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

Abstract

An effective dimensionality dp 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 dp is directly related to percolation critical exponents, and that 1<or=dp<or=d, which in turn places bounds on certain scaling powers and critical exponents. The exponents-when renormalised according to Suzuki's 'extended universality' prescription-have an appealingly simple form in terms of dp; in particular, the renormalised mean cluster size exponent is dp, while both the order parameter and 'decay of correlation' exponents are given by the co-dimension d-dp.