Fractal Dimension of 3-Blocks in 4d, 5d, and 6d Percolation Systems

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Using Monte Carlo simulations we study the distributions of the 3-block mass N₃ in 4d, 5d, and 6d 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, d₃, in 2d 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 d₃=0.7± 0.2 (4d) and 0.5± 0.2 (5d). At the upper critical dimension of percolation, dc=6, our simulations are consistent with d₃=0 with logarithmic corrections to power-law scaling.