Chip-firing on the Lattice of Nonnegative Integer Points

TL;DR

This paper investigates a chip-firing game on a lattice's Hasse diagram, analyzing stable configurations and intermediate firing patterns, revealing structural properties and relationships between configurations.

Contribution

It introduces a novel analysis of chip-firing on lattice structures, detailing the properties of intermediate and stable configurations and their interrelations.

Findings

Nonzero entries of stable configurations correspond to odd entries in intermediate configurations.

The intermediate configuration has a three-part structure: top triangle, midsection, bottom triangle.

Properties of each row and difference tables of the intermediate configuration are characterized.

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.