Critical exponents and universal excess cluster number of percolation in four and five dimensions

Abstract

We study critical bond percolation on periodic four-dimensional (4D) and five-dimensional (5D) hypercubes by Monte Carlo simulations. By classifying the occupied bonds into branches, junctions and non-bridges, we construct the whole, the leaf-free and the bridge-free clusters using the breadth-first-search algorithm. From the geometric properties of these clusters, we determine a set of four critical exponents, including the thermal exponent $y_{\mathrm{t}}\equiv 1/\nu$, the fractal dimension $d_{\mathrm{f}}$, the backbone exponent $d_{\mathrm{B}}$ and the shortest-path exponent $d_{\min }$. We also obtain an estimate of the excess cluster number $b$ which is a universal quantity related to the finite-size scaling of the total number of clusters. The results are $y_{\mathrm{t}}=1.461(5)$, $d_{\mathrm{f}}=3.0446(7)$, $d_{\mathrm{B}}=1.9844(11)$, $d_{\min }=1.6042(5)$, $b=0.62(1)$ for 4D; and $y_{\mathrm{t}}=1.743(10)$, $d_{\mathrm{f}}=3.5260(14)$, $d_{\mathrm{B}}=2.0226(27)$, $d_{\min }=1.8137(16)$, $b=0.62(2)$ for 5D. The values of the critical exponents are compatible with or improving over the existing estimates, and those of the excess cluster number $b$ have not been reported before. Together with the existing values in other spatial dimensions $d$, the $d$-dependent behavior of the critical exponents is obtained, and a local maximum of $d_{\mathrm{B}}$ is observed near $d\approx 5$. It is suggested that, as expected, critical percolation clusters become more and more dendritic as $d$ increases.

11 pages, 10 figures

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• Precise bond percolation thresholds on several four-dimensional lattices
• Percolation in the canonical ensemble
• $N$-cluster correlations in four- and five-dimensional percolation
• Universal Critical Wrapping Probabilities in the Canonical Ensemble
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• Field-theoretic Analysis of Dynamic Isotropic Percolation: Three-loop Approximation
• Universal correlations in percolation
• On the monotonicity of the critical time in the Constrained-degree percolation model
• Universality of the complete-graph Potts model with $0<q\le 2$
• Extended-range percolation in five dimensions