The chemical distance distribution in percolation clusters

Abstract

The authors study the conditional probability density p(r mod l), for the geometrical distance r corresponding to a given chemical distance l for percolation clusters in two dimensions. They argue that (i) p(r mod l)=A_lx^g exp(-ax^delta ) where x=r/l^nu , g=2.5+or-0.3, delta =9.8+or-0.5, and nu =0.88+or-0.02, and (ii) there is a relation delta =(1- nu )^-1 which is in good agreement with numerical data. These results are derived by considering a special class of self-avoiding walks consisting of chains which are chemical paths (shortest paths) on critical percolation clusters.