Near-critical percolation and the geometry of diffusion fronts
Pierre Nolin (ENS-Paris)
We discuss a model of inhomogeneous medium as "Gradient Percolation", which is an inhomogeneous percolation process where the density of occupied sites depends on the location in space. This model was first introduced by the physicists Gouyet, Rosso and Sapoval in 1985 to show numerical evidence that diffusion fronts are fractal. The macroscopic interface-separating occupied sites and vacant sites - that appears remains localized in regions where the density of occupied sites is close to the percolation threshold Pc, and its behavior can be described using properties of near-critical standard percolation.
This allows to study a simple two-dimensional model where a large number of particles that start at a given site diffuse independently. As the particles evolve, a concentration gradient appears and we observe a macroscopic interface. We exhibit in particular a regime where this (properly rescaled) interface is fractal with dimension 7/4: this model thus provides a natural setting where fractal geometry spontaneously arises, as predicted by physicists.