On lengths of burn-off chip-firing games

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 \( R \) 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.