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

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  • Zhang, Zhongjin
  • Hou, Pengcheng
  • Fang, Sheng
  • Hu, Hao
  • Deng, Youjin

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 yt≡1∕ν, the fractal dimension df, the backbone exponent dB and the shortest-path exponent dmin. 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 yt=1.461(5), df=3.0446(7), dB=1.9844(11), dmin=1.6042(5), b=0.62(1) for 4D; and yt=1.743(10), df=3.5260(14), dB=2.0226(27), dmin=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 dB is observed near d≈5. It is suggested that, as expected, critical percolation clusters become more and more dendritic as d increases.

Suggested Citation

  • Zhang, Zhongjin & Hou, Pengcheng & Fang, Sheng & Hu, Hao & Deng, Youjin, 2021. "Critical exponents and universal excess cluster number of percolation in four and five dimensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 580(C).
    • Handle: RePEc:eee:phsmap:v:580:y:2021:i:c:s0378437121003976
    DOI: 10.1016/j.physa.2021.126124
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    References listed on IDEAS

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    1. Daniel Tiggemann, 2001. "Simulation Of Percolation On Massively-Parallel Computers," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 12(06), pages 871-878.
    2. C. Moukarzel, 1998. "A Fast Algorithm for Backbones," International Journal of Modern Physics C (IJMPC), World Scientific Publishing Co. Pte. Ltd., vol. 9(06), pages 887-895.
    3. Ziff, Robert M & Lorenz, Christian D & Kleban, Peter, 1999. "Shape-dependent universality in percolation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 266(1), pages 17-26.
    4. Grassberger, Peter, 1999. "Conductivity exponent and backbone dimension in 2-d percolation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 262(3), pages 251-263.
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    Cited by:

    1. Balankin, Alexander S., 2024. "A survey of fractal features of Bernoulli percolation," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).

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