Algebraic Combinatorics Seminar - David Wagner

Title: Discrete diffusion on graphs and real hyperplane arrangements

Abstract:

In 2016, Duffy et al. introduced the following process on a graph.  Initially, each vertex has some integer number of ``chips'' placed there (possibly negative).  Thereafter, in discrete time steps, if an edge has more chips at one end than at the other, then one chip moves along that edge from the richer to the poorer end.  All edges are processed in parallel at each time step.  Duffy et al. observed experimentally that the dynamics of this process was eventually periodic of period one or two.  This was proven in 2017 by Long and Narayanan.  I will give their proof generalized to the context of real hyperplane arrangements, explain the analogies with the heat equation, and present some conjectures about what happens when the system is held out of equilibrium by some external sources and sinks of chips.