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Critical surfaces for general bond percolation problems.

Christian R Scullard1, Robert M Ziff

  • 1Department of Geophysical Sciences, University of Chicago, Chicago, Illinois 60637, USA. scullard@uchicago.edu

Physical Review Letters
|June 4, 2008
PubMed
Summary
This summary is machine-generated.

A new general method accurately predicts bond percolation thresholds for 2D periodic lattices. This approach reproduces known results and offers precise approximations for unsolved lattice problems.

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Area of Science:

  • Physics
  • Materials Science
  • Statistical Mechanics

Background:

  • Percolation theory studies the connectivity of random networks.
  • Determining percolation thresholds is crucial for understanding material properties and phase transitions.
  • Exact solutions for many lattice types remain elusive.

Purpose of the Study:

  • To introduce a versatile method for predicting bond percolation thresholds.
  • To calculate critical surfaces for various 2D periodic lattices.
  • To validate the method against known exact results and unsolved problems.

Main Methods:

  • Development of a general computational framework.
  • Application to diverse two-dimensional periodic lattice structures.
  • Comparison of predicted values with existing numerical and analytical data.

Main Results:

  • The method successfully reproduces numerous known exact bond percolation thresholds.
  • It provides highly accurate approximations for several previously unsolved lattices.
  • Predictions for checkerboard and inhomogeneous bow-tie lattices show agreement to over six decimal places.

Conclusions:

  • The presented method is a powerful tool for percolation threshold prediction.
  • It offers a reliable approach for both solved and unsolved lattice systems.
  • The high accuracy suggests potential for exactness in certain lattice types.