Deterministic Abelian Sandpile Models and Patterns

In this thesis we want to study the ASM in connection with its capability to produce interesting patterns. it is a surprising example of model that shows the emergence of patterns but maintains the property of being analytically tractable. Then it is qualitatively different from other typical growth models --like Eden model, the diffusion limit aggregation, or the surface deposition -- indeed while in these models the growth of the patterns is confined on the surfaces and the inner structures, once formed, are frozen and do not evolve anymore, in the ASM the patterns formed grow in size but at the same time the internal structures aquire structure, as it has been noted in several papers. There have been several earlier studies of the spatial patterns in sandpile models. The first of them was by Liu et.al. The asymptotic shape of the boundaries of the patterns produced in centrally seeded sandpile model on different periodic backgrounds was discussed in a work of Dhar of 1999. Borgne et.al. obtained bounds on the rate of growth of these boundaries, and later these bounds were improved by Fey et.al. and Levine et.al. An analysis of different periodic structures found in the patterns were first carried out by Ostojic who also first noted the exact quadratic nature of the toppling function within a patch. Wilson et.al. have developed a very efficient algorithm to generate patterns for a large numbers of particles added, which allows them to generate pictures of patterns with N up to 2^26. There are other models, which are related to the Abelian Sandpile Model,e.g., the Internal Diffusion-Limited Aggregation (IDLA), Eulerian walkers (also called the rotor-router model), and the infinitely-divisible sandpile, which also show similar structure. For the IDLA, Gravner and Quastel showed that the asymptotic shape of the growth pattern is related to the classical Stefan problem in hydrodynamics, and determined the exact radius of the pattern with a single point source. Levine and Peres have studied patterns with multiple sources in these models, and proved the existence of a limit shape. Limiting shapes for the non-Abelian sandpile has recently been studied by Fey et.al. The results of our investigation toward a comprehension of the patterns emerging in the ASM are reported along the thesis. In chapter 3 we will introduce some new algebraic operators, $a^\dagger_i$ and $\Pi_i$ in addition to $a_i$, over the space of the sandpile configurations, that will be in the following basic ingredients in the creation of patterns in the sandpile. We derive some Temperley-Lieb like relations they satisfy. At the end of the chapter we show how do they are closely related to multitopplings and which consequences has that relation on the action of $\Pi_i$ on recurrent configurations. In chapter 4 we search for a closed formula to characterize the Identity configuration of the ASM. At this scope we study the ASM on the square lattice, in different geometries, and in a variant with directed edges, the F-lattice or pseudo-Manhattan lattice. Cylinders, through their extra symmetry, allow an easy characterization of the identity which is a homogeneous function. In the directed version, the pseudo-Manhattan lattice, we see a remarkable exact self-similar structure at different sizes, which results in the possibility to give a closed formula for the identity, this work has been published. In chapter 5 we reach the cardinal point of our study, here we present the theory of strings and patches. The regions of a configuration periodic in space, called patches, are the ingredients of pattern formation. In a last paper of Dhar, a condition on the shape of patch interfaces has been established, and proven at a coarse-grained level. We discuss how this result is strengthened by avoiding the coarsening, and describe the emerging fine-level structures, including linear interfaces and rigid domain walls with a residual one-dimensional translational invariance. These structures, that we shall call strings, are macroscopically extended in their periodic direction, while showing thickness in a full range of scales between the microscopic lattice spacing and the macroscopic volume size. We first explore the relations among these objects and then we present full classification of them, which leads to the construction and explanation of a Sierpinski triangular structure, which displays patterns of all the possible patches.