Article

# Pattern formation in growing sandpiles

EPL (Europhysics Letters) (Impact Factor: 2.26). 08/2008; DOI: 10.1209/0295-5075/85/48002

Source: arXiv

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**ABSTRACT:**An interesting feature of growth in animals is that different parts of the body grow at approximately the same rate. This property is called proportionate growth. In this paper, we review our recent work on patterns formed by adding $N$ grains at a single site in the abelian sandpile model. These simple models show very intricate patterns, show proportionate growth, and sometimes having a striking resemblance to natural forms. We give several examples of such patterns. We discuss the special cases where the asymptotic pattern can be determined exactly. The effect of noise in the background or in the rules on the patterns is also discussed.Journal of Statistical Mechanics Theory and Experiment 10/2013; · 1.87 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In Dhar's model of abelian distributed processors, finite automata occupy the vertices of a graph and communicate via the edges. A local commutativity condition ensures that the final output does not depend on the order in which the automata process their inputs. We consider the halting problem for such networks and the critical group, an invariant that governs the behavior of the network on large inputs. Our main results are 1. A finite abelian network halts on all inputs if and only if all eigenvalues of its production matrix lie in the open unit disk; 2. The critical group of an irreducible abelian network acts freely and transitively on recurrent states of the network; 3. The critical group is a quotient of a free abelian group by a subgroup containing the image of the Laplacian, with equality in the case that the network is rectangular. We also estimate the running time of an abelian network up to a constant additive error.09/2013; - [Show abstract] [Hide abstract]

**ABSTRACT:**We study the growing patterns in the rotor-router model formed by adding $N$ walkers at the center of a $L \times L$ two-dimensional square lattice, starting with a periodic background of arrows, and relaxing to a stable configuration. The pattern is made of large number of triangular regions, where in each region all arrows point in the same direction. The square circumscribing the region, where all the arrows have been rotated atleast one full circle, may be considered as made up of smaller squares of different sizes, all of which grow linearly with $N$, for $ 1 \ll N < 2 L$. We use the Brooks-Smith-Stone-Tutte theorem relating tilings of squares by smaller squares to resistor networks, to determine the exact relative sizes of the different elements of the asymptotic pattern. We also determine the scaling limit of the function describing the variation of number of visits to a site with its position in the pattern. We also present evidence that deviations of the sizes of different small squares from the linear growth for finite $N$ are bounded and quasiperiodic functions of $N$.12/2013;

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