13 research outputs found
Driving sandpiles to criticality and beyond
A popular theory of self-organized criticality relates driven dissipative
systems to systems with conservation. This theory predicts that the stationary
density of the abelian sandpile model equals the threshold density of the
fixed-energy sandpile. We refute this prediction for a wide variety of
underlying graphs, including the square grid. Driven dissipative sandpiles
continue to evolve even after reaching criticality. This result casts doubt on
the validity of using fixed-energy sandpiles to explore the critical behavior
of the abelian sandpile model at stationarity.Comment: v4 adds referenc
Growth Rates and Explosions in Sandpiles
We study the abelian sandpile growth model, where n particles are added at
the origin on a stable background configuration in Z^d. Any site with at least
2d particles then topples by sending one particle to each neighbor. We find
that with constant background height h <= 2d-2, the diameter of the set of
sites that topple has order n^{1/d}. This was previously known only for h<d.
Our proof uses a strong form of the least action principle for sandpiles, and a
novel method of background modification.
We can extend this diameter bound to certain backgrounds in which an
arbitrarily high fraction of sites have height 2d-1. On the other hand, we show
that if the background height 2d-2 is augmented by 1 at an arbitrarily small
fraction of sites chosen independently at random, then adding finitely many
particles creates an explosion (a sandpile that never stabilizes).Comment: 19 pages, 4 figures, to appear in Journal of Statistical Physics. v2
corrects the proof of the outer bound of Theorem 4.1 of arXiv:0704.068
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
Limiting shapes for deterministic internal growth models
We study the rotor router model and two deterministic sandpile models. For the rotor router model in Zd, 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 h (the initial number of particles at each site), allows to prove a number of new limiting shape results in any dimension d ≥ 1. For the rotor router model, the limiting shape is a sphere for all values of h. For one of the sandpile models, and h = 2d − 2 (the maximal value), the limiting shape is a cube. For both sandpile models, the limiting shape is a sphere in the limit h → −∞. Finally, we prove that the rotor router shape contains a diamond, which is a new result even in the case studied by Levine and Peres.