Self-organized critical state of sandpile automaton models

Predicting extreme avalanches in self-organized critical sandpiles

In a finite-size Abelian sandpile model, extreme avalanches are repelling each other. Taking a time series of the avalanche size and using a decision variable derived from that, we predict the occurrence of a particularly large avalanche in the next time step. The larger the magnitude of these target avalanches, the better is their predictability. The predictability which is based on a finite-size effect, is discussed as a function of the system size.

Self-organized criticality of a catalytic reaction network under flow

Self-organized critical behavior in a catalytic reaction network system induced by smallness in the molecule number is reported. The system under a flow of chemicals is shown to undergo a transition from a stationary to an intermittent reaction phase when the flow rate is decreased. In the intermittent reaction phase, two temporal regimes with active and halted reactions alternate. The number frequency of reaction events at each active regime and its duration time are shown to obey a universal power law with the exponents 4/3 and 3/2, respectively, independently of the parameters and network structure. These power laws are explained by a one-dimensional random-walk representation of the number of catalytically active chemicals. Possible relevance of the result to reaction dynamics in artificial and biological cells is briefly discussed.

Earthquake size-frequency statistics in a forest-fire model of individual faults

The earthquake size-frequency distribution of individual seismic faults commonly differs from the Gutenberg-Richter law of regional seismicity by the presence of an excess of large earthquakes. Here we present a cellular automaton of the forest-fire model type that is able to reproduce several size-frequency distributions depending on the number and location of asperities on the fault plane. The model describes a fault plane as a two-dimensional array of cells where each cell can be either a normal site or a trigger site. Earthquakes start on trigger sites. Asperities appear as the dual entities of the trigger sites. We study the effect that the number and distribution of asperities (the dual of the set of trigger sites), the earthquake rate, and the aspect ratio of the fault have on the size-frequency distribution. Size-frequency distributions have been grouped into subcritical, critical, and supercritical, and the relationship between the model parameters and these three kinds of distributions is presented in the form of phase maps for each of the five asperity types tested. We also study the connection between the model parameters and the aperiodicity of the large earthquakes. For this purpose the concept of aperiodicity spectrum is introduced. The aperiodicity in the recurrence of the large earthquakes in a fault shows an interesting variability that can be potentially useful for prediction purposes.

Universal scaling of optimal current distribution in transportation networks

Transportation networks are inevitably selected with reference to their global cost which depends on the strengths and the distribution of the embedded currents. We prove that optimal current distributions for a uniformly injected d -dimensional network exhibit robust scale-invariance properties, independently of the particular cost function considered, as long as it is convex. We find that, in the limit of large currents, the distribution decays as a power law with an exponent equal to (2d-1)/(d-1). The current distribution can be exactly calculated in d=2 for all values of the current. Numerical simulations further suggest that the scaling properties remain unchanged for both random injections and by randomizing the convex cost functions.

Geometric percolation thresholds of interpenetrating plates in three-dimensional space

The geometric percolation thresholds for circular, elliptical, square, and triangular plates in the three-dimensional space are determined precisely by Monte Carlo simulations. These geometries represent oblate particles in the limit of zero thickness. The normalized percolation points, which are estimated by extrapolating the data to zero radius, are etac=0.961 4+/-0.000 5, 0.864 7+/-0.000 6, and 0.729 5+/-0.000 6 for circles, squares, and equilateral triangles, respectively. These results show that the noncircular shapes and corner angles in the plate geometry tend to increase the interparticle connectivity and therefore reduce the percolation point. For elliptical plates, the percolation threshold is found to decrease moderately, when the aspect ratio epsilon is between 1 and 1.5, but decrease rapidly for epsilon greater than 1.5. For the binary dispersion of circular plates with two different radii, etac is consistently larger than that of equisized plates, with the maximum value located at around r1/r2=0.5.

Direct evidence for conformal invariance of avalanche frontiers in sandpile models

Appreciation of stochastic Loewner evolution (SLE_{kappa}) , as a powerful tool to check for conformal invariant properties of geometrical features of critical systems has been rising. In this paper we use this method to check conformal invariance in sandpile models. Avalanche frontiers in Abelian sandpile model are numerically shown to be conformally invariant and can be described by SLE with diffusivity kappa=2 . This value is the same as value obtained for loop-erased random walks. The fractal dimension and Schramm's formula for left passage probability also suggest the same result. We also check the same properties for Zhang's sandpile model.

Confirming and extending the hypothesis of universality in sandpiles

Stochastic sandpiles self-organize to an absorbing-state critical point with scaling behavior different from directed percolation (DP) and characterized by the presence of an additional conservation law. This is usually called the C-DP or Manna universality class. There remains, however, an exception to this universality principle: a sandpile automaton introduced by Maslov and Zhang, which was claimed to be in the DP class despite the existence of a conservation law. We show, by means of careful numerical simulations as well as by constructing and analyzing a field theory, that (contrarily to what was previously thought) this sandpile is also in the C-DP or Manna class. This confirms the hypothesis of universality for stochastic sandpiles and gives rise to a fully coherent picture of self-organized criticality in systems with conservation. In passing, we obtain a number of results for the C-DP class and introduce a strategy to easily discriminate between DP and C-DP scaling.

Dynamics of the one-dimensional self-organized forest-fire model

We examine the dynamical evolution of the one-dimensional self-organized forest-fire model (FFM), when the system is far from its statistically steady state. In particular, we investigate situations in which conditions change on a time scale that is faster than, or of the order of the typical time needed for relaxation. An analytical approach is introduced based on a hierarchy of first-order nonlinear differential equations. This hierarchy can be closed at any level, yielding a sequence of successively more accurate descriptions of the dynamics. It is found that our approximate description can yield a faithful description of the FFM dynamics, even when a low order truncation is used. Employing both full simulations of the FFM and our approximate descriptions, we examine the time scales and cluster-size-dependent dynamics of relaxation to the statistical equilibrium. As an example of changing external conditions in a natural forest, the effects of a time-dependent lightning frequency are considered.

Numerical renormalization-group approach to a sandpile

Multiple topplings and grain redistribution are two essential features in sandpile dynamics. A renormalization group (RG) approach incorporating these features is investigated. The full enumeration of all relaxations involving such an approach is difficult. Instead, we developed an efficient procedure to sample the relaxations. We applied this RG scheme to a square lattice and a triangular lattice. As shown by the fixed point analysis on a square lattice, the effect of multiple topplings leads the resultant height probabilities towards the exact solution while the effect of grain redistribution does not.

Analytical approach to directed sandpile models on the Apollonian network

We investigate a set of directed sandpile models on the Apollonian network, which are inspired by the work of Dhar and Ramaswamy [Phys. Rev. Lett. 63, 1659 (1989)] on Euclidian lattices. They are characterized by a single parameter q , which restricts the number of neighbors receiving grains from a toppling node. Due to the geometry of the network, two- and three-point correlation functions are amenable to exact treatment, leading to analytical results for avalanche distributions in the limit of an infinite system for q=1,2 . The exact recurrence expressions for the correlation functions are numerically iterated to obtain results for finite-size systems when larger values of q are considered. Finally, a detailed description of the local flux properties is provided by a multifractal scaling analysis.

Asymptotic behavior and synchronizability characteristics of a class of recurrent neural networks

We propose an approach to the analysis of the influence of the topology of a neural network on its synchronizability in the sense of equal output activity rates given by a particular neural network model. The model we introduce is a variation of the Zhang model. We investigate the time-asymptotic behavior of the corresponding dynamical system (in particular, the conditions for the existence of an invariant compact asymptotic set) and apply the results of the synchronizability analysis on a class of random scale free networks and to the classical random networks with Poisson connectivity distribution.

Self-organizing behavior in a lattice model for co-evolution of virus and immune systems

We propose a lattice model for the co-evolution of a virus population and an adaptive immune system. We show that, under some natural assumptions, both probability distribution of the virus population and the distribution of activity of the immune system tend during the evolution to a self-organized critical state.

Effects of bulk dissipation on the critical exponents of a sandpile

Bulk dissipation of a sandpile on a square lattice with the periodic boundary condition is investigated through a dissipating probability f during each toppling process. We find that the power-law behavior is broken for f>10(-1) and not evident for 10(-1)}>f>10(-2). In the range 10(-2)>or=f>or=10(-5), numerical simulations for the toppling size exponents of all, dissipative, and last waves have been studied. Two kinds of definitions for exponents are considered: the exponents obtained from the direct fitting of data and the exponents defined by the simple scaling. Our result shows that the exponents from these two definitions may be different. Furthermore, we propose analytic expressions of the exponents for the direct fitting, and it is consistent with the numerical result. Finally, we point out that small dissipation drives the behavior of this model toward the simple scaling.

Void percolation and conduction of overlapping ellipsoids

The void percolation and conduction problems for equisized overlapping ellipsoids of revolution are investigated using the discretization method. The method is validated by comparing the estimated percolation threshold of spheres with the precise result found in literature. The technique is then extended to determine the threshold of void percolation as a function of the geometric aspect ratio of ellipsoidal particles. The finite element method is also applied to evaluate the equivalent conductivity of the void phase in the system. The results confirm that there are no universalities for void percolation threshold and conductivity in particulate systems, and these properties are clearly dependent on the geometrical shape of particles. As a consequence, void percolation and conduction associated with ellipsoidal particles of large aspect ratio should be treated differently from spheres.

Dissipative sandpile models with universal exponents

We consider a dissipative variant of the stochastic-Abelian sandpile model on a two-dimensional lattice. The boundaries are closed and the dissipation is due to the fact that each toppled grain is removed from the lattice with probability epsilon. It is shown that the scaling properties of this model are in the universality class of the stochastic-Abelian models with conservative dynamics and open boundaries. In particular, the dissipation rate epsilon can be adjusted according to a suitable function epsilon = f(L), such that the avalanche size distribution will coincide with that of the conservative model on a finite lattice of size L.

Self-organized criticality and coevolution of network structure and dynamics

We investigate, by numerical simulations, how the avalanche dynamics of the Bak-Tang-Wiesenfeld sandpile model can induce emergence of scale-free networks and how this emerging structure affects dynamics of the system.

Exact solution of the one-dimensional deterministic fixed-energy sandpile

By reason of the strongly nonergodic dynamical behavior, universality properties of deterministic fixed-energy sandpiles are still an open and debated issue. We investigate the one-dimensional model, whose microscopical dynamics can be solved exactly, and provide a deeper understanding of the origin of the nonergodicity. By means of exact arguments, we prove the occurrence of orbits of well-defined periods and their dependence on the conserved energy density. Further statistical estimates of the size of the attraction's basins of the different periodic orbits lead to a complete characterization of the activity vs energy density phase diagram in the limit of large system's size.

Universality class of one-dimensional directed sandpile models

A general n-state directed "sandpile" model is introduced. The stationary properties of the n-state model are derived for n<infinity, and analytical arguments based on a central limit theorem show that the model belongs to the universality class of the totally asymmetric Oslo model, with a crossover to uncorrelated branching process behavior for small system sizes. Hence, the central limit theorem allows us to identify the existence of a large universality class of one-dimensional directed sandpile models.

Precise toppling balance, quenched disorder, and universality for sandpiles

A single sandpile model with quenched random toppling matrices captures the crucial features of different models of self-organized criticality. With symmetric matrices avalanche statistics falls in the multiscaling Bak-Tang-Wiesenfeld universality class. In the asymmetric case the simple scaling of the Manna model is observed. The presence or absence of a precise toppling balance between the amount of sand released by a toppling site and the total quantity the same site receives when all its neighbors topple once determines the appropriate universality class.

Four height variables, boundary correlations, and dissipative defects in the Abelian sandpile model

We analyze the two-dimensional Abelian sandpile model, and demonstrate that the four height variables have different field identifications in the bulk, and along closed boundaries, but become identical, up to rescaling, along open boundaries. We consider two-point boundary correlations in detail, and discuss a number of complications that arise in the mapping from sandpile correlations to spanning tree correlations; the structure of our results suggests a conjecture that could greatly simplify future calculations. We find a number of three-point functions along closed boundaries, and propose closed boundary field identifications for the height variables. We analyze the effects of dissipative defect sites, at which the number of grains is not conserved, and show that dissipative defects along closed boundaries, and in the bulk, have no effect on any weakly allowed cluster variables, or on their correlations. Along open boundaries, we find a particularly simple field structure; we calculate all n-point correlations, for any combinations of height variables and dissipative defect sites, and find that all heights and defects are represented by the same field operator.

Conformal field theory correlations in the Abelian sandpile model

We calculate all multipoint correlation functions of all local bond modifications in the two-dimensional Abelian sandpile model, both at the critical point, and in the model with dissipation. The set of local bond modifications includes, as the most physically interesting case, all weakly allowed cluster variables. The correlation functions show that all local bond modifications have scaling dimension 2, and can be written as linear combinations of operators in the central charge -2 logarithmic conformal field theory, in agreement with a form conjectured earlier by Mahieu and Ruelle in Phys. Rev. E 64, 066130 (2001). We find closed form expressions for the coefficients of the operators, and describe methods that allow their rapid calculation. We determine the fields associated with adding or removing bonds, both in the bulk, and along open and closed boundaries; some bond defects have scaling dimension 2, while others have scaling dimension 4. We also determine the corrections to bulk probabilities for local bond modifications near open and closed boundaries.

Conserved lattice gas model with infinitely many absorbing states in one dimension

The conserved lattice gas model with infinitely many absorbing states is studied on a chain and on a ladder. In both one-dimensional lattices it exhibits a phase transition from an absorbing phase to an active state. The model defined on a chain is solved exactly and shows a critical behavior with classical critical exponents. However, the model defined on a ladder shows a critical behavior, obtained from numerical simulation, that places the model in the same universality class as the Manna model.

Exact multileg correlation functions for the dense phase of branching polymers in two dimensions

We consider branching polymers on the planar square lattice with open boundary conditions and exactly calculate correlation functions of k polymer chains that connect two lattice sites with a large distance r apart for odd number of polymer chains k. We find that besides the standard power-law factor the leading term also has a logarithmic multiplier.

Self-erasing perturbations of Abelian sandpiles

We investigate generalized seeding of the attracting states of Abelian sandpile automata and find there exists a class of global perturbations of such automata that are completely removed by the natural local dynamics. We derive a general form for such self-erasing perturbations and demonstrate that they can be highly nontrivial. This phenomenon provides a different conceptual framework for studying such automata and suggests possible applications for data protection and encryption.

Boundary conditions and defect lines in the Abelian sandpile model

We add a defect line of dissipation, or crack, to the Abelian sandpile model. We find that the defect line renormalizes to separate the two-dimensional plane into two half planes with open boundary conditions. We also show that varying the amount of dissipation at a boundary of the Abelian sandpile model does not affect the universality class of the boundary condition. We demonstrate that a universal coefficient associated with height probabilities near the defect can be used to classify boundary conditions.

Behavior of coupled automata

We study the nature of statistical correlations that develop between systems of interacting self-organized critical automata (sandpiles). Numerical and analytical findings are presented describing the emergence of "synchronization" between sandpiles and the dependency of this synchronization on factors such as variations in coupling strength, toppling rule probabilities, symmetric versus asymmetric coupling rules, and numbers of sandpiles.

Numerical study of the Langevin theory for fixed-energy sandpiles

The recently proposed Langevin equation, aimed to capture the relevant critical features of stochastic sandpiles and other self-organizing systems, is studied numerically. The equation is similar to the Reggeon field theory, describing generic systems with absorbing states, but it is coupled linearly to a second conserved and static (nondiffusive) field. It has been claimed to represent a different universality class, including different discrete models: the Manna as well as other sandpiles, reaction-diffusion systems, etc. In order to integrate the equation, and surpass the difficulties associated with its singular noise, we follow a numerical technique introduced by Dickman. Our results coincide remarkably well with those of discrete models claimed to belong to this universality class, in one, two, and three dimensions. This provides a strong backing for the Langevin theory of stochastic sandpiles, and to the very existence of this meagerly understood universality class.

Calculation of the transition matrix and of the occupation probabilities for the states of the Oslo sandpile model

The Oslo sandpile model, or if one wants to be precise, ricepile model, is a cellular automaton designed to model experiments on granular piles displaying self-organized criticality. We present an analytic treatment that allows the calculation of the transition probabilities between the different configurations of the system; from here, using the theory of Markov chains, we can obtain the stationary occupation distribution, which tells us that the phase space is occupied with probabilities that vary in many orders of magnitude from one state to another. Our results show how the complexity of this simple model is built as the number of elements increases, and allow, for small system sizes, the exact calculation of the avalanche-size distribution and other properties related to the profile of the pile.

Sandpile models and random walkers on finite lattices

Abelian sandpile models, both deterministic, such as the Bak, Tang, Wiesenfeld (BTW) model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] and stochastic, such as the Manna model [S.S. Manna, J. Phys. A 24, L363 (1991)] are studied on finite square lattices with open boundaries. The avalanche size distribution P(L)(n) is calculated for a range of system sizes, L. The first few moments of this distribution are evaluated numerically and their dependence on the system size is examined. The sandpile models are conservative in the sense that grains are conserved in the bulk and can leave the system only through the boundaries. It is shown that the conservation law provides an interesting connection between the sandpile models and random-walk models. Using this connection, it is shown that the average avalanche sizes <n>(L) for the BTW and Manna models are equal to each other, and both are equal to the average path length of a random walker starting from a random initial site on the same lattice of size L. This is in spite of the fact that the sandpile models with deterministic (BTW) and stochastic (Manna) toppling rules exhibit different critical exponents, indicating that they belong to different universality classes.

Analytical approximation of the two-dimensional percolation threshold for fields of overlapping ellipses

Percolation of particle arrays is of high interest in microstructural design of materials. While there have been numerous contributions to theoretical modeling of percolation in particulate systems, no analytical approximation for the generalized problem of variable aspect-ratio ellipses has been reported. In the present work, we (1) derive, and verify through simulation, an analytical percolation approach capable of identifying the percolation point in two-phase materials containing generalized ellipses of uniform shape and size; and (2) explore the dependence of percolation on the particle aspect ratio. We validate our technique with simulations tracking both cluster sizes and percolation status, in networks of elliptical and circular particles. We also outline the steps needed to extend our approach to three-dimensional particles (ellipsoids). For biological materials, we ultimately aim to provide direct insight into the contribution of each single phase in multiphase tissues to mechanical or conductive properties. For engineered materials, we aim to provide insight into the minimum amount of a particular phase needed to strongly influence properties.

Finite-size effects of avalanche dynamics

We study the avalanche dynamics of a system of globally coupled threshold elements receiving random input. The model belongs to the same universality class as the random-neighbor version of the Olami-Feder-Christensen stick-slip model. A closed expression for avalanche size distributions is derived for arbitrary system sizes N using geometrical arguments in the system's configuration space. For finite systems, approximate power-law behavior is obtained in the nonconservative regime, whereas for N--> infinity, critical behavior with an exponent of -3/2 is found in the conservative case only. We compare these results to the avalanche properties found in networks of integrate-and-fire neurons, and relate the different dynamical regimes to the emergence of synchronization with and without oscillatory components.

Continuously varying critical exponents in a sandpile model with internal disorder

A sandpile model with an internal disorder is presented. The updating of critical sites is done according to a stochastic rule (with a probabilistic toppling q). Using a unified mean-field theory and numerical simulations, we have shown that the criticality is ensured for any value of q. The static critical exponents have been calculated and found to be the same as those obtained for the deterministic sandpile model, which is a particular case of the stochastic model. They have a universal q-independent behavior. In the limit of slow driving, we have developed a relation between our model and the branching process in order to compute the size exponent tau. It presents a continuous variation with the parameter of toppling q.

Renormalization-group approach to an Abelian sandpile model on planar lattices

One important step in the renormalization-group (RG) approach to a lattice sandpile model is the exact enumeration of all possible toppling processes of sandpile dynamics inside a cell for RG transformations. Here we propose a computer algorithm to carry out such exact enumeration for cells of planar lattices in the RG approach to the Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett. 59, 381 (1987)] and consider both the reduced-high RG equations proposed by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. 72, 1690 (1994)], and the real-height RG equations proposed by Ivashkevich [Phys. Rev. Lett. 76, 3368 (1996)]. Using this algorithm, we are able to carry out RG transformations more quickly with large cell size, e.g., 3x3 cell for the square (SQ) lattice in PVZ RG equations, which is the largest cell size at the present, and find some mistakes in a previous paper [Phys. Rev. E 51, 1711 (1995)]. For SQ and plane triangular (PT) lattices, we obtain the only attractive fixed point for each lattice and calculate the avalanche exponent tau and the dynamical exponent z. Our results suggest that the increase of the cell size in the PVZ RG transformation does not lead to more accurate results. The implication of such result is discussed.

Granular superconductors and a sandpile model with intrinsic spatial randomness

We present a model for investigation of self-organized criticality applicable to a real physical system: a granular superconductor. The model demonstrates self-organized behavior even in circumstances when other models do not.

Self-organized criticality: robustness of scaling exponents

We investigate a deterministic, conservative, undirected, critical height sandpile model with dissipation of an energy at boundaries that can simulate avalanche dynamics. It was derived from the Bak-Tang-Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] introducing an additional second-higher threshold so the model has two distinct thresholds. Our computer simulations for a two-dimensional lattice show that scaling properties of the model depend on the higher-threshold values and site concentrations. These results are not therefore consistent with the present self-organized criticality hypothesis where the scaling properties are independent of the model parameters.

Anomalous scaling and Lee-Yang zeros in self-organized criticality

We show that the generating functions of probability distributions in self-organized criticality (SOC) models exhibit a Lee-Yang phenomenon [Phys. Rev. 87, 404 (1952)]. Namely, their zeros pinch the real axis at z=1, as the system size goes to infinity. This establishes a new link between the classical theory of critical phenomena and SOC. A scaling theory of the Lee-Yang zeros is proposed in this setting.

Scaling fields in the two-dimensional Abelian sandpile model

We consider the unoriented two-dimensional Abelian sandpile model from a perspective based on two-dimensional (conformal) field theory. We compute lattice correlation functions for various cluster variables (at and off criticality), from which we infer the field-theoretic description in the scaling limit. We find perfect agreement with the predictions of a c=-2 conformal field theory and its massive perturbation, thereby providing direct evidence for conformal invariance and more generally for a description in terms of a local field theory. The question of the height 2 variable is also addressed, with, however, no definite conclusion yet.

Exact phase diagram for an asymmetric avalanche process

The Bethe ansatz method and an iterative procedure based on detailed balance are used to obtain exact results for an asymmetric avalanche process on a ring. The average velocity of particle flow, v, is derived as a function of the toppling probabilities and the density of particles, rho. As rho increases, the system shows a transition from intermittent to continuous flow, and v diverges at a critical point rho(c) with exponent alpha. The exact phase diagram of the transition is obtained and alpha is found to depend on the toppling rules.

Lyapunov exponents and transport in the Zhang model of self-organized criticality

We discuss the role played by Lyapunov exponents in the dynamics of Zhang's model of self-organized criticality. We show that a large part of the spectrum (the slowest modes) is associated with energy transport in the lattice. In particular, we give bounds on the first negative Lyapunov exponent in terms of the energy flux dissipated at the boundaries per unit of time. We then establish an explicit formula for the transport modes that appear as diffusion modes in a landscape where the metric is given by the density of active sites. We use a finite size scaling ansatz for the Lyapunov spectrum, and relate the scaling exponent to the scaling of quantities such as avalanche size, duration, density of active sites, etc.

Evidence for universality within the classes of deterministic and stochastic sandpile models

Recent numerical studies have provided evidence that within the family of conservative, undirected sandpile models with short range dynamic rules, deterministic models such as the Bak-Tang-Wiesenfeld model [P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59, 381 (1987)] and stochastic models such as the Manna model [S. S. Manna, J. Phys. A 24, L363 (1991)] belong to different universality classes. In this paper we examine the universality within each of the two classes in two dimensions by numerical simulations. To this end we consider additional deterministic and stochastic models and use an extended set of critical exponents, scaling functions, and geometrical features. Universal behavior is found within the class of deterministic Abelian models, as well as within the class of stochastic models (which includes both Abelian and non-Abelian models). In addition, it is observed that deterministic but non-Abelian models exhibit critical exponents that depend on a parameter, namely they are nonuniversal.

Large-scale synchrony in weakly interacting automata

We study the behavior of two spatially distributed (sandpile) models which are weakly linked with one another. Using a Monte Carlo implementation of the renormalization-group and algebraic methods, we describe how large-scale correlations emerge between the two systems, leading to synchronized behavior.

Dissipative Abelian sandpiles and random walks

We show that the dissipative Abelian sandpile on a graph L can be related to a random walk on a graph that consists of L extended with a trapping site. From this relation it can be shown, using exact results and a scaling assumption, that the correlation length exponent nu of the dissipative sandpiles always equals 1/d(w), where d(w) is the fractal dimension of the random walker. This leads to a new understanding of the known result that nu=1/2 on any Euclidean lattice. Our result is, however, more general, and as an example we also present exact data for finite Sierpinski gaskets, which fully confirm our predictions.

Exact solution of a stochastic directed sandpile model

We introduce and analytically solve a directed sandpile model with stochastic toppling rules. The model clearly belongs to a different universality class from its counterpart with deterministic toppling rules, previously solved by Dhar and Ramaswamy. The critical exponents are D(//)=7/4, tau=10/7 in two dimensions and D(//)=3/2, tau=4/3 in one dimension. The upper critical dimension of the model is three, at which the exponents apart from logarithmic corrections reach their mean-field values D(//)=2, tau=3/2.

Time-Inhomogeneous Fokker-Planck Equation for Wave Distributions in the Abelian Sandpile Model

The time and size distributions of the waves of topplings in the Abelian sandpile model are expressed as the first arrival at the origin distribution for a scale invariant, time-inhomogeneous Fokker-Plank equation. Assuming a linear conjecture for the time inhomogeneity exponent as a function of a loop-erased random walk (LERW) critical exponent, suggested by numerical results, this approach allows one to estimate the lower critical dimension of the model and the exact value of the critical exponent for LERW in three dimensions. The avalanche size distribution in two dimensions is found to be the difference between two closed power laws.