Critical behavior of the sandpile model as a self-organized branching process

Annealed and quenched disorder in sand-pile models with local violation of conservation

In this paper we consider the Bak, Tang, and Wiesenfeld (BTW) sand-pile model with local violation of conservation through annealed and quenched disorder. We have observed that the probability distribution functions of avalanches have two distinct exponents, one of which is associated with the usual BTW model and another one which we propose to belong to a new fixed point; that is, a crossover from the original BTW fixed point to a new fixed point is observed. Through field theoretic calculations, we show that such a perturbation is relevant and takes the system to a new fixed point.

Renormalization-group approach to the Manna sandpile

In this paper a renormalization group (RG) scheme for the q -state Manna model is proposed based on the stochastic characteristic and the similarity of topplings at different scales. A full enumeration of the RG evolution events inside a 2 x 2 RG cell for q=2 , 3, and 4 was carried out. A fixed point analysis shows that the resulting height probabilities are very close to the results obtained by the numerical simulations. The calculations of the toppling number exponent tau and the dynamical exponent z are also provided. It was found that the RG values of tau and z for q=4 are very close to the simulation values.

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.

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.

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.

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.

Scaling of waves in the bak-tang-wiesenfeld sandpile model

We study probability distributions of waves of topplings in the Bak-Tang-Wiesenfeld model on hypercubic lattices for dimensions D>/=2. Waves represent relaxation processes which do not contain multiple toppling events. We investigate bulk and boundary waves by means of their correspondence to spanning trees, and by extensive numerical simulations. While the scaling behavior of avalanches is complex and usually not governed by simple scaling laws, we show that the probability distributions for waves display clear power-law asymptotic behavior in perfect agreement with the analytical predictions. Critical exponents are obtained for the distributions of radius, area, and duration of bulk and boundary waves. Relations between them and fractal dimensions of waves are derived. We confirm that the upper critical dimension D(u) of the model is 4, and calculate logarithmic corrections to the scaling behavior of waves in D=4. In addition, we present analytical estimates for bulk avalanches in dimensions D>/=4 and simulation data for avalanches in D</=3. For D=2 they seem not easy to interpret.

Modified renormalization strategy for sandpile models

Following the renormalization-group scheme recently developed by Pietronero et al. [Phys. Rev. Lett. 72, 1690 (1994)] we introduce a simplifying strategy for the renormalization of the relaxation dynamics of sandpile models. In our scheme, five subcells at a generic scale b form the renormalized cell at the next larger scale. Now the fixed point has a unique nonzero dynamical component that allows for a great simplification in the computation of the critical exponent z. The values obtained are in good agreement with both numerical and theoretical results previously reported.

Dynamical real space renormalization group applied to sandpile models

A general framework for the renormalization group analysis of self-organized critical sandpile models is formulated. The usual real space renormalization scheme for lattice models when applied to nonequilibrium dynamical models must be supplemented by feedback relations coming from the stationarity conditions. On the basis of these ideas the dynamically driven renormalization group is applied to describe the boundary and bulk critical behavior of sandpile models. A detailed description of the branching nature of sandpile avalanches is given in terms of the generating functions of the underlying branching process.