Pattern formation in growing sandpiles

EPL (Europhysics Letters) (Impact Factor: 2.1). 08/2008; 85(4). DOI: 10.1209/0295-5075/85/48002
Source: arXiv


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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Available from: Deepak Dhar, Feb 04, 2013
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    • "Let us briefly remark on the consequences of this corollary for our understanding of the limit sandpile. As observed in [8] [16] and visible in Figure 3, the sandpile s n for large n features many clearly visible patches, each with its own characteristic periodic pattern of sand (sometimes punctuated by one-dimensional 'defects' which are not relevant to the weak-* limit of the sandpile). Empirically, we observe that triples of touching regions of these kinds are always regions where the observed finite ¯ v n correspond (away from the one-dimensional defects) exactly to minimal representatives in the sense of (1.1) of quadratic forms 1 2 x t Ax + bx where the A's for each region are always as required by Corollary 1.4. "
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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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    • "The abelian sandpile model in Z 2 produces beautiful examples of pattern formation that remain far from understood [8] [12]. Using Lemma 2.3 of [12], it should be possible to characterize the sandpile odometer function in a manner similar to Theorem 1. "
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    ABSTRACT: We give an algorithm that computes the final state of certain growth models without computing all intermediate states. Our technique is based on a "least action principle" which characterizes the odometer function of the growth process. Starting from an approximation for the odometer, we successively correct under- and overestimates and provably arrive at the correct final state. Internal diffusion-limited aggregation (IDLA) is one of the models amenable to our technique. The boundary fluctuations in IDLA were recently proved to be at most logarithmic in the size of the growth cluster, but the constant in front of the logarithm is still not known. As an application of our method, we calculate the size of fluctuations over two orders of magnitude beyond previous simulations, and use the results to estimate this constant.
    Random Structures and Algorithms 06/2010; 42(2). DOI:10.1007/978-3-642-22935-0_47 · 0.92 Impact Factor
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