Gap Independence and Lacunarity in Percolation Clusters

The gaps between occupied sites on linear cuts of two and three dimensional critical percolation clusters are found to be closely described as statistically independent, with a universal scaling distribution close to that of positive Lévy flights. The moments of the mass distribution of Lévy flights obey $〈m^k〉/〈m{〉}^k=k!\left[\gamma \right(\alpha +1\mathrm{}\left){]}^k/\gamma \right(k\alpha +1\mathrm{})$, where $\alpha$ is their fractal dimension. Our data on linear cuts of critical percolation clusters are consistent (within the numerical error bars) with these predictions. The property of statistical independence of the gaps characterizes the lacunarity of the percolation clusters as being neutral.