Dynamic Space Filling

Dynamic space filling (DSF) is a stochastic process defined on any connected graph. Each vertex can host an arbitrary number of particles forming a pile, with every arriving particle landing on the top of the pile. Particles in a pile, except for the particle at the bottom, can hop to neighboring vertices. Eligible particles hop independently and stochastically, with the overall hopping rate set to unity. When the number of vertices in a graph is equal to the total number of particles, the evolution stops when a single particle occupies every vertex. We determine the halting time distribution on complete graphs. Using the mapping of the DSF into a two-species annihilation process, we argue that on $d$-dimensional tori with $N\gg 1$ vertices, the average halting time scales with the number of vertices as $N^{4/d}$ when $d\leq 4$ and as $N$ when $d>4$.