The Abelian Sandpile Model on Fractal Graphs

We study the Abelian sandpile model (ASM), a process where grains of sand are placed on a graph’s vertices. When the number of grains on a vertex is at least its degree, one grain is distributed to each neighboring vertex. This model has been shown to form fractal patterns on the integer lattice, and using these fractal patterns as motivation, we consider the model on graph approximations of post critically finite (p.c.f) fractals. We determine asymptotic behavior of the diameter of sites toppled and characterize graphs which exhibit a periodic number of grains with respect to the initial placement.