Diffusion on Graphs Is Eventually Periodic

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.