Extreme and Smooth Gradient Percolation

Bernard Sapoval
École Polytechnique
Directeur de Recherches

Classically, percolation critical exponents are linked to power laws that characterize cluster fractal properties. We find here that the gradient percolation power laws are conserved even for extreme gradient values for which the frontier of the infinite cluster is no longer fractal. In particular the exponent 7/4 which was recently shown to be the exact value for the dimension of the so-called ``hull'' or external perimeter of the incipient percolation cluster, keeps its value in describing the width and length of gradient percolation frontiers whatever the gradient value. This indicates that independently of SLE arguments, the value 7/4 may originates from direct combinatorial analysis. The comparison between the numerical and the exact results that can be obtained analytically for extreme values of the gradient suggests that there exists a unique power law, from size 2 to infinity, that describes the gradient percolation frontier. These results provides an intrinsic method to find whether a rough interface belongs to gradient percolation without knowledge of the gradient and can be considered as resulting from a new conservation law for diffusion on a lattice.
We will also discuss a new modality of percolation problems that we call “smooth percolation”. In smooth percolation, percolation criticality occurs as a function of an arbitrary continuous variable. It extends the concept of critical fluctuations to a much wider class of physical situations.

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