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Short-range correlations in percolation at criticality.

Hao Hu1, Henk W J Blöte2, Robert M Ziff3

  • 1Hefei National Laboratory for Physical Sciences at Microscale, Department of Modern Physics, University of Science and Technology of China, Hefei 230027, China.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|November 7, 2014
PubMed
Summary

We calculated critical connectivity values for bond percolation on square, honeycomb, and triangular lattices. Monte Carlo simulations confirmed these results, providing insights into lattice connectivity and scaling behavior.

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Area of Science:

  • Statistical Physics
  • Condensed Matter Physics
  • Network Science

Background:

  • Percolation theory studies the formation of connected clusters in random networks.
  • Understanding critical connectivity is crucial for characterizing phase transitions in various systems.
  • Previous work has established theoretical frameworks for lattice connectivity but requires numerical validation.

Purpose of the Study:

  • To derive and confirm critical nearest-neighbor connectivity values for different lattice types.
  • To numerically determine critical next-nearest-neighbor connectivity and surface connectivity.
  • To investigate the finite-size scaling behavior of connectivity and related quantities at criticality.

Main Methods:

  • Analytical derivation of critical connectivity for square, honeycomb, and triangular lattices.
  • Monte Carlo simulations to confirm derived critical connectivity values.
  • Numerical determination of next-nearest-neighbor and surface connectivity.
  • Analysis of finite-size scaling laws for connectivity and specific-heat-like quantities.

Main Results:

  • Exact critical nearest-neighbor connectivity values derived for square (3/4), honeycomb, and triangular lattices.
  • Numerical confirmation of next-nearest-neighbor connectivity (11/16) and surface connectivity (5/8) on the square lattice.
  • Demonstration of connectivity scaling as ~L^(yt-d) and specific-heat-like scaling as ~L^(2yt-d)ln(L/L0) at criticality.

Conclusions:

  • The study provides precise critical connectivity values for key lattice structures.
  • Numerical results validate theoretical conjectures and offer new insights into surface phenomena.
  • The observed logarithmic scaling factor at criticality is explained within a recent theoretical framework.