The Abelian Sandpile Model on Fractal Graphs

TL;DR

This paper investigates the Abelian sandpile model on fractal graphs, analyzing its asymptotic behavior and periodicity, extending understanding from lattice patterns to complex fractal structures.

Contribution

It extends the Abelian sandpile model analysis to fractal graphs, characterizing toppling behavior and periodicity on p.c.f fractals, which was not previously explored.

Findings

Asymptotic behavior of toppled site diameter determined

Graphs with periodic grain configurations characterized

Fractal patterns influence the sandpile dynamics

Abstract

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.