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 lambda and phi describing the scaling of fragmentation rate and the distribution of fragments' masses produced by a binary fragmentation. Our results for lambda and phi 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 at. 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.