Diffusion-driven spreading phenomena: the structure of the hull of the visited territory

We study the hull of the territory visited by N random walkers after t time steps. The walkers move on two-dimensional substrates, starting all from the same position. For the substrate, we consider (a). a square lattice and (b). a percolation cluster at criticality. On the square lattice, we (c). also allow for birth and death processes, where at every time step, alphaN walkers die and are removed from the substrate, and simultaneously the same number of walkers is added randomly at the positions of the remaining walkers, such that the total numbers of walkers is constant in time. We perform numerical simulations for the three processes and find that for all of them, the structure of the hull is self-similar and described by a fractal dimension d(H) that slowly approaches, with an increasing number of time steps, the value d(H)=4/3. For process (c), however, the time to approach the asymptotic value increases drastically with increasing fraction of N/alpha, and can be observed numerically only for sufficiently small values of N/alpha.