Conformal map modeling of the pinning transition in Laplacian growth

In Laplacian growth processes pinning may be expected due to a nonlinear response of a material during dielectric breakdown, or due to stick-slip boundary conditions in two-fluid flow in a porous medium, while thermal noise will lead to depinning. Using a method recently proposed by Hastings and Levitov, the size R(max) approximately E(-alpha)(c) of the pinned pattern is shown to scale with the critical field E(c) (electric field for dielectric breakdown, pressure gradient for fluid flow). These pinned patterns have a lower effective fractal dimension d(f) than diffusion-limited aggregation due to the enhancement of growth at the hot tips of the developing pattern. At finite temperature, thermal noise leads to depinning and growth of patterns with a shape and dimensionality dependent on both E(c) and the thermal noise. Using multifractal analysis, scaling expressions are established for this dependency.