Fragmentation scaling of percolation clusters in two and three dimensions: Large-cell Monte Carlo RG approach

The scaling behavior for a binary fragmentation of critical percolation clusters is investigated by a large-cell Monte Carlo real-space renormalization group method in two and three dimensions. We obtain accurate values of critical exponents λ and ϕ describing the scaling of fragmentation rate and the distribution of fragments' masses produced by a binary fragmentation. Our results for λ and ϕ show that the fragmentation rate is proportional to the size of mother cluster, and the scaling relation $\sigma = 1 + \lambda - \phi$ conjectured by Edwards et al. to be valid for all dimensions is satisfied in two and three dimensions, where $\sigma $ is the crossover exponent of the average cluster number in percolation theory, which excludes the other scaling relations