Chip-Firing on Infinite $k$-ary Trees

TL;DR

This paper investigates the chip-firing process on an infinite k-ary tree with a self-loop at the root, analyzing stable configurations and fire counts to understand the process's behavior.

Contribution

It provides a detailed analysis of chip-firing dynamics on infinite k-ary trees, including stable configurations and fire count sequences, which are novel in this context.

Findings

Characterization of stable configurations

Formulas for number of fires at each vertex

Properties of fire count sequences as functions of N

Abstract

We use an infinite $k$-ary tree with a self-loop at the root as our underlying graph. We consider a chip-firing process starting with $N$ chips at the root. We describe the stable configurations. We calculate the number of fires for each vertex and the total number of fires. We study a sequence of the number of root fires for a given $k$ as a function of $N$ and study its properties. We do the same for the total number of fires.