Shapes, Surfaces, and Interfaces in Percolation Clusters

Percolation theory is reviewed. Intuitive arguments are given to derive scaling and hyperscaling relations. Above six dimensions the breakdown of hyperscaling is related to the interpenetration of the critical large clusters, and to the appearence at p c of an infinite number of infinite clusters of zero density with fractal dimension d f = 4. The structure of the percolating cluster made of links and blobs is characterized by an infinite set of exponents related to the anomalous voltage distribution in a random resistor network at p c . The surface structure of critical clusters below pc, which is relevant to the study of random superconducting networks, is also discussed. In particular, an exact result is presented which shows that in any dimension the interface of two critical clusters diverge as (p c -p)-1 as the percolation threshold is approached.

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