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Chip-firing on the Lattice of Nonnegative Integer Points

Ryota Inagaki, Tanya Khovanova, Austin Luo

TL;DR

This work analyzes chip-firing on the quadrant lattice, the Hasse diagram of $\mathbb{Z}_{\ge 0}^2$, with $2^n$ chips at the origin. The authors introduce the intermediate firing configuration $F(x,y)$ and prove that the stable configuration’s nonzero entries coincide with odd $F$-entries, decomposing $F$ into a top triangular region, a middle expanse, and a bottom triangular tail. They establish a Pascal-triangle scaling for the top rows, prove symmetry and parity constraints, and investigate the middle and bottom regions, including row-length behavior and the structure of the minimal bottom rows, supported by extensive computational data and the analysis of difference tables $F'$. The results illuminate the global shape and internal structure of chip-firing on a non-tree poset, revealing Gaussian-like profiles in the middle and a rigid, minimal bottom boundary, with conjectures about bottom-triangle length and row counts guiding future work. This advances understanding of Abelian-like chip-firing beyond trees and provides a framework for analyzing similar processes on posets with reconvergent flows.

Abstract

Chip-firing on a directed graph is a game in which chips, a discrete commodity, are placed on the vertices of the graph and are transferred between vertices. In this paper, we study a chip-firing game on the Hasse diagram of the lattice of nonnegative integer points on the plane, where we start with $2^n$ chips at the origin. When we fire a vertex $v$, we send one chip to each out-neighbor. We fire until we reach a stable configuration, a distribution of chips where no vertex can fire. We study the intermediate firing configuration: a table that assigns to each vertex the total number of chips that pass through it. We prove that the nonzero entries of the stable configuration correspond to the odd entries of the intermediate configuration. The intermediate configuration consists of three parts: the top triangle, the midsection, and the bottom triangle. We describe properties of each part. We study properties of each row and the number of rows of the intermediate configuration. We also explore properties of the difference tables, which are tables of first differences of each row of the intermediate firing configuration.

Chip-firing on the Lattice of Nonnegative Integer Points

TL;DR

This work analyzes chip-firing on the quadrant lattice, the Hasse diagram of , with chips at the origin. The authors introduce the intermediate firing configuration and prove that the stable configuration’s nonzero entries coincide with odd -entries, decomposing into a top triangular region, a middle expanse, and a bottom triangular tail. They establish a Pascal-triangle scaling for the top rows, prove symmetry and parity constraints, and investigate the middle and bottom regions, including row-length behavior and the structure of the minimal bottom rows, supported by extensive computational data and the analysis of difference tables . The results illuminate the global shape and internal structure of chip-firing on a non-tree poset, revealing Gaussian-like profiles in the middle and a rigid, minimal bottom boundary, with conjectures about bottom-triangle length and row counts guiding future work. This advances understanding of Abelian-like chip-firing beyond trees and provides a framework for analyzing similar processes on posets with reconvergent flows.

Abstract

Chip-firing on a directed graph is a game in which chips, a discrete commodity, are placed on the vertices of the graph and are transferred between vertices. In this paper, we study a chip-firing game on the Hasse diagram of the lattice of nonnegative integer points on the plane, where we start with chips at the origin. When we fire a vertex , we send one chip to each out-neighbor. We fire until we reach a stable configuration, a distribution of chips where no vertex can fire. We study the intermediate firing configuration: a table that assigns to each vertex the total number of chips that pass through it. We prove that the nonzero entries of the stable configuration correspond to the odd entries of the intermediate configuration. The intermediate configuration consists of three parts: the top triangle, the midsection, and the bottom triangle. We describe properties of each part. We study properties of each row and the number of rows of the intermediate configuration. We also explore properties of the difference tables, which are tables of first differences of each row of the intermediate firing configuration.
Paper Structure (18 sections, 21 theorems, 46 equations, 10 figures)

This paper contains 18 sections, 21 theorems, 46 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unlabeled Chip-Firing
  4. Our Setup
  5. The Total Number of Firings
  6. The Intermediate Firing Configuration
  7. General facts
  8. Symmetry
  9. A Computed Example and Observations
  10. Pascal's Triangle and the Intermediate Firing Configuration
  11. The Middle of the Intermediate Firing Configuration
  12. General Structure of Rows
  13. Row length
  14. The number of nonzero rows of the intermediate firing configuration
  15. The Bottom Triangle
  16. ...and 3 more sections

Key Result

Theorem 2.1

For a directed graph $G$ and initial configuration $\mathcal{C}$ of chips on the graph, the unlabeled chip-firing game will either run forever or end after the same number of moves and at the same stable configuration. Furthermore, the number of times each vertex fires is the same regardless of the

Figures (10)

  • Figure 1: The quadrant lattice graph with labeled vertices.
  • Figure 2: Example of unlabeled chip-firing on an infinite directed quadrant lattice
  • Figure 3: Distance distribution for $n=15$.
  • Figure 4: Representation of the stable configuration for $n=9$. Filled, bold dots represent positions that have a single chip. Unfilled dots represent positions that do not have any chips.
  • Figure 5: Three significant rows in the intermediate configuration
  • ...and 5 more figures

Theorems & Definitions (49)

  • Theorem 2.1: Theorem 1.1 of MR1203679
  • Example 1
  • Example 2
  • Proposition 3.1
  • proof
  • Example 3
  • Example 4
  • Theorem 4.1
  • proof
  • Proposition 4.2
  • ...and 39 more