Percolation through self-affine surfaces

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Abstract
We study the percolation transition in a long-range correlated system: a self-affine surface. For all relevant physical cases (i.e. positive roughness exponents), it is found that the onset of percolation is governed by the largest wavelength of the height distribution, and thus self-averaging breaks down. Self-averaging is recovered for negative roughness exponents (i.e. power-law decay of the height pair correlation function) and, in this case, the critical exponents that characterize the transition are explicitly dependent on the roughness exponent above a threshold value. Below this threshold, the spatial correlations are no longer relevant. The problem is analytically investigated for a hierarchical network and by means of numerical simulations in two dimensions. Finally, we discuss the application of those properties to mercury porosimetry in cracks.

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