Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Random rotor walks and i.i.d. sandpiles on Sierpinski graphs

Robin Kaiser, Ecaterina Sava-Huss

Abstract

We prove that, on the infinite Sierpinski gasket graph SG, rotor walk with random initial configuration of rotors is recurrent. We also give a necessary condition for an i.i.d. sandpile to stabilize. In particular, we prove that an i.i.d. sandpile with expected number of chips per site greater or equal to three does not stabilize almost surely. Furthermore, the proof also applies to divisible sandpiles and shows that divisible sandpile at critical density one does not stabilize almost surely on SG.

Random rotor walks and i.i.d. sandpiles on Sierpinski graphs

Abstract

We prove that, on the infinite Sierpinski gasket graph SG, rotor walk with random initial configuration of rotors is recurrent. We also give a necessary condition for an i.i.d. sandpile to stabilize. In particular, we prove that an i.i.d. sandpile with expected number of chips per site greater or equal to three does not stabilize almost surely. Furthermore, the proof also applies to divisible sandpiles and shows that divisible sandpile at critical density one does not stabilize almost surely on SG.
Paper Structure (4 sections, 7 theorems, 15 equations, 4 figures)

This paper contains 4 sections, 7 theorems, 15 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notation
  3. Uniform rotor walks on the Sierpiński gasket
  4. Stabilization of i.i.d. sandpiles

Key Result

Theorem 1.1

The uniform rotor walk $(R_t)_{t\in\mathbb{N}}$ on the doubly-infinite Sierpiński gasket graph $\mathsf{SG}$, with anticlockwise ordering of the neighbours of each vertex, is recurrent.

Figures (4)

  • Figure 1: Doubly-infinite Sierpinski gasket graph $\mathsf{SG}$.
  • Figure 2: The sets $SG_2$ and $S_2=SG_2\setminus\{x_2,y_2,z_2,t_2\}$ together with the rotors at the outer boundary of $S_2$ such that $S_2$ has reflecting boundary.
  • Figure 3: The sets $SG_3$ and $S_3=SG_3\setminus\{x_3,y_3,z_3,t_3\}$ together with the rotors at the outer boundary of $S_3$ such that $S_3$ has reflecting boundary.
  • Figure 4: An illustration of $\mathsf{SG}_{2,4}$. The picture shows the graph $\mathsf{SG}_4^+$ with its three corner vertices $o,x_4$ and $y_4$. The part colored in red shows $\mathsf{SG}_{2,4}$ with its six corner vertices $\{x_2,y_2,a_2,b_2,c_2,d_2\}$.

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Definition 3.1
  • proof : Proof of Theorem \ref{['thm:rec']}
  • Proposition 3.2
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • ...and 5 more