2020/07/19 by P. Mark Kayll, Kayll, P. Mark, Dave Perkins +1
Mathematics · Physics and Astronomy · #05C05 #05C25 #60J20 #68R10 #91A43 #Combinatorics (math.CO) #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Primary: 05C57 #Secondary: 05C85 #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.2007.09732
openalex publication_date 2020/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.