The chemical distance distribution in percolation clusters

We study the conditional probability density p (r/l), for the geometrical distance r corresponding to a given chemical distance l for percolation clusters in two dimensions. We argue that (i) p(r/l) = A₁x^g exp(-ax^δ) where x = r/l^v, g = 2.5 ± 0.3, δ = 9.8 ± 0.5, and v = 0.88 ± 0.02, and (ii) there is a relation δ = (l - v)^-1 which is in good agreement with our 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.