Diffusion: Quiescence and Perturbation

Abstract: Originally proposed by Duffy et al., Diffusion is a variant of chip-firing in which chips from flow from places of high concentration to places of low concentration. In the variant, Perturbation Diffusion, the first step involves a "perturbation" in which some number of vertices send chips to each of their respective neighbours even though the rules of Diffusion only permit for chips to be sent from richer vertices to poorer vertices. Perturbation Diffusion allows us to expand our study of Diffusion by asking new questions such as ``Given an initial configuration, which vertices, when perturbed, will return the initial configuration after some number of steps in Diffusion." We give some results in this paper that begin to answer this question in the specific case of every vertex initially having 0 chips. We characterize some of the ways a graph can reach such a state in Perturbation Diffusion before focusing on paths in particular with more specific results.