Sep 29 , 3:30pm, Wean 8220

Wes Pegden, New York University

The fractal nature of the Abelian Sandpile

Wes Pegden, New York University

The fractal nature of the Abelian Sandpile

Abstract:

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.