Conservation laws for strings in the Abelian Sandpile Model

Abstract

The Abelian Sandpile generates complex and beautiful patterns and seems to display allometry. On the plane, beyond patches, patterns periodic in both dimensions, we remark the presence of structures periodic in one dimension, that we call strings. We classify completely their constituents in terms of their principal periodic vector k, that we call momentum. We derive a simple relation between the momentum of a string and its density of particles, E, which is reminiscent of a dispersion relation, E=k^2. Strings interact: they can merge and split and within these processes momentum is conserved. We reveal the role of the modular group SL(2,Z) behind these laws.

4 pages, 4 figures in color

References in corpus (6)

• The abelian sandpile and related models
• Chip-Firing and Rotor-Routing on Directed Graphs
• Growth Rates and Explosions in Sandpiles
• Triangular Trimers on the Triangular Lattice: an Exact Solution
• Pattern formation in growing sandpiles
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Cited by in corpus (8)

• Self-Organized Criticality and Pattern Emergence through the lens of Tropical Geometry
• Multiple and inverse topplings in the Abelian Sandpile Model
• Harmonic dynamics of the Abelian sandpile
• Tropical curves in sandpiles
• Pattern formation in fast-growing sandpiles
• Sandpile solitons via smoothing of superharmonic functions
• Sandpile solitons in higher dimensions
• Deterministic Abelian Sandpile and square-triangle tilings