New directions arising from a special diffusion process on the integer lattice

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The so-called Abelian sandpile represents a striking example of surprising mathematics arising from a simple setting. The rules of the sandpile are simple: given a configuration of "chips" on an infinite chessboard (i.e., the integer lattice), one can "topple" any square with at least 4 chips, by sending one chip from this square to each of its four neighboring squares. When begun from a single large stack of chips and continued until no square has more than 3 chips, the result is a striking fractal configuration. Studying this process has lead to the discovery of surprising connections between seemingly disparate areas of mathematics, and this project aims to leverage these new connections both for the sake of a deeper understanding of the sandpile process, and to enrich our understandings of these other areas as well. This area of mathematics has connections with phase transitions and conformal field theory in physics as well.More specifically, this project concerns the so-called Abelian sandpile model of Bak, Tang, and Wiesenfeld. By identifying new connections between integer superharmonic functions, Apollonian circle packings, and certain regular tilings of the Euclidean plane, we are now able to characterize the scaling limit of the sandpile and analyze its local fractal structure. This project has two primary goals. The first is to extend our knowledge of the sandpile: for example, we would like to strengthen the kind of convergence we can prove the sandpile process admits, characterize solutions to long-studied instances of the Dirichlet problem for the sandpile, and extend our knowledge of the scaling limit of the sandpile process beyond the universe of the square lattice. On the other hand, we would like to bring new perspectives developed for our analysis of the sandpile to bear on other areas, with the possibility of confirming number-theoretic conjectures regarding Apollonian circle packings, or characterizing tilings of the plane with certain symmetry properties.