Dispersion processes

Abstract

We study a synchronous process called dispersion. Initially M particles are placed at a distinguished origin vertex of a graph G. At each time step, at each vertex v occupied by more than one particle at the beginning of this step, each of these particles moves to a neighbor of v chosen independently and uniformly at random. The process ends at the first step when no vertex is occupied by more than one particle. For the complete graph Kn, for any constant δ > 1, with high probability, if M ≤ n/2(1 − δ), the dispersion process finishes in steps, whereas if M ≥ n/2(1 + δ), the process needs eΩ(n) steps to complete, if ever. A lazy variant of the process exhibits analogous behavior but at a higher threshold, thus allowing faster dispersion of more particles. For paths, trees, grids, hypercubes, and Abelian Cayley graphs of large enough size, we give bounds on the time to finish and the maximum distance traveled from the origin as a function of M.