Two-dimensional Abelian Sandpile Model Research Articles

Multipoint correlators in the Abelian sandpile model

We revisit the calculation of height correlations in the two-dimensional Abelian sandpile model by taking advantage of a technique developed recently by Kenyon and Wilson. The formalism requires to equip the usual graph Laplacian, ubiquitous in the context of cycle-rooted spanning forests, with a complex connection. In the case at hand, the connection is constant and localized along a semi-infinite defect line (zipper). In the appropriate limit of a trivial connection, it allows one to count spanning forests whose components contain prescribed sites, which are of direct relevance for height correlations in the sandpile model. Using this technique, we first rederive known 1- and 2-site lattice correlators on the plane and upper half-plane, more efficiently than what has been done so far. We also compute explicitly the (new) next-to-leading order in the distances ( for 1-site on the upper half-plane, for 2-site on the plane). We extend these results by computing new correlators involving one arbitrary height and a few heights 1 on the plane and upper half-plane, for the open and closed boundary conditions. We examine our lattice results from the conformal point of view, and confirm the full consistency with the specific features currently conjectured to be present in the associated logarithmic conformal field theory.

Functional scale-free networks in the two-dimensional Abelian sandpile model.

Recently, the similarity of the functional network of the brain and the Ising model was investigated by Chialvo [Nat. Phys. 6, 744 (2010)]. This similarity supports the idea that the brain is a self-organized critical system. In this study we derive a functional network of the two-dimensional Bak-Tang-Wiesenfeld sandpile model as a self-organized critical model, and compare its characteristics with those of the functional network of the brain, obtained from functional magnetic resonance imaging.

Logarithmic two-point correlators in the Abelian sandpile model

We present the detailed calculations of the asymptotics of two-site correlation functions forheight variables in the two-dimensional Abelian sandpile model. By using combinatorialmethods for the enumeration of spanning trees, we extend the well-known result for thecorrelation of minimal heights h1 = h2 = 1 to σ1, h = P1, h − P1Ph forheight values h = 2, 3, 4. These results confirm the dominant logarithmic behaviour for large r, predicted by logarithmic conformal field theory based on field identifications obtainedpreviously. We obtain, from our lattice calculations, the explicit values for the coefficientsch anddh (the latter are new).

Scaling and complex avalanche dynamics in the Abelian sandpile model

We study the two-dimensional Abelian Sandpile Model on a square lattice of linear size L. We introduce the notion of avalanche's fine structure and compare the behavior of avalanches and waves of toppling. We show that according to the degree of complexity in the fine structure of avalanches, which is a direct consequence of the intricate superposition of the boundaries of successive waves, avalanches fall into two different categories. We propose scaling ans\"{a}tz for these avalanche types and verify them numerically. We find that while the first type of avalanches has a simple scaling behavior, the second (complex) type is characterized by an avalanche-size dependent scaling exponent. This provides a framework within which one can understand the failure of a consistent scaling behavior in this model.

The Abelian sandpile model on the honeycomb lattice

We check the universality properties of the two-dimensional Abelian sandpile model bycomputing some of its properties on the honeycomb lattice. Exact expressions for unit-heightcorrelation functions in the presence of boundaries and for different boundary conditionsare derived. Also, we study the statistics of the boundaries of avalanche waves by using thetheory of SLE and suggest that these curves are conformally invariant and described bySLE2.

Conformal covariance of the Abelian sandpile height one field

We study the scaling limit for the height one field of the two-dimensional Abelian sandpile model. The scaling limit for the covariance having height one at two macroscopically distant sites, more generally the centred height one joint moment of a finite number of macroscopically distant sites, is identified and shown to be conformally covariant. The result is based on a representation of the height one joint intensities that is close to a block-determinantal structure.

Pair correlations in sandpile model: A check of logarithmic conformal field theory

We compute the correlations of two height variables in the two-dimensional Abelian sandpile model. We extend the known result for two minimal heights to the case when one of the heights is bigger than one. We find that the most dominant correlation logr/r4 exactly fits the prediction obtained within the logarithmic conformal approach.

Height variables in the Abelian sandpile model: scaling fields and correlations

We compute the lattice one-site probabilities, on the upper half-plane, of the four heightvariables in the two-dimensional Abelian sandpile model. We find their exact scaling formwhen the insertion point is far from the boundary, and when the boundary is either open orclosed. Comparing with the predictions of a logarithmic conformal theory with central chargec = −2, we find a full compatibility with the following field assignments: the heights2, 3 and 4 behave like (an unusual realization of) the logarithmic partner of aprimary field with scaling dimension 2, the primary field itself being associated withthe height 1 variable. Finite size corrections are also computed and successfullycompared with numerical simulations. Relying on these field assignments, weformulate a conjecture for the scaling form of the lattice two-point correlationsof the height variables on the plane, which remain as yet unknown. The wayconformal invariance is realized in this system points to a local field theory withc = −2 which is different from the triplet theory.

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.

Logarithmic scaling for height variables in the Abelian sandpile model

We report on the exact computation of the scaling form of the 1-point function, on the upper-half plane, of the height 2 variable in the two-dimensional Abelian sandpile model. By comparing the open versus the closed boundary condition, we find that the scaling field associated to the height 2 is a logarithmic scalar field of scaling dimension 2, belonging to a c=−2 logarithmic conformal field theory. This identification is confirmed by numerical simulations and extended to the height 3 and 4 variables, which exhibit the same scaling form. Using the conformal setting, we make precise proposals for the bulk 2-point functions of all height variables.

Pre-logarithmic and logarithmic fields in a sandpile model

We consider the unoriented two-dimensional Abelian sandpile model onthe half-plane with open and closed boundary conditions, and relateit to the boundary logarithmic conformal field theory with central chargec = −2. Building on previous results, we first perform a complementary latticeanalysis of the operator effecting the change of boundary conditionbetween open and closed, which confirms that this operator is a weight−1/8 boundary primary field, whose fusion agrees with lattice calculations. We then consider theoperators corresponding to the unit height variable and to a mass insertion at an isolatedsite of the upper half-plane and compute their one-point functions in the presenceof a boundary containing the two kinds of boundary conditions. We show thatthe scaling limit of the mass insertion operator is a weight 0 logarithmic field.

A c=−2 boundary changing operator for the Abelian sandpile model

We consider the unoriented two-dimensional Abelian sandpile model on the half-plane with open and closed boundary conditions. We show that the operator effecting the change from closed to open, or from open to closed, is a boundary primary field of weight −1/8, belonging to a c=−2 logarithmic conformal field theory.

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.

Expansion and contraction of avalanches in the two-dimensional Abelian sandpile

We present a detailed analysis of large scale simulations of avalanches in the two-dimensional Abelian sandpile model. We compare statistical properties of two different decompositions of avalanches into clusters of topplings and waves of topplings. Auxiliary critical exponents are introduced and the existence of the exponent governing the contraction of avalanches claimed in our previous work [Priezzhev et al., Phys. Rev. Lett. 76, 2093 (1996)] is confirmed. We also give more elaborate argumentation for the exact values of the exponents characterizing the statistics of waves.

Avalanches and waves in the Abelian sandpile model

We numerically study avalanches in the two-dimensional Abelian sandpile model in terms of a sequence of waves of toppling events. Priezzhev et al. [Phys. Rev. Lett. 76, 2093 (1996)] have recently proposed exact results for the critical exponents in this model based on the existence of a proposed scaling relation for the difference in sizes of subsequent waves, $\ensuremath{\Delta}{s=s}_{k}\ensuremath{-}{s}_{k+1},$ where the size of the previous wave ${s}_{k}$ was considered to be almost always an upper bound for the size of the next wave ${s}_{k+1}.$ Here we show that the significant contribution to $\ensuremath{\Delta}s$ comes from waves that violate the bound; the average $〈\ensuremath{\Delta}{s(s}_{k})〉$ is actually negative and diverges with the system size, contradicting the proposed solution.

Boundary height correlations in a two-dimensional Abelian sandpile

Boundary height distributions of the two-dimensional Abelian sandpile model are studied in the self-organized critical state. All height probabilities are calculated explicitly both at open and closed boundaries. The leading asymptotic term of the corresponding correlation functions is observed to behave as r-4 when r to infinity . On the basis of conformal field theory predictions the bulk height correlators are shown to have the same critical exponents as boundary ones. All heights seem to be identified with appropriate counterparts of the local energy operator in the zero-component Potts model.

Structure of two-dimensional sandpile. I. Height probabilities

The height probabilities of the two-dimensional Abelian sandpile model are the fractionial numbers of lattice sites having heights 1, 2, 3, 4. A combinatorial method for evaluation of these quantities is proposed. The method is based on mapping the set of allowed sandpile configurations onto the set of spanning trees covering a given lattice. Exact analytical expressions for all probabilities are obtained.