Cluster structure near the percolation threshold

Abstract

Derives exact relations that allow one to describe unambiguously and quantitatively the structure of clusters near the percolation threshold p c . In particular, the author proves the relations p(dp ij /dp)=( lambda ij ) where p is the bond density, p ij is the pair connectedness function and ( lambda ij ) is the average number of cutting bonds between i and j. From this relation it follows that the average number of cutting bonds between two points separated by a distance of the order of the connectedness length xi , diverges as mod p-p c mod -1 . The remaining (multiply connected) bonds in the percolating backbone, which lump together in 'blobs', diverge with a dimensionality-dependent exponent. He also shows that in the cell renormalisation group of Reynolds et al. (1978, 1980) the 'thermal' eigenvalue is simply related to the average number of cutting bonds in the spanning cluster. He discusses a percolation model in which the 'blobs' can be controlled by varying a parameter, and study the influence on the critical exponents.