Non-Stabilizing Parallel Chip-Firing Games

TL;DR

This paper investigates the behavior of parallel chip-firing games at a specific activity level, proving non-stabilization and certain period properties within a critical range of chip counts, extending previous results to bipartite graphs.

Contribution

It extends analysis of chip-firing game activity levels to bipartite graphs, proving non-stabilization and specific period constraints in the middle activity staircase.

Findings

Games with $2|E|-|V|< |\sigma|< 2|E|$ are non-stabilizing.

All such games have periods not equal to 3 or 4.

Conjecture: these games have period 2 and activity 1/2, proven for specific graph classes.

Abstract

In 2010, Kominers and Kominers proved that any parallel chip-firing game on $G(V,\,E)$ with $|\sigma|\geq 4|E|-|V|$ chips stabilizes. Recently, Bu, Choi, and Xu made the bound exact: all games with $|\sigma|< |E|$ chips or $|\sigma|> 3|E|-|V|$ chips stabilize. Meanwhile, Levine found a "devil's staircase'' pattern in the plot of the activity of parallel chip-firing games against their density of chips. The stabilizing bound of Bu, Choi, and Xu corresponds to the top and bottom stairs of this staircase, in which the activity is 1 and 0, respectively. In this paper, we analyze the middle stair of the staircase, corresponding to activity $\frac{1}{2}$. We prove that all parallel chip-firing games with $2|E|-|V|< |\sigma|< 2|E|$ have period $T\neq 3,\,4$. In fact, this is exactly the range of $|\sigma|$ for which all games are non-stabilizing. We conjecture that all parallel chip-firing games with $2|E|-|V|< |\sigma|<2|E|$ have $T=2$ and thus activity $\frac{1}{2}$. This conjecture has been proven for trees by Bu, Choi, and Xu, cycles by Dall'asta, and complete graphs by Levine. We extend Levine's method of conjugate configurations to prove the conjecture on complete bipartite graphs $K_{a,a}$.