A sandpile model for proportionate growth

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.Comment: 18 pages, 14 figures, to appear in a special issue of JSTAT dedicated to Statphys2

A growth model based on the arithmetic ZZ-game

We present an evolutionary self-governing model based on the numerical atomic rule $Z(a,b)=ab/\gcd(a,b)^2$, for $a,b$ positive integers. Starting with a sequence of numbers, the initial generation $Gin$, a new sequence is obtained by applying the $Z$-rule to any neighbor terms. Likewise, applying repeatedly the same procedure to the newest generation, an entire matrix $T_{Gin}$ is generated. Most often, this matrix, which is the recorder of the whole process, shows a fractal aspect and has intriguing properties. If $Gin$ is the sequence of positive integers, in the associated matrix remarkable are the distinguished geometrical figures called the $Z$-solitons and the sinuous evolution of the size of numbers on the western edge. We observe that $T_{\mathbb{N}^*}$ is close to the analogue free of solitons matrix generated from an initial generation in which each natural number is replaced by its largest divisor that is a product of distinct primes. We describe the shape and the properties of this new matrix. N. J. A. Sloane raised a few interesting problems regarding the western edge of the matrix $T_{\mathbb{N}^*}$. We solve one of them and present arguments for a precise conjecture on another.Comment: 20 pages, 9 figure

Discrete Balayage and Boundary Sandpile

We introduce a new lattice growth model, which we call boundary sandpile. The model amounts to potential-theoretic redistribution of a given initial mass on $\mathbb{Z}^d$ ($d\geq 2$) onto the boundary of an (a priori) unknown domain. The latter evolves through sandpile dynamics, and has the property that the mass on the boundary is forced to stay below a prescribed threshold. Since finding the domain is part of the problem, the redistribution process is a discrete model of a free boundary problem, whose continuum limit is yet to be understood. We prove general results concerning our model. These include canonical representation of the model in terms of the smallest super-solution among a certain class of functions, uniform Lipschitz regularity of the scaled odometer function, and hence the convergence of a subsequence of the odometer and the visited sites, discrete symmetry properties, as well as directional monotonicity of the odometer function. The latter (in part) implies the Lipschitz regularity of the free boundary of the sandpile. As a direct application of some of the methods developed in this paper, combined with earlier results on classical Abelian sandpile, we show that the boundary of the scaling limit of Abelian sandpile is locally a Lipschitz graph.Comment: 34 pages, 3 figures. Version to appear in Journal d'Analyse Mathematiqu