Size distributions of the largest hole in the largest percolation cluster and backbone

At criticality in two dimensions, the size distribution of holes within a large percolation cluster or backbone was recently found to be governed by a power law with the exponent given by a hyperscaling relation. It was also found that, though the boundary of the largest hole is a fractal, the size of the largest hole itself is proportional to Ld, where L represents the linear system size and d=2 is the spatial dimension. In this work, we study the probability distribution of the size of the largest hole in the largest percolation cluster and that in the largest backbone, and compare them with the probability distribution of the size of the largest percolation cluster and that of the largest backbone themselves. We obtain these four size distributions in the critical region by Monte Carlo simulations for the two-dimensional bond percolation model on the square lattice with periodic boundary conditions. It is found that each of these distributions can be described by a different single-variable function P(C,L)dC=P̃(x)dx, with x≡C∕LdC, where C is the size of the structure whose (fractal) dimension is dC. The function P̃(x) is generally not unimodal, and interestingly, for the largest cluster hole, symmetric properties are found in its size distribution due to duality, e.g., P̃(x) is symmetric around the mean value x=1 at the percolation threshold pc. We characterize these size distributions by calculating the variance and different dimensionless cumulant ratios, which exhibit rich critical behaviors at the pseudocritical and critical points. For example, the extremum of the excess kurtosis and the zero point of the skewness locate at the same (pseudocritical) point, whose values are different for the four structures but approach pc in the limit L→∞. We also perform wrapping analyses for each size distribution, showing that different wrapping types contribute to the overall distribution in different forms.