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Area of Science:

  • Statistical physics
  • Complex systems
  • Fractal geometry

Background:

  • Shortest paths in critical percolation clusters are complex to analyze.
  • Schramm-Loewner evolution (SLE) describes random fractal curves.
  • A connection between these two areas is not well-established.

Purpose of the Study:

  • To investigate the statistical properties of shortest paths on critical percolation clusters.
  • To determine if these properties align with predictions from Schramm-Loewner evolution (SLE).
  • To explore the implications of this connection for theoretical modeling and curve generation.

Main Methods:

  • Numerical simulations of critical percolation clusters.
  • Analysis of shortest path properties: winding angle, left passage probability, and driving function.
  • Comparison of numerical results with theoretical predictions for SLE curves.

Main Results:

  • Statistical properties of shortest paths are consistent with SLE curves for κ = 1.04 ± 0.02.
  • Fractal dimension of shortest paths matches SLE predictions.
  • SLE provides a framework for generating statistically equivalent shortest path curves.

Conclusions:

  • The study establishes a significant link between shortest paths on percolation clusters and SLE.
  • This connection allows for the application of SLE's theoretical framework to shortest path analysis.
  • It opens avenues for generating shortest path-like curves using Brownian motion and exploring conformal invariance properties.