The ACO Seminar (2011-2012)

The Abelian Sandpile is a diffusion process for configurations of chips on the integer lattice, in which a vertex with at least 4 chips can "topple", distributing one of its chips to each of its 4 neighbors. This process can be shown to Abelian in the sense that if topplings are performed until no more topplings are possible, the terminal configuration depends only on the initial distribution of chips and not on the order in which we choose to perform topplings.

Though the sandpile has been the object of study from a diverse set of perspectives, even some of the most basic questions about the terminal configurations produced by the process remain unanswered. One of the most striking features of the sandpile is that when begun from a large concentration of n chips, the resulting terminal configurations seem to converge to a peculiar fractal pattern as n goes to infinity. In this talk, we will discuss a new mathematical explanation for the fractal nature of the sandpile.

Joint work with Charles Smart and Lionel Levine.