Cluster shapes at the percolation threshold: and effective cluster dimensionality and its connection with critical-point exponents

Abstract

An effective dimensionality d_p is introduced for the purpose of providing a quantitative characterisation of the degree of ramification of the clusters that occur at the percolation threshold. It is found that d_p is directly related to percolation critical exponents, and that 1pp; in particular, the renormalised mean cluster size exponent is d_p, while both the order parameter and 'decay of correlation' exponents are given by the co-dimension d-d_p.