Dispersion processes

We study a synchronous dispersion process in which M particles are initially 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 neighbour of v chosen independently and uniformly at random. The dispersion process ends at the first step when each vertex has at most one particle. For the complete graph Kn and star graph Sn, we show that for any constant δ > 1, with high probability, if M ≤ n/2(1 − δ), then the process finishes in O(log n) steps, whereas if M ≥ n/2(1 + δ), then the process needs e steps to complete (if ever). We also show that an analogous lazy variant of the process exhibits the same behaviour but for higher thresholds, allowing faster dispersion of more particles. For paths, trees, grids, hypercubes and Cayley graphs of large enough sizes (in terms of M) we give bounds on the time to finish and the maximum distance traveled from the origin as a function of the number of particles M .