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Bond percolation between k separated points on a square lattice.

S S Manna1, Robert M Ziff2

  • 1Satyendra Nath Bose National Centre for Basic Sciences, Block JD, Sector III, Salt Lake, Kolkata 700106, India.

Physical Review. E
|July 22, 2020
PubMed
Summary

New percolation thresholds, p[over ¯]_{ck}, were identified for k-point or center-to-boundary connections in lattice systems. These thresholds are above the standard percolation threshold and indicate specific supercritical states, remaining consistent even with random point selection.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Network Science

Background:

  • Percolation theory studies the formation of connected clusters in random networks.
  • Standard percolation thresholds (p_c) define the point at which a giant cluster emerges.
  • Understanding connectivity in systems with specific connection requirements is crucial for various applications.

Purpose of the Study:

  • To define and investigate new percolation thresholds (p[over ¯]_{ck}) for simultaneous k-point connections or center-to-boundary connections.
  • To determine if these new thresholds differ from the standard percolation threshold (p_c).
  • To analyze the behavior and scaling properties of these novel thresholds in large systems.

Main Methods:

  • Simulations of bond percolation on a square lattice.
  • Calculation of average thresholds (p[over ¯]_{ck}) for specific connection events.
  • Numerical measurement of the probability of a point connecting to the infinite cluster (P_∞(p)).
  • Analysis of finite-size corrections and three-point correlations.

Main Results:

  • New thresholds p[over ¯]_{ck} were found to be above the standard percolation threshold (p_c = 0.5).
  • Specific values for p[over ¯]_{c1}, p[over ¯]_{c2}, p[over ¯]_{c3}, and p[over ¯]_{c4} were numerically determined for large systems.
  • Finite-size corrections were shown to scale as L^{-1/ν_{k}}, with ν_{k} dependent on critical exponents.
  • Percolation thresholds remain invariant under random selection of the k points.

Conclusions:

  • The identified percolation thresholds (p[over ¯]_{ck}) represent distinct supercritical states.
  • The broadness of the p_{ck} distribution indicates these are not sharp thresholds.
  • The scaling behavior provides insights into the critical phenomena associated with these new connection types.