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Chip-Firing and Rotor-Routing on Directed Graphs

Alexander E. Holroyd, Lionel Levine, Karola Meszaros, Yuval Peres, James Propp, David B. Wilson

TL;DR

This paper surveys the abelian sandpile (chip-firing) and rotor-router models on finite directed graphs, clarifying their connections through the sandpile group $\mathcal{S}(G)$ and recurrent configurations, and highlighting the abelian property that underpins both dynamics. It develops a unified framework linking chip-firing, rotor-routing, Eulerian structures, and cycle-popping, with central results including the Matrix-Tree Theorem and a natural isomorphism between the sandpile group and the rotor-router group acting on oriented spanning trees. It provides specialized results for Eulerian digraphs, including burning and superstability criteria, and shows how rotor routing yields Eulerian tours and bijections to spanning trees. It also outlines several open conjectures and directions, from asymptotic shape questions in aggregation models to cycle-period properties in non-Eulerian graphs.

Abstract

We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.

Chip-Firing and Rotor-Routing on Directed Graphs

TL;DR

This paper surveys the abelian sandpile (chip-firing) and rotor-router models on finite directed graphs, clarifying their connections through the sandpile group and recurrent configurations, and highlighting the abelian property that underpins both dynamics. It develops a unified framework linking chip-firing, rotor-routing, Eulerian structures, and cycle-popping, with central results including the Matrix-Tree Theorem and a natural isomorphism between the sandpile group and the rotor-router group acting on oriented spanning trees. It provides specialized results for Eulerian digraphs, including burning and superstability criteria, and shows how rotor routing yields Eulerian tours and bijections to spanning trees. It also outlines several open conjectures and directions, from asymptotic shape questions in aggregation models to cycle-period properties in non-Eulerian graphs.

Abstract

We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.

Paper Structure

This paper contains 6 sections, 44 theorems, 39 equations, 11 figures.

Table of Contents

  1. Introduction
  2. Chip-Firing
  3. Rotor-Routing
  4. Eulerian Graphs
  5. Stacks and Cycle-Popping
  6. Conjectures and Open Problems

Key Result

Lemma 2.2

Let $G$ be any digraph, let $\sigma_0$, $\sigma_1, \ldots, \sigma_n$ be a sequence of chip configurations on $G$, each of which is a successor of the one before, and let $\sigma'_0, \sigma'_1, \ldots, \sigma'_m$ be another such sequence with $\sigma'_0=\sigma_0$.

Figures (11)

  • Figure 1: Some chip configurations eventually stabilize, while others never stabilize.
  • Figure 2: Commutation of the chip-firing operations.
  • Figure 3: Two stable chip configurations in the equivalence class of the identity.
  • Figure 4: The identity element of the sandpile group of the $L\times L$ square grid for different values of $L$, namely $L=128$ (upper left), $198$ (upper right), $243$ (lower left), and $521$ (lower right). The color scheme is as follows: orange=$0$ chips, red=$1$ chip, green=$2$ chips, and blue=$3$ chips.
  • Figure 5: The identity element of the sandpile group of the $100\times100$ directed torus (left) and the $500\times500$ directed torus (right). The color scheme is as follows: white=$0$ chips, black=$1$ chip, and the sink, which is at the lower-left corner, is shown in red.
  • ...and 6 more figures

Theorems & Definitions (92)

  • Example 2.1
  • Lemma 2.2: D,DF
  • proof
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • Definition 2.7
  • ...and 82 more