Deterministic Abelian Sandpile Models and Patterns by Guglielmo Paoletti

By Guglielmo Paoletti

The version investigated during this paintings, a specific mobile automaton with stochastic evolution, used to be brought because the least difficult case of self-organized-criticality, that's, a dynamical procedure which exhibits algebraic long-range correlations with none tuning of parameters. the writer derives certain effects that are in all probability additionally attention-grabbing open air the realm of serious phenomena. certain capability additionally site-by-site and never in simple terms ensemble typical or coarse graining. Very advanced and amazingly appealing periodic styles are usually generated by means of the dynamics concerned, particularly in deterministic protocols during which the sand is further at selected websites. for instance, the writer reports the looks of allometric constructions, that's, styles which develop within the related approach of their entire physique, and never simply close to their obstacles, as normally happens. The neighborhood conservation legislation which govern the evolution of those styles also are awarded. This paintings has already attracted curiosity, not just in non-equilibrium statistical mechanics, but in addition in arithmetic, either in likelihood and in combinatorics. There also are fascinating connections with quantity idea. finally, it additionally poses new questions on an outdated topic. As such, will probably be of curiosity to machine practitioners, demonstrating the simplicity with which fascinating styles may be received, in addition to to researchers operating in lots of different areas.

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E. 5 Algebraic Aspects We report here some features of the abelian group G associated to the ASM. In particular we determine scalar function, invariant under toppling, and the rank of the group for the square lattice. 39) That is, the group is isomorphic to the direct product of g cyclic groups of order d1 , d2 , . . , dg . Moreover the integers d1 ≥ d2 ≥ . . ≥ dg > 1 can be chosen such that di is an integer multiple of di+1 and, under this condition, the decomposition is unique. In the following we determine the canonical decomposition of the group.

Several equivalent characterizations of recurrency exist, some of which extend naturally to S+ , and even to the full space Zn (this is also our choice). In particular, we give three definitions, all valid in Zn . Under all definitions, a configuration is transient if it is not recurrent. 1 A configuration z is recurrent by identity test if there exists a permutation σ ∈ Sn such that tσ (n) · · · tσ (2) tσ (1) (z + b+ ) = z is a valid toppling sequence. 2 A configuration z is recurrent by toppling covering if there exists a configuration u, such that z = tik · · · ti1 u is a valid toppling sequence, and at least one toppling is performed at each site.

1 The Extended Configuration Space 39 A site i is positively-unstable (or just unstable) if z i > z i . In this case, and only in this case, a toppling can be performed at i. Note that, after the toppling, it is still z i ≥ z i (more precisely, z i > z i − ii ), while z j ’s for j = i have not decreased, thus the topplings leave stable the space S+ . The relaxation operator R is the map from S+ to S, coinciding with the identity on S, that associates to a configuration z ∈ S+ the unique configuration R(z) ∈ S resulting from the application of topplings at unstable sites.

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