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Percolation fractal exponents without fractal geometry

By Agnès Desolneux and Bernard Sapoval


Classically, percolation critical exponents are linked to the power laws that characterize percolation cluster fractal properties. It is found here that the gradient percolation power laws are conserved even for extreme gradient values for which the frontier of the infinite cluster is no more fractal. In particular the exponent 7/4 which was recently demonstrated 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 numerical results and the exact results that can be obtained analytically for extreme values of the gradient suggests that there exist a unique power law from size 1 to infinity which describes the gradient percolation frontier, this law becoming a scaling law in the large system limit. Spreading of objects in space with a gradient of probability is most common. From chemical composition gradients to the distribution of plants which depends of their solar exposure, probability gradients exist in many inhomogeneous systems. In fact inhomogeneit

Year: 2008
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