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On the structure of the sandpile identity element on Sierpinski gasket graphs

Robin Kaiser, Ecaterina Sava-Huss, Julia Überbacher

Abstract

We consider the identity of the abelian sandpile group of finite approximation graphs of the Sierpinski gasket, and we show that the second-order term in the scaling limit converges to the path distance to the nearest corner on the Sierpinski gasket. The proof relies on a decomposition of the identity of the sandpile group into the sum of a constant function and the Laplacian of the graph distance on the approximating graphs.

On the structure of the sandpile identity element on Sierpinski gasket graphs

Abstract

We consider the identity of the abelian sandpile group of finite approximation graphs of the Sierpinski gasket, and we show that the second-order term in the scaling limit converges to the path distance to the nearest corner on the Sierpinski gasket. The proof relies on a decomposition of the identity of the sandpile group into the sum of a constant function and the Laplacian of the graph distance on the approximating graphs.
Paper Structure (6 sections, 8 theorems, 53 equations, 5 figures)

This paper contains 6 sections, 8 theorems, 53 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Potential Theory on the Sierpiński Gasket
  4. Sandpiles on the Sierpiński gasket graphs
  5. Decomposition of the sandpile identity
  6. Scaling limit of the sandpile identity

Key Result

Theorem 1.1

Let $\mathsf{id}_n$ be the identity of the sandpile group of the $n$-th approximation graph $SG_n$ of the Sierpiński gasket $SG$, for $n\in \mathbb{N}$. Denote by $g_n$ the Green function of the simple random walk on $SG_n$ stopped at the corners, and let $G$ be the Green function on the Sierpiński

Figures (5)

  • Figure 1: The first three Sierpiński gasket approximation graphs $SG_0$, $SG_1$ and $SG_2$.
  • Figure 2: The Sierpiński gasket graphs $SG_0$, $SG_1$ and $SG_2$ with normal boundary conditions.
  • Figure 3: The identity element $\mathsf{id}_n$ on $SG_n$ for $n \in \{2,3,4,5\}$. Blue dots are vertices with 3 chips and red dots are vertices with 2 chips.
  • Figure 4: Illustration of the cutpoints $p_1,p_2,p_3$ and the subtriangles $\triangle_n^i$ for $i\in\{1,2,3\}$ in $SG_n$.
  • Figure 5: The values of the graph distance function in the subtriangle $\bigtriangleup_{n+1}^2$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Lemma 2.1
  • Proposition 3.1
  • proof
  • Proposition 4.1
  • proof
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 5 more