On lengths of burn-off chip-firing games

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.