3 research outputs found
Explicit characterization of the identity configuration in an Abelian Sandpile Model
Since the work of Creutz, identifying the group identities for the Abelian
Sandpile Model (ASM) on a given lattice is a puzzling issue: on rectangular
portions of Z^2 complex quasi-self-similar structures arise. We study the ASM
on the square lattice, in different geometries, and a variant with directed
edges. Cylinders, through their extra symmetry, allow an easy determination of
the identity, which is a homogeneous function. The directed variant on square
geometry shows a remarkable exact structure, asymptotically self-similar.Comment: 11 pages, 8 figure
Multiple and inverse topplings in the Abelian Sandpile Model
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.Comment: 22 pages, proceedings of the SigmaPhi2011 Conferenc
Deterministic Abelian Sandpile and square-triangle tilings
The Abelian Sandpile Model, seen as a deterministic lattice automaton, on two-dimensional periodic graphs, generates complex regular patterns displaying (fractal) self-similarity. In particular, on a variety of lattices and initial conditions, at all sizes, there appears what we call an exact Sierpinski structure: the volume is filled with periodic patterns, glued together along straight lines, with the topology of a triangular Sierpinski gasket. Various lattices (square, hexagonal, kagome, \u2026), initial conditions, and toppling rules show Sierpinski structures which are apparently unrelated and involve different mechanisms. As will be shown elsewhere, all these structures fall under one roof, and are in fact different projections of a unique mechanism pertinent to a family of deterministic surfaces in a four-dimensional lattice. This short note gives a description of this surface, and of the combinatorics associated to its construction