6 research outputs found
A note on the abelian sandpile in Z^d
We analyse the abelian sandpile model on \mathbbm{Z}^d for the starting
configuration of particles in the origin and particles otherwise. We
give a new short proof of the theorem of Fey, Levine and Peres \cite{FLP} that
the radius of the toppled cluster of this configuration is
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