Our goal is to understand the volume, boundary size, and other structural properties of growth sets (indexed by time) arising from the competition among random walkers. Consider a discrete torus of large side length n with fixed dimension d and k many walkers, each starting from random points on the torus. Each walker follows an independent random walk and whoever arrives first at a vertex owns that vertex (breaking ties at random). At each time, we have k sets of vertices owned by the different walkers. The size (or boundary size) of these k sets gives random processes indexed by time. Using simulation, we will try to understand these processes' continuum scaling limit and fluctuation properties for large n.
Math 461 (Probability Theory), Math 416 (Linear algebra), Python