Average height for Abelian sandpiles and the looping constant on Sierpinski graphs

TL;DR

This paper analyzes the Abelian sandpile model on Sierpinski graphs, calculating average heights, height probabilities, and the looping constant, providing new algorithms and exact values for these statistics on fractal structures.

Contribution

It introduces an algorithmic method to compute height probabilities and explicitly calculates the expected average height and looping constant on Sierpinski graphs.

Findings

Expected average height of recurrent sandpiles on Sierpinski gasket

Algorithm for calculating height probabilities

Relation between average height and looping constant

Abstract

For the Abelian sandpile model on Sierpinski graphs, we investigate several statistics such as average height, height probabilities and looping constant. In particular, we calculate the expected average height of a recurrent sandpile on the finite iterations of the Sierpinski gasket and we also give an algorithmic approach for calculating the height probabilities of recurrent sandpiles under stationarity by using the connection between recurrent configurations of the Abelian sandpile Markov chain and uniform spanning trees. We also calculate the expected fraction of vertices of height $i$ for $i\in\{0,1,2,3\}$ of sandpiles under stationarity and relate the bulk average height to the looping constant on the Sierpinski gasket.