The approach to criticality in 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.

30 pages, 8 figures, long version of arXiv:0912.3206

References in corpus (5)

• Confirming and extending the hypothesis of universality in sandpiles
• Driving sandpiles to criticality and beyond
• Stabilizability and percolation in the infinite volume sandpile model
• Self-organized Criticality and Absorbing States: Lessons from the Ising Model
• Diffusion in stochastic sandpiles

Cited by in corpus (10)

• On the critical nature of plastic flow: one and two dimensional models
• The Oslo model, hyperuniformity, and the quenched Edwards-Wilkinson model
• Self-organized quantization and oscillations on continuous fixed-energy sandpiles
• Multiple and inverse topplings in the Abelian Sandpile Model
• Numerical Study of the Correspondence Between the Dissipative and Fixed Energy Abelian Sandpile Models
• Exact integration of height probabilities in the Abelian Sandpile Model
• Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle
• The looping constant of Z^d
• The hockey-stick conjecture for activated random walk
• Sandpiles and Dominos