Approach to criticality in sandpiles

Area of Science:

• Complex Systems
• Statistical Physics
• Graph Theory

Background:

• Self-organized criticality (SOC) models, like the Abelian sandpile model, are widely studied for their emergent complex behaviors.
• A key prediction, the "density conjecture," posits that the stationary density in driven-dissipative SOC sandpiles equals the threshold density in fixed-energy sandpiles.
• This conjecture has been a cornerstone for understanding sandpile dynamics across various mathematical structures.

Purpose of the Study:

• To rigorously investigate and disprove the density conjecture for Abelian sandpile models on specific graph structures.
• To elucidate the relationship between driven-dissipative and fixed-energy sandpile models.
• To identify critical points governing sand loss in driven-dissipative systems.

Main Methods:

• Theoretical analysis involving rigorous mathematical proofs.
• Extensive computational simulations on various graph types (Z2, complete graphs, Cayley trees, ladder graphs, bracelet graphs, flower graphs).
• Comparison of stationary densities and threshold densities between different sandpile models.

Main Results:

• The density conjecture is demonstrated to be false for Abelian sandpiles on the tested graph classes.
• Driven-dissipative sandpiles exhibit continuous evolution and sand loss even beyond the conjectured stationary density.
• A direct relationship is proven: the fixed-energy sandpile's threshold density marks the onset of macroscopic sand loss in the driven-dissipative model.

Conclusions:

• The widely accepted density conjecture for Abelian sandpiles is invalidated on several common graph structures.
• The threshold density of fixed-energy sandpiles serves as a critical indicator for the stability and sand loss behavior of driven-dissipative sandpiles.
• This research refines our understanding of self-organized criticality and sandpile dynamics, offering new insights into system behavior.