On lengths of burn-off chip-firing games

TL;DR

This paper analyzes the distribution of game lengths in burn-off chip-firing games on graphs, providing exact counts and probabilistic insights into how long such games last under random seed selection.

Contribution

It introduces exact enumeration methods for game lengths and offers a new bijective proof of the correspondence between relaxed configurations and spanning trees.

Findings

Exact counts for pairs of configurations and seeds for each game length

Probability distribution of game lengths in long sequences

A new bijective proof linking relaxed configurations and spanning trees

Abstract

We continue our studies of burn-off chip-firing games from [Discrete Math. Theor. Comput. Sci. 15 (2013), no. 1, 121-132; MR3040546] and [Australas. J. Combin. 68 (2017), no. 3, 330-345; MR3656659]. The latter article introduced randomness by choosing successive seeds uniformly from the vertex set of a graph $G$. The length of a game is the number of vertices that fire (by sending a chip to each neighbor and annihilating one chip) as an excited chip configuration passes to a relaxed state. This article determines the probability distribution of the game length in a long sequence of burn-off games. Our main results give exact counts for the number of pairs $(C,v)$, with $C$ a relaxed legal configuration and $v$ a seed, corresponding to each possible length. In support, we give our own proof of the well-known equicardinality of the set $\mathcal{R}$ of relaxed legal configurations on $G$ and the set of spanning trees in the cone $G^*$ of $G$. We present an algorithmic, bijective proof of this correspondence.