Growth and roughness of the interface for ballistic deposition

In ballistic deposition (BD), (d+1)-dimensional particles fall sequentially at random towards an initially flat, large but bounded d-dimensional surface, and each particle sticks to the first point of contact. For both lattice and continuum BD, a law of large numbers in the thermodynamic limit establishes convergence of the mean height and surface width of the interface to constants h(t) and w(t), respectively, depending on time t. We show that h(t) is asymptotically linear in t, while w(t) grows at least logarithmically in t when d = 1. We also give duality results saying that the height above the origin for deposition onto an initially flat surface is equidistributed with the maximum height for deposition onto a surface growing from a single site.