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Related Experiment Videos

Efficient Monte Carlo algorithm and high-precision results for percolation.

M E Newman1, R M Ziff

  • 1Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA.

Physical Review Letters
|November 1, 2000
PubMed
Summary
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A new Monte Carlo algorithm efficiently studies percolation on any lattice. This method precisely determines the percolation threshold for square lattices and confirms theoretical predictions for spanning probabilities.

Area of Science:

  • Statistical Physics
  • Computational Physics

Background:

  • Percolation theory models critical phenomena in disordered systems.
  • Efficient simulation of large-scale lattice systems remains a computational challenge.

Purpose of the Study:

  • Introduce a novel, efficient Monte Carlo algorithm for site and bond percolation.
  • Accurately calculate percolation properties across the full probability range.
  • Determine the precise percolation threshold and spanning probability behavior.

Main Methods:

  • Developed a linear-time Monte Carlo algorithm for percolation studies.
  • Algorithm applicable to any lattice structure.
  • Calculates cluster size distribution and spanning probability in a single run.

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Main Results:

  • Determined the site percolation threshold on the square lattice: p(c) = 0.592 746 21(13).
  • Provided numerical confirmation of 4/3-power stretched-exponential tails in spanning probability functions.
  • Algorithm's runtime scales linearly with the number of lattice sites.

Conclusions:

  • The new Monte Carlo algorithm offers significant efficiency gains for percolation simulations.
  • The results validate theoretical predictions for critical exponents and tail behavior.
  • This method facilitates comprehensive studies of percolation phenomena on diverse lattices.