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 equicar-dinality of the set 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.