocity limit are given.     

Percolation analysis of stochastic models of galactic evolution

The stochastic star formation model of galactic evolution can be cast as a problem of directed percolation, the time dimension being that along which the directed bonds exist. We study various aspects of this percolation, those of general interest for the percolation phase transition and those of particular importance for the astrophysical application. Both analytical calculations and computer simulations are provided and the results compared. Among the properties are: value of the percolation threshold, critical indices, percolation probability (star density) near and away from the critical point, local density, cluster sizes, effects of rotation (for disk galaxy models) on the percolation threshold. Astrophysical consequences of some of these properties are discussed, in particular the way in which general phase transition behavior contributes to spiral arm morphology. We look at 1 (space) + 1 (time), 2 + 1 and + 1 dimensions, the 2 + 1 case being of interest for disk galaxies.     

New universality classes in “Percolative” dynamics

We study a site analogue of directed percolation. Random trajectories are generated and their critical behavior is studied. The critical behavior corresponds to that of simple percolation in some of the parameter space, but elsewhere the exponents reveal new universality classes. As a byproduct, we use the model to make an improved estimate of the percolation hull exponents and to calculate the site percolation probability for the square lattice.     

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 ≈ 0.655 is found between the directed and spiral percolation thresholds. Infinite percolation clusters are fractals of dimension d _f ≈ 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 → 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 behaviour 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. Received 10 November 2002 / Received in final form 20 February 2003 Published online 23 May 2003 RID="a" ID="a"e-mail: santra@iitg.ernet.in     

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.     

Adsorbing and Collapsing Directed Animals

A model of a self-interacting directed animal, which also interacts with a solid wall, is studied as a model of a directed branched polymer which can undergo both a collapse and an adsorption transition. The directed animal is confined to a 45° wedge, and it interacts with one of the walls of this wedge. The existence of a thermodynamic limit in this model shown, and the presence of an adsorption transition is demonstrated by using constructive techniques. By comparing this model to a process of directed percolation, we show that there is also a collapse or -transition in this model. We examine directed percolation in a wedge to show that there is a collapse phase present for arbitrary large values of the adsorption activity. The generating function of adsorbing directed animals in a half-space is found next from which we find the tricritical exponents associated with the adsorption transition. A full solution for a collapsing directed animal seems intractible, so instead we examine the collapse transition of a model of column convex directed animals with a contact activity next.     

Infinite-range mean-field percolation: Transfer matrix study of longitudinal correlation length

Two infinite-range directed percolation models, equivalent also to epidemic models, are considered for a finite population (finite number of sites)N at each time (directed axis) step. The general features of the transfer matrix spectrum (evolution operator spectrum for the epidemic) are studied numerically, and compared with analytical predictions in the limitN = . One of the models is devised to allow numerical results to be obtained forN as high as nearly 800 for the largest longitudinal percolation correlation length (relaxation time for epidemic). The finite-N behavior of this correlation length is studied in detail, including scaling near the percolation transition, exponential divergence (withN) above the percolation transition, as well as other noncritical and critical-point properties.     

Dynamic critical properties of a one-dimensional probabilistic cellular automaton

Dynamic properties of a one-dimensional probabilistic cellular automaton are studied by Monte Carlo simulation near a critical point which marks a second-order phase transition from an active state to an effectively unique absorbing state. Values obtained for the dynamic critical exponents indicate that the transition belongs to the universality class of directed percolation. Finally the model is compared with a previously studied one to show that a difference in the nature of the absorbing states places them in different universality classes. Received: 6 February 1998 / Revised and Accepted: 17 February 1998     

Criticality of Parasitic Disease Transmission in a Diffusive Population

Through using the methods of finite-size effect and short time dynamic scaling, we study the critical behavior of parasitic disease spreading process in a diffusive population mediated by a static vector environment. Through comprehensive analysis of parasitic disease spreading we find that this model presents a dynamical phase transition from disease-free state to endemic state with a finite population density. We determine the critical population density, above which the system reaches an epidemic spreading stationary state. We also perform a scaling analysis to determine the order parameter and critical relaxation exponents. The results show that the model does not belong to the usual directed percolation universality class and is compatible with the class of directed percolation with diffusive and conserved fields.     

Percolation in Media with Columnar Disorder

We study a generalization of site percolation on a simple cubic lattice, where not only single sites are removed randomly, but also entire parallel columns of sites. We show that typical clusters near the percolation transition are very anisotropic, with different scaling exponents for the sizes parallel and perpendicular to the columns. Below the critical point there is a Griffiths phase where cluster size distributions and spanning probabilities in the direction parallel to the columns have power-law tails with continuously varying non-universal powers. This region is very similar to the Griffiths phase in subcritical directed percolation with frozen disorder in the preferred direction, and the proof follows essentially the same arguments as in that case. But in contrast to directed percolation in disordered media, the number of active (“growth”) sites in a growing cluster at criticality shows a power law, while the probability of a cluster to continue to grow shows logarithmic behavior.     

Scaling behavior of the directed percolation universality class

In this work we consider five different lattice models which exhibit continuous phase transitions into absorbing states. By measuring certain universal functions, which characterize the steady state as well as the dynamical scaling behavior, we present clear numerical evidence that all models belong to the universality class of directed percolation. Since the considered models are characterized by different interaction details the obtained universal scaling plots are an impressive manifestation of the universality of directed percolation.     

Asymmetrically coupled directed percolation systems

We introduce a dynamical model of coupled directed percolation systems with two particle species. The two species A and B are coupled asymmetrically in that A particles branch B particles, whereas B particles prey on A particles. This model may describe epidemic spreading controlled by reactive immunization agents. We study nonequilibrium phase transitions with attention focused on the multicritical point where both species undergo the absorbing phase transition simultaneously. In one dimension, we find that the inhibitory coupling from B to A is irrelevant and the model belongs to the unidirectionally coupled directed percolation class. On the contrary, a mean-field analysis predicts that the inhibitory coupling is relevant and a new universality appears with a variable dynamic exponent. Numerical simulations on small-world networks confirm our predictions.     

Directed polymer – directed percolation transition: the strong disorder case

The transition of physical properties in disordered systems from strong disorder characteristics to weak disorder characteristics is studied for the directed polymer case. It is shown analytically that this transition is governed by the ratio (pc)/k, where is the probability density of the maximal bond of the optimal Min-Max path, pc is the critical probability of directed percolation, and k is the degree of disorder. This analytic result is found to be in agreement with numerical results related to this transition.