Chip Firing on Directed $k$-ary Trees

TL;DR

This paper analyzes a chip-firing game on an infinite directed $k$-ary tree, determining the stable configurations when chips are distinguishable, expanding understanding of combinatorial dynamics on such structures.

Contribution

It provides an exact characterization of stable configurations in a chip-firing game on directed $k$-ary trees with distinguishable chips, a novel analysis in this context.

Findings

Exact number of stable configurations determined

Properties of stable configurations characterized

Analysis specific to distinguishable chips

Abstract

Chip-firing is a combinatorial game played on a graph in which we place and disperse chips on vertices until a stable state is reached. We study a chip-firing variant played on an infinite rooted directed $k$-ary tree, where we place $k^\ell$ chips on the root for some positive integer $\ell$, and we say a vertex $v$ can fire if it has at least $k$ chips. A vertex fires by dispersing one chip to each out-neighbor. Once every vertex has less than $k$ chips, we reach a stable configuration since no vertex can fire. We determine the exact number and properties of the possible stable configurations of chips in the setting where chips are distinguishable.