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Continuum percolation thresholds in two dimensions.

Stephan Mertens1, Cristopher Moore

  • 1Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA. mertens@ovgu.de

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 2, 2013
PubMed
Summary
This summary is machine-generated.

Researchers computed percolation thresholds in continuum models using microcanonical simulations and conformal field theory. This approach precisely determined transition points for various shapes, confirming theoretical predictions.

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Area of Science:

  • Statistical Physics
  • Computational Physics

Background:

  • Percolation theory studies connectivity in random systems.
  • Lattice percolation uses powerful methods like union-find and conformal field theory.
  • Continuum percolation models involve objects with continuous positions and orientations.

Purpose of the Study:

  • To apply advanced lattice percolation methods to continuum models.
  • To precisely calculate percolation thresholds for various geometric shapes.
  • To verify the predictions of conformal field theory in continuum percolation.

Main Methods:

  • Microcanonical simulations with the union-find algorithm for cluster identification.
  • Leveraging conformal field theory for critical probability calculations in 2D.
  • Applying these methods to continuum models with disks, squares, and rotated sticks.

Main Results:

  • Precise percolation transition values were determined for disks, squares, rotated squares, and rotated sticks in 2D.
  • The observed transitions align with conformal field theory predictions.
  • The algorithm demonstrated near-linear time and memory efficiency at criticality.

Conclusions:

  • The combined approach of simulation and conformal field theory is effective for continuum percolation.
  • Continuum percolation transitions exhibit behaviors consistent with theoretical frameworks.
  • The developed algorithm is efficient for analyzing large systems of objects.