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

TL;DR

This study uses advanced Monte Carlo simulations to analyze the fractal dimensions of 3-blocks in higher-dimensional percolation systems, revealing a decrease in fractal dimension with increasing dimension and behavior at the critical dimension.

Contribution

The paper introduces a statistical enhancement method to accurately estimate the fractal dimension of 3-blocks in 4d, 5d, and 6d percolation systems, extending previous 2d and 3d results.

Findings

Fractal dimension of 3-blocks decreases rapidly in higher dimensions.

Estimated $d_3$ is 0.7±0.2 in 4d and 0.5±0.2 in 5d.

At 6d, results are consistent with $d_3=0$ with logarithmic corrections.

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.