Nature of largest cluster size distribution at the percolation threshold

Abstract

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, $P_{tot}(m)$ does not follow 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.