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One-dimensional long-range percolation: A numerical study.

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This study investigates bond percolation on a 1D chain with long-range interactions. A new Monte Carlo algorithm confirms mean-field predictions for critical exponents, showing no correlation effects.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Network Science

Background:

  • Percolation theory studies the formation of connected clusters in random systems.
  • Long-range interactions in one-dimensional systems present unique theoretical challenges.
  • Understanding critical phenomena is crucial for diverse fields like materials science and epidemiology.

Purpose of the Study:

  • To analyze bond percolation on a 1D chain with power-law bond probability.
  • To introduce and validate an efficient order-N Monte Carlo algorithm.
  • To determine critical values and exponents for varying system parameters.

Main Methods:

  • Development and application of an order-N Monte Carlo algorithm.
  • Analysis of bond percolation on a one-dimensional chain with power-law bond probability C/r^{d+σ}.
  • Comparison of numerical results with theoretical predictions (mean-field, ɛ-expansion).

Main Results:

  • The critical value C_c for percolation was determined as a function of σ.
  • Critical exponents were reported for 0 < σ < 1.
  • Numerical precision confirmed agreement with mean-field anomalous dimension η=2-σ, indicating no correlation effects.

Conclusions:

  • The developed Monte Carlo algorithm is efficient and accurate for studying long-range percolation.
  • The study validates mean-field predictions for critical exponents in this specific 1D system.
  • The algorithm's formulation for general graphs suggests broad applicability in complex systems research.