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

Abstract

Using Monte Carlo simulations we study the distributions of the 3-block mass $N_3$ 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, $d_3$, in 2d and 3d were found to be $1.20\pm 0.1$ and $1.15\pm 0.1$, 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 $d_3=0.7\pm 0.2$ (4d) and $0.5\pm 0.2$ (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.

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References in corpus (5)

• Go with the Winners: a General Monte Carlo Strategy
• Percolation Threshold, Fisher Exponent, and Shortest Path Exponent for 4 and 5 Dimensions
• Percolation in real Wildfires
• A Transfer Matrix for the Backbone Exponent of Two-Dimensional Percolation
• Beyond Blobs in Percolation Cluster Structure: The Distribution of 3-Blocks at the Percolation Threshold