On Chip-Firing on Undirected Binary Trees

TL;DR

This paper analyzes chip-firing dynamics on an infinite binary tree with a self-loop at the root, providing bounds on stable configurations and counting vertex fires for various initial chip counts.

Contribution

It introduces new bounds on stable configurations and explicit fire counts for chip-firing on an infinite binary tree with a root self-loop, extending prior work.

Findings

Upper bound for stable configurations with $2^ ext{ell}-1$ chips at root

Exact number of fires for arbitrary initial chips at root

Total number of fires for different initial conditions

Abstract

Chip-firing is a combinatorial game played on an undirected graph in which we place chips on vertices. We study chip-firing on an infinite binary tree in which we add a self-loop to the root to ensure each vertex has degree 3. A vertex can fire if the number of chips placed on it is at least its degree. In our case, a vertex can fire if it has at least 3 chips, and it fires by dispersing $1$ chip to each neighbor. Motivated by a 2023 paper by Musiker and Nguyen on this setting of chip-firing, we give an upper bound for the number of stable configurations when we place $2^\ell - 1$ labeled chips at the root. When starting with $N$ chips at the root where $N$ is a positive integer, we determine the number of times each vertex fires when $N$ is not necessarily of the form $2^\ell - 1$. We also calculate the total number of fires in this case.