Nature of largest cluster size distribution at the percolation threshold

Details Nature of largest cluster size distribution at the percolation threshold Nature of largest cluster size distribution at the percolation threshold

2001-09-13

arxiv.org/abs/cond-mat/0109229v1

Physics

Condensed Matter

Disordered Systems and Neural Networks

7 pages revtex, figures included; to be published in J. Phys. A

Scientific paper

10.1088/0305-4470/34/41/306

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.

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