Fractal dimension of 3-blocks in four-, five-, and six-dimensional percolation systems

Using Monte Carlo simulations, we study the distributions of the 3-block mass N3 in four-, five-, and six-dimensional percolation systems. Because the probability of creating large 3-blocks in these dimensions is very small, we use a "go with the winners" method of statistical enhancement to simulate configurations having probability as small as 10(-30). In earlier work, the fractal dimensions of 3-blocks, d(3), in 2D (two dimensional) and 3D were found to be 1.20+/-0.1 and 1.15+/-0.1, respectively, consistent with the possibility that the fractal dimension might be the same in all dimensions. We find that the fractal dimension of 3-blocks decreases rapidly in higher dimensions, and estimate d(3)=0.7+/-0.2 (4D) and 0.5+/-0.2 (5D). At the upper critical dimension of percolation, d(c)=6, our simulations are consistent with d(3)=0 with logarithmic corrections to power-law scaling.