Title: Discrete diffusion on graphs and real hyperplane arrangements
|Affiliation:||University of Waterloo|
|Zoom:||Please email Emma Watson|
|To view the slides:||Click here|
In 2016, Duffy, Lidbetter, Messinger, and Nowakowski introduced the following variation of a chip-firing model on a graph. At time zero, there is an integer number of "chips" at each vertex. Time proceeds in discrete steps. At each step, each edge is examined (in parallel) -- one chip is moved from the greater end to the lesser end if the ends are not equal. What happens? The dynamics is governed by a discrete -- slightly non-linear -- analogue of the heat equation, and behaves accordingly: the sequence of states is eventually periodic. Somewhat surprisingly, the period is either one or two time steps (Long and Narayanan, 2017). The whole set-up can be generalized to real hyperplane arrangements (and beyond), and an external "forcing current" can be applied to hold the system out of equilibrium. What happens? The effect of the forcing current is easily explained, but there remain some small mysteries about the equilibrium steady state.