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Critical percolation in high dimensions.

Peter Grassberger1

  • 1John-von-Neumann Institute for Computing, Forschungszentrum Jülich, Germany.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|April 12, 2003
PubMed
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We provide precise Monte Carlo estimates for percolation thresholds in high-dimensional hypercubic lattices. Our advanced methods significantly improve accuracy for dimensions 6 and above, offering new insights into critical phenomena.

Area of Science:

  • Statistical Physics
  • Computational Physics
  • Materials Science

Background:

  • Percolation theory studies the connectivity of random networks.
  • Hypercubic lattices are fundamental structures in statistical mechanics.
  • Accurate determination of percolation thresholds is crucial for understanding phase transitions.

Purpose of the Study:

  • To compute highly precise Monte Carlo estimates for site and bond percolation thresholds.
  • To investigate percolation in simple hypercubic lattices across dimensions 4-13.
  • To propose a new scaling law for finite cluster size corrections.

Main Methods:

  • Utilized fast hashing for efficient simulation of large clusters (millions of sites).
  • Employed a histogram method to gather data for multiple percolation probabilities (p) from single simulations.

Related Experiment Videos

  • Implemented a variance reduction technique, particularly effective in high dimensions, reducing error bars by up to ~30x.

    • Main Results:

    • Achieved 20 to 10^4 times greater precision for percolation thresholds in dimensions d ≥ 6 compared to previous estimates.
    • Preliminary estimates for dimensions d < 6 were also obtained.
    • The developed methods allowed simulations on computers with limited memory (<500 Mbytes).

    Conclusions:

    • The study presents the most precise percolation threshold estimates to date for high-dimensional lattices.
    • A novel scaling law for finite cluster size corrections has been proposed based on the obtained data.
    • The computational techniques offer a powerful approach for future studies in percolation theory and related fields.