Pattern formation in growing sandpiles

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

ABSTRACT Adding grains at a single site on a flat substrate in the Abelian sandpile models produce beautiful complex patterns. We study in detail the pattern produced by adding grains on a two-dimensional square lattice with directed edges (each site has two arrows directed inward and two outward), starting with a periodic background with half the sites occupied. The size of the pattern formed scales with the number of grains added $N$ as $\sqrt{N}$. We give exact characterization of the asymptotic pattern, in terms of the position and shape of different features of the pattern. Comment: 5 pages, 4 figures, submitted to Phys. Rev. Lett

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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
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    ABSTRACT: We state a conjecture relating integer-valued superharmonic functions on $\mathbb{Z}^2$ to an Apollonian circle packing of $\mathbb{R}^2$. The conjecture is motivated by the Abelian sandpile process, which evolves configurations of chips on the integer lattice by toppling any vertex with at least 4 chips, distributing one of its chips to each of its 4 neighbors. When begun from a large stack of chips, the terminal state of the sandpile has a curious fractal structure which has remained unexplained. Our conjecture implies that the Sandpile PDE recently shown to characterize the continuum limit of the sandpile is equivalent to the Apollonian PDE, and we use the special geometric structure of the latter to prove that it admits certain fractal solutions. Boundary condition evidence from finite sandpiles suggest that these solutions exactly correspond to regions of the limiting sandpile, leading to precise geometric conjectures on the Abelian sandpile's fractal behavior.
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    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 $N$.

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