The approach to criticality in sandpiles

TL;DR

This paper investigates the density conjecture in self-organized criticality, demonstrating through simulations and proofs that it fails on various graphs, challenging existing theoretical links between driven dissipative and fixed-energy sandpile models.

Contribution

The paper provides the first comprehensive evidence that the density conjecture does not hold on multiple graph structures, questioning its universality in sandpile models.

Findings

Density conjecture is false on Z^2, K_n, Cayley tree, ladder, bracelet, and flower graphs.

Driven dissipative sandpiles evolve even after significant sand loss.

Results challenge the use of fixed-energy sandpiles to understand criticality in abelian sandpiles.

Abstract

A popular theory of self-organized criticality relates the critical behavior of driven dissipative systems to that of systems with conservation. In particular, this theory predicts that the stationary density of the abelian sandpile model should be equal to the threshold density of the corresponding fixed-energy sandpile. This "density conjecture" has been proved for the underlying graph Z. We show (by simulation or by proof) that the density conjecture is false when the underlying graph is any of Z^2, the complete graph K_n, the Cayley tree, the ladder graph, the bracelet graph, or the flower graph. Driven dissipative sandpiles continue to evolve even after a constant fraction of the sand has been lost at the sink. These results cast doubt on the validity of using fixed-energy sandpiles to explore the critical behavior of the abelian sandpile model at stationarity.