BOUNDARY CONDITION DEPENDENCE OF CLUSTER SIZE RATIOS IN RANDOM PERCOLATION

Abstract

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 - pc) L1/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.