Average height for Abelian sandpiles and the looping constant on Sierpiński graphs.

Abstract

For the Abelian sandpile model on Sierpiński 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 Sierpiński 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 Sierpiński gasket.