# Growth Rates and Explosions in Sandpiles

@article{Fey2009GrowthRA, title={Growth Rates and Explosions in Sandpiles}, author={Anne Fey and Lionel Levine and Yuval Peres}, journal={Journal of Statistical Physics}, year={2009}, volume={138}, pages={143-159} }

We study the abelian sandpile growth model, where n particles are added at the origin on a stable background configuration in ℤd. 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 n1/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…

## 77 Citations

Pattern formation in fast-growing sandpiles.

- Computer SciencePhysical review. E, Statistical, nonlinear, and soft matter physics
- 2012

This paper describes the unexpected finding of an interesting class of backgrounds in two dimensions that show an intermediate behavior: for any N, the avalanches are finite, but the diameter of the pattern increases as N(α), for large N, with 1/2<α≤1.

Tropical curves in sandpile models

- Mathematics
- 2015

A sandpile is a cellular automata on a subgraph $\Omega_h$ of ${h}\mathbb Z^2$ which evolves by the toppling rule: if the number of grains at a vertex is at least four, then it sends one grain to…

Convergence of the Abelian sandpile

- Mathematics
- 2011

The Abelian sandpile growth model is a diffusion process for configurations of chips placed on vertices of the integer lattice $\mathbb{Z}^d$, in which sites with at least 2d chips {\em topple},…

Laplacian growth and sandpiles on the Sierpiński gasket: limit shape universality and exact solutions

- Mathematics
- 2018

We establish quantitative spherical shape theorems for rotor-router aggregation and abelian sandpile growth on the graphical Sierpinski gasket ($SG$) when particles are launched from the corner…

Discrete Balayage and Boundary Sandpile

- MathematicsJournal d'Analyse Mathématique
- 2019

We introduce a new lattice growth model, which we call the boundary sandpile. The model amounts to potential-theoretic redistribution of a given initial mass on ℤd (d ≥ 2) onto the boundary of an (a…

The effect of noise on patterns formed by growing sandpiles

- Mathematics
- 2011

We consider patterns generated by adding large numbers of sand grains at a single site in an Abelian sandpile model with a periodic initial configuration, and relaxing. The patterns show…

Approach to criticality in sandpiles.

- MathematicsPhysical review. E, Statistical, nonlinear, and soft matter physics
- 2010

It is shown that driven-dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink, and is proved that the density conjecture is false when the underlying graph is any of Z2, the complete graph K(n), the Cayley tree, the ladder graph, the bracelet graph, or the flower graph.

On the emergence of regularities on one-dimensional decreasing sandpiles

- MathematicsTheor. Comput. Sci.
- 2020

Apollonian structure in the Abelian sandpile

- Geology
- 2012

The Abelian sandpile process evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun…

Multiple and inverse topplings in the Abelian Sandpile Model

- Mathematics
- 2012

The Abelian Sandpile Model is a cellular automaton whose discrete dynamics reaches an out-of-equilibrium steady state resembling avalanches in piles of sand. The fundamental moves defining the…

## References

SHOWING 1-10 OF 25 REFERENCES

Stabilizability and percolation in the infinite volume sandpile model

- Mathematics
- 2007

We study the sandpile model in infinite volume on Z d . In particular, we are interested in the question whether or not initial configurations, chosen according to a stationary measure μ, are…

Limiting Shapes for Deterministic Centrally Seeded Growth Models

- Mathematics
- 2007

Abstract
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.…

Strong Spherical Asymptotics for Rotor-Router Aggregation and the Divisible Sandpile

- Mathematics
- 2008

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We prove that the asymptotic shape of this model is…

Organized versus self-organized criticality in the abelian sandpile model

- Mathematics
- 2004

It is shown that for high enough densities, a probability measure cannot be stabilized and that in some sense the thermodynamic limit of the uniform measures on the recurrent configurations of the abelian sandpile model is a maximal element of the set of stabilizable measures.

Scaling limits for internal aggregation models with multiple sources

- Mathematics
- 2010

We study the scaling limits of three different aggregation models on ℤd: internal DLA, in which particles perform random walks until reaching an unoccupied site; the rotor-router model, in which…

Mathematical aspects of the abelian sandpile model

- Physics
- 2005

In 1988, Bak, Tang and Wiesenfeld (BTW) introduced a lattice model of what they called “self-organized criticality”. Since its appearance, this model has been studied intensively, both in the physics…

Studying Self-Organized Criticality with Exactly Solved Models

- Physics
- 1999

These lecture-notes are intended to provide a pedagogical introduction to the abelian sandpile model of self-organized criticality, and its related models. The abelian group structure of the algebra…

Self-organized critical state of sandpile automaton models.

- Computer Science, PhysicsPhysical review letters
- 1990

The critical state is characterized, and its entropy for an arbitrary finite lattice in any dimension is determined, and the two-point correlation function is shown to satisfy a linear equation.

The Abelian sandpile : a mathematical introduction

- Mathematics
- 2001

We give a simple rigourous treatment of the classical results of the abelian sandpile model. Although we treat results which are well-known in the physics literature, in many cases we did not find…