Infinite clusters in percolation models

The qualitative nature of infinite clusters in percolation models is investigated. The results, which apply to both independent and correlated percolation in any dimension, concern the number and density of infinite clusters, the size of their external surface, the value of their (total) surface-to-volume ratio, and the fluctuations in their density. In particular it is shown thatN0, the number of distinct infinite clusters, is either 0, 1, or ∞ and the caseN0=∞ (which might occur in sufficiently high dimension) is analyzed.

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