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Rigorous confidence intervals for critical probabilities.

Oliver Riordan1, Mark Walters

  • 1Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, Cambridge, CB3 0WB, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|August 7, 2007
PubMed
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This study provides highly accurate confidence intervals for critical probabilities in site and bond percolation on Archimedean lattices. The research emphasizes simple, verifiable computer calculations for reliable results.

Area of Science:

  • Statistical Physics
  • Probability Theory
  • Computational Mathematics

Background:

  • Percolation theory studies the behavior of connected components in random graphs.
  • Archimedean lattices are regular structures with applications in various scientific fields.
  • Determining critical probabilities is essential for understanding phase transitions in these systems.

Purpose of the Study:

  • To calculate rigorous 99.9999% confidence intervals for critical probabilities.
  • To analyze both site and bond percolation on 11 Archimedean lattices.
  • To employ a method focused on simplicity and verification.

Main Methods:

  • Utilizing the Balister, Bollobás, and Walters method.
  • Performing computer calculations with an emphasis on ease of verification.

Related Experiment Videos

  • Generating high-confidence intervals for critical percolation probabilities.

    • Main Results:

    • Achieved rigorous 99.9999% confidence intervals for critical probabilities.
    • Successfully analyzed 11 Archimedean lattices for both site and bond percolation.
    • Obtained narrow confidence intervals with a maximum width of 0.0005.

    Conclusions:

    • The study successfully established precise confidence intervals for percolation critical probabilities.
    • The chosen computational approach ensures simplicity and verifiability.
    • Results provide a reliable foundation for further research in percolation theory on Archimedean lattices.