Multiple and inverse topplings in the Abelian Sandpile Model

Abstract

The Abelian Sandpile Model is a cellular automaton whose discrete dynamics reaches an out-of-equilibrium steady state resembling avalanches in piles of sand. The fundamental moves defining the dynamics are encoded by the toppling rules. The transition monoid corresponding to this dynamics in the set of stable configurations is abelian, a property which seems at the basis of our understanding of the model. By including also antitoppling rules, we introduce and investigate a larger monoid, which is not abelian anymore. We prove a number of algebraic properties of this monoid, and describe their practical implications on the emerging structures of the model.

22 pages, proceedings of the SigmaPhi2011 Conference

References in corpus (3)

• Grassmann Integral Representation for Spanning Hyperforests
• Explicit characterization of the identity configuration in an Abelian Sandpile Model
• Conservation laws for strings in the Abelian Sandpile Model

Cited by in corpus (12)

• Holographic applications of logarithmic conformal field theories
• Tropical curves in sandpiles
• Exact integration of height probabilities in the Abelian Sandpile Model
• Sandpile solitons via smoothing of superharmonic functions
• Abelian sandpile model and Biggs-Merino polynomial for directed graphs
• Sandpile solitons in higher dimensions
• Sandpiles on the heptagonal tiling
• One-skeleta of $G$-parking function ideals: resolutions and standard monomials
• A Bijection Between the Recurrent Configurations of a Hereditary Chip-Firing Model and Spanning Trees
• Compatible Recurrent Identities of the Sandpile Group and Maximal Stable Configurations
• Natural Measures on Polyominoes Induced by the Abelian Sandpile Model
• Trees, parking functions, and standard monomials of skeleton ideals