Nature of the largest cluster size distribution at the percolation threshold

Two distinct distribution functions P-sp(m) and P-ns(m) of the scaled largest cluster sizes m are obtained at the percolation threshold by numerical simulations, depending on the condition whether the lattice is actually spanned or not. With R(p(c)) the spanning probability, the total distribution of the largest cluster is given by P-tot(m) = R(p(c))P-sp(m) + (1 - R(p(c)))P-ns(m). The three distributions apparently have similar forms in three and four dimensions while in two dimensions, Ptot(m) does not exhibit a familiar form. By studying the first and second cumulants of the distribution functions, the different behaviour of P-tot(m) in different dimensions may be quantified.