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

Nico Heizmann, Robin Kaiser, Ecaterina Sava-Huss

TL;DR

This work analyzes the Abelian sandpile model on finite Sierpiński graphs, focusing on height statistics and their relation to loop-erased random walks. By exploiting the burning bijection and a recursive decomposition of spanning trees and forests, the authors derive algorithmic methods to compute height probabilities and the expected height under stationarity, and they establish exact limiting values for bulk quantities. A central result is the explicit looping constant $\zeta=\frac{7259}{5616}$ and the induced bulk average height $\sigma=(\zeta+3)/2\approx 2.15$, linking loop-erased paths to recurrent sandpile configurations on fractal graphs. The paper thus bridges Abelian sandpiles, uniform spanning structures, and loop-erased dynamics on the fractal Sierpiński gasket, with potential extensions to infinite graphs and avalanche statistics.

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.

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

TL;DR

This work analyzes the Abelian sandpile model on finite Sierpiński graphs, focusing on height statistics and their relation to loop-erased random walks. By exploiting the burning bijection and a recursive decomposition of spanning trees and forests, the authors derive algorithmic methods to compute height probabilities and the expected height under stationarity, and they establish exact limiting values for bulk quantities. A central result is the explicit looping constant and the induced bulk average height , linking loop-erased paths to recurrent sandpile configurations on fractal graphs. The paper thus bridges Abelian sandpiles, uniform spanning structures, and loop-erased dynamics on the fractal Sierpiński gasket, with potential extensions to infinite graphs and avalanche statistics.

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 for of sandpiles under stationarity and relate the bulk average height to the looping constant on the Sierpinski gasket.
Paper Structure (17 sections, 6 theorems, 96 equations, 7 figures, 1 table)

This paper contains 17 sections, 6 theorems, 96 equations, 7 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Abelian sandpile model
  4. Burning algorithm
  5. Sierpiński graphs
  6. Construction of the finite iterations of the Sierpiński graph.
  7. Spanning trees and forests of $\mathsf{SG}_n$.
  8. Choice of the sink.
  9. Height probabilities
  10. Probabilities at corner points
  11. Neighbours in the same component
  12. Probabilities at cut points
  13. Probabilities for arbitrary vertices
  14. Expected height
  15. Connection to the looping constant
  16. ...and 2 more sections

Key Result

Theorem 1.1

For any $n\in\mathbb{N}$ and $v\in\mathsf{SG}_n$, let where LERW is the loop erased random walk on $\mathsf{SG}_n$ stopped after hitting either the bottom right vertex $A^n_3$ or the top corner vertex $A^n_2$. Further let and denote $\zeta:= lim_{n\to\infty}\zeta_n$. We then have

Figures (7)

  • Figure 1: The graphs $\mathsf{SG}_0$, $\mathsf{SG}_1$ and $\mathsf{SG}_2$.
  • Figure 2: The burning bijection: the spanning tree with statistics $a_T, b_T$ (left) and its corresponding recurrent sandpile (right). The total ordering of $E_v$ is determined by the number of clockwise rotations by $\pi/3$ needed to align the edge with $(1,0)$, with fewer rotations indicating a lower position in the ordering.
  • Figure 3: Pictograms for one, two, and three component forests
  • Figure 4: All possible configurations for $\mathcal{T}_{n}$
  • Figure 5: All possible configurations for $\mathcal{S}_{n}^2$
  • ...and 2 more figures

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: dhar-burning
  • Lemma 2.2: heights-on-zd
  • Lemma 5.1
  • proof
  • Proposition 5.1
  • proof