2017/04/13 by Jason Long, Long, Jason, Bhargav Narayanan +1
Mathematics · Physics and Astronomy · #05C57 (Primary) #91A43 (Secondary) #Combinatorics (math.CO) #Complex Network Analysis Techniques #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Stochastic processes and statistical mechanics #math.CO #msc:05C57 #msc:91A43
paper · pdf · doi:10.48550/arxiv.1704.04295
7 pages, submitted
openalex publication_date 2017/04/13 · arxiv created 2017/06/05 · arxiv updated 2017/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study a variant of the chip-firing game called diffusion. In diffusion on a graph, each vertex of the graph is initially labelled with an integer interpreted as the number of chips at that vertex, and at each subsequent step, each vertex simultaneously fires one chip to each of its neighbours with fewer chips. Since this firing rule may result in negative labels, diffusion, unlike the parallel chip-firing game, is not obviously periodic. In 2016, Duffy, Lidbetter, Messinger and Nowakowski nevertheless conjectured that diffusion is always eventually periodic, and moreover, that the process eventually has period either 1 or 2. Here, we establish this conjecture.