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Critical polynomials in the nonplanar and continuum percolation models.

Wenhui Xu1,2, Junfeng Wang3, Hao Hu1

  • 1School of Physics and Materials Science, Anhui University, Hefei, Anhui 230601, China.

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Summary

The critical polynomial method precisely determines percolation thresholds in lattice and continuum models. This approach minimizes finite-size corrections, enabling high-precision calculations for complex systems.

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Area of Science:

  • Statistical Mechanics
  • Computational Physics

Background:

  • Percolation models are fundamental in statistical mechanics for understanding connectivity.
  • Exact thresholds are crucial but challenging to compute, especially for complex systems.

Purpose of the Study:

  • To introduce and validate the critical polynomial (P_B) as a precise tool for determining percolation thresholds.
  • To apply the P_B method to nonplanar lattice and continuum percolation models.

Main Methods:

  • Utilized the critical polynomial P_B(p,L) for threshold determination.
  • Employed Monte Carlo simulations and finite-size scaling analysis.
  • Assessed the universality of P_B across different lattice types.

Main Results:

  • Achieved high-precision thresholds for equivalent-neighbor percolation, confirming asymptotic behavior.
  • Demonstrated minimal finite-size corrections for P_B in continuum percolation.
  • Estimated the critical density for disk percolation with high accuracy.

Conclusions:

  • The critical polynomial method offers superior precision and reduced finite-size effects compared to traditional methods.
  • P_B is a powerful and versatile tool for studying percolation in diverse systems.
  • This method advances the precise calculation of critical phenomena in statistical mechanics.