31 research outputs found
Introduction to the Sandpile Model
This article is based on a talk given by one of us (EVI) at the conference
``StatPhys-Taipei-1997''. It overviews the exact results in the theory of the
sandpile model and discusses shortly yet unsolved problem of calculation of
avalanche distribution exponents. The key ingredients include the analogy with
the critical reaction-diffusion system, the spanning tree representation of
height configurations and the decomposition of the avalanche process into waves
of topplings
Scaling fields in the two-dimensional abelian sandpile model
We consider the isotropic 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 a 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.Comment: 22 pages, 1 figure (eps), uses revte
Infinite volume limit of the Abelian sandpile model in dimensions d >= 3
We study the Abelian sandpile model on Z^d. In dimensions at least 3 we prove
existence of the infinite volume addition operator, almost surely with respect
to the infinite volume limit mu of the uniform measures on recurrent
configurations. We prove the existence of a Markov process with stationary
measure mu, and study ergodic properties of this process. The main techniques
we use are a connection between the statistics of waves and uniform
two-component spanning trees and results on the uniform spanning tree measure
on Z^d.Comment: First version: LaTeX; 29 pages. Revised version: LaTeX; 29 pages. The
main result of the paper has been extended to all dimensions at least 3, with
a new and simplyfied proof of finiteness of the two-component spanning tree.
Second revision: LaTeX; 32 page
Dynamically Driven Renormalization Group Applied to Sandpile Models
The 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.Comment: 18 RevTeX pages, 5 figure
Fine Structure of Avalanches 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.Comment: 10 page
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 RG approach to Bak-Tang-Wiesenfeld sandpile model [Phys. Rev. Lett.
{\bf 59}, 381 (1987)] and consider both the reduced-high RG equations proposed
by Pietronero, Vespignani, and Zapperi (PVZ) [Phys. Rev. Lett. {\bf 72}, 1690
(1994)] and the real-height RG equations proposed by Ivashkevich [Phys. Rev.
Lett. {\bf 76}, 3368 (1996)]. Using this algorithm we are able to carry out RG
transformations more quickly with large cell size, e.g. 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 {\bf 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
and the dynamical exponent . 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.Comment: 29 pages, 6 figure
N-Site approximations and CAM analysis for a stochastic sandpile
I develop n-site cluster approximations for a stochastic sandpile in one
dimension. A height restriction is imposed to limit the number of states: each
site can harbor at most two particles (height z_i \leq 2). (This yields a
considerable simplification over the unrestricted case, in which the number of
states per site is unbounded.) On the basis of results for n \leq 11 sites, I
estimate the critical particle density as zeta_c = 0.930(1), in good agreement
with simulations. A coherent anomaly analysis yields estimates for the order
parameter exponent [beta = 0.41(1)] and the relaxation time exponent (nu_||
\simeq 2.5).Comment: 12 pages, 7 figure
A probabilistic approach to Zhang's sandpile model
The current literature on sandpile models mainly deals with the abelian
sandpile model (ASM) and its variants. We treat a less known - but equally
interesting - model, namely Zhang's sandpile. This model differs in two aspects
from the ASM. First, additions are not discrete, but random amounts with a
uniform distribution on an interval . Second, if a site topples - which
happens if the amount at that site is larger than a threshold value
(which is a model parameter), then it divides its entire content in equal
amounts among its neighbors. Zhang conjectured that in the infinite volume
limit, this model tends to behave like the ASM in the sense that the stationary
measure for the system in large volumes tends to be peaked narrowly around a
finite set. This belief is supported by simulations, but so far not by
analytical investigations.
We study the stationary distribution of this model in one dimension, for
several values of and . When there is only one site, exact computations
are possible. Our main result concerns the limit as the number of sites tends
to infinity, in the one-dimensional case. We find that the stationary
distribution, in the case , indeed tends to that of the ASM (up
to a scaling factor), in agreement with Zhang's conjecture. For the case ,
we provide strong evidence that the stationary expectation tends to
.Comment: 47 pages, 3 figure
Limiting shapes for deterministic centrally seeded growth models
We study the rotor router model and two deterministic sandpile models. For
the rotor router model in , Levine and Peres proved that the
limiting shape of the growth cluster is a sphere. For the other two models,
only bounds in dimension 2 are known. A unified approach for these models with
a new parameter (the initial number of particles at each site), allows to
prove a number of new limiting shape results in any dimension .
For the rotor router model, the limiting shape is a sphere for all values of
. For one of the sandpile models, and (the maximal value), the
limiting shape is a cube. For both sandpile models, the limiting shape is a
sphere in the limit . Finally, we prove that the rotor router
shape contains a diamond.Comment: 18 pages, 3 figures, some errors corrected and more explanation
added, to appear in Journal of Statistical Physic
On the scaling behavior of the abelian sandpile model
The abelian sandpile model in two dimensions does not show the type of
critical behavior familar from equilibrium systems. Rather, the properties of
the stationary state follow from the condition that an avalanche started at a
distance r from the system boundary has a probability proportional to 1/sqrt(r)
to reach the boundary. As a consequence, the scaling behavior of the model can
be obtained from evaluating dissipative avalanches alone, allowing not only to
determine the values of all exponents, but showing also the breakdown of
finite-size scaling.Comment: 4 pages, 5 figures; the new version takes into account that the
radius distribution of avalanches cannot become steeper than a certain power
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