Laplacian Needle Growth

We study the scaling properties of a forest of one-dimensional needles that grow from a d -dimensional substrate by the aggregation of individual random walkers. Using opacity arguments we establish the existence of an upper critical dimension d c such that for d ⩾ d c the decay of the needle density ρ( h ) as a function of the height h above the substrate is correctly described by a continuum mean-field theory. Below d c the decay of the density profile can be inferred from the competition between two needles. Scaling arguments in combination with a conformal mapping calculation indicate that ρ( h ) ∼ ln h/h in d = 1, in agreement with extensive simulations.