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 greater than or equal to d(c) the decay of the needle density rho(h) as a function of the height h above the substrate is correctly described by a continuum mean-field theory. Below d, 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 rho(h) similar to In h/h in d = 1, in agreement with extensive simulations.