Narrow Results

We study the ratio of the number of sites in the largest and second largest clusters in random percolation. Using the scaling hypothesis that the ratio <M₁>/<M₂> of the mean cluster sizes M₁ and M₂ scales as f ((p - p_c) L^1/ν), we employ finite-size scaling analysis to find that <M₁>/<M₂> is nonuniversal with respect to the boundary conditions imposed. The mean <M₁/M₂> of the ratios behaves similarly although with a distinct critical value reflecting the relevance of mass fluctuations at the percolation threshold. These zero exponent ratios also allow for reliable estimates of the critical parameters at percolation from relatively small lattices.

The three-dimensional 3-state Potts model has been investigated by examining the average cluster size distributions. Cluster distributions give evidence for the weak first-order nature of the transition for lattice sizes as small as 16³, where the energy histogram method fails to exhibit the true nature of the transition.