Labeled Chip-Firing on Directed $k$-ary Trees and Where Chips Land

TL;DR

This paper analyzes a chip-firing game on infinite directed $k$-ary trees, characterizing stable configurations and chip landing patterns, with implications for understanding combinatorial dynamics on such structures.

Contribution

It introduces a novel chip-firing variant on directed $k$-ary trees and characterizes the stable configurations and chip landing behaviors.

Findings

Chips land within specific vertex ranges

Lengths of landing ranges are determined

Possible stable configurations are described

Abstract

Chip-firing is a combinatorial game played on a graph, in which chips are placed and dispersed on the vertices until a stable configuration is achieved. We study a chip-firing variant on an infinite, rooted directed $k$-ary tree, where we place $k^n$ chips labeled $1,2,3,\dots, k^n$ on the root for some nonnegative integer $n$. A vertex $v$ can fire if it has at least $k$ chips; when it fires, $k$ chips are selected, and the chip with the $i$th smallest label is sent to the $i$th leftmost child of $v$. A stable configuration is reached when no vertices can fire. In this paper, we prove numerous properties of the stable configuration, such as that chips land on vertices in ranges and the lengths of those ranges. We also describe where each chip can land. This helps us describe possible stable configurations of the game.