Percolation fractal exponents without fractals and a new conservation law in diffusion

¹ Mathématiques Appliquées à Paris 5, CNRS UMR 8145, Université Paris 5 75006 Paris, France
² Centre de Mathématiques et de leurs Applications, CNRS UMR 8536 École Normale Supérieure - 94235 Cachan, France
³ Laboratoire de Physique de la Matière Condensée, CNRS UMR 7643 École Polytechnique, 91128 Palaiseau, France

Received: 14 September 2005
Accepted: 13 October 2005

Abstract

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. Its origin is then not to be found in the thermodynamic limit. 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.