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Related Experiment Videos

Convergence of threshold estimates for two-dimensional percolation.

R M Ziff1, M E J Newman

  • 1Michigan Center for Theoretical Physics and Department of Chemical Engineering, University of Michigan, Ann Arbor, Michigan 48109-2136, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|September 21, 2002
PubMed
Summary

Researchers studied percolation threshold convergence using a new algorithm. The average-probability estimate showed nontrivial scaling, while median and cell-to-cell estimates converged faster without this scaling dependence.

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Area of Science:

  • Statistical Physics
  • Computational Physics
  • Materials Science

Background:

  • Percolation theory is crucial for understanding phase transitions in disordered systems.
  • Accurate estimation of the percolation threshold is essential for various scientific and engineering applications.
  • Previous studies have explored different methods for estimating percolation thresholds, but convergence behavior requires further investigation.

Purpose of the Study:

  • To investigate the system size convergence of percolation threshold estimates.
  • To analyze the behavior of different estimation methods, including average-probability, median, and cell-to-cell estimates.
  • To determine the correction-to-scaling exponent for the average-probability estimate in site percolation on a square lattice.

Main Methods:

Related Experiment Videos

  • Simulation of site percolation on a square lattice using a recently introduced algorithm for microcanonical (fixed-occupancy) samples.
  • Analysis of the convergence of various percolation threshold estimates with increasing system size.
  • Measurement of the correction-to-scaling exponent for the average-probability estimate.

Main Results:

  • The convergence of the average-probability estimate is described by a nontrivial correction-to-scaling exponent (0.90+/-0.02).
  • The median and cell-to-cell estimates exhibit convergence independent of this exponent.
  • These alternative estimates show slightly faster convergence with a trivial analytic leading exponent.

Conclusions:

  • The study provides valuable insights into the finite-size scaling behavior of percolation threshold estimates.
  • Different estimation methods display distinct convergence properties, impacting their reliability and efficiency.
  • The findings contribute to a deeper understanding of critical phenomena in disordered systems and inform the selection of appropriate simulation and analysis techniques.