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Loop-erased random walk on a percolation cluster is compatible with Schramm-Loewner evolution.

E Daryaei1

  • 1Department of Physics, Faculty of Basic Sciences, University of Neyshabur, P.O. Box 91136-899, Neyshabur, Iran.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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We studied loop-erased random walks on percolation clusters. Results show these walks are described by Schramm-Loewner evolution, revealing new insights into fractal geometry and conformal invariance.

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Area of Science:

  • Statistical Physics
  • Probability Theory
  • Complex Systems

Background:

  • Loop-erased random walks (LERW) are fundamental models in statistical physics.
  • Percolation theory describes the behavior of connected clusters in random systems.
  • Schramm-Loewner evolution (SLE) is a powerful tool for analyzing random curves in 2D.

Purpose of the Study:

  • To investigate the scaling limit of planar LERW on percolation clusters.
  • To determine the applicability of SLE to LERW on percolation clusters for various occupation probabilities.
  • To explore the crossover behavior from Euclidean to fractal geometry in LERW.

Main Methods:

  • Numerical simulations of planar LERW on percolation clusters.
  • Application of geometrical tests to analyze LERW paths.
  • Analysis of winding angles and crossover exponents.

Main Results:

  • LERW on percolation clusters for p>p(c) is described by SLE with a diffusivity coefficient kappa near that of Euclidean LERW.
  • LERW on critical incipient percolation clusters is compatible with SLE, but with a distinct kappa value (1.732±0.016).
  • This kappa value falls outside the standard 2≤κ≤8 range, indicating unique behavior.

Conclusions:

  • The study confirms SLE's applicability to LERW on percolation clusters, extending beyond standard ranges.
  • A crossover phenomenon from Euclidean to fractal geometry was observed and quantified.
  • Findings contribute to understanding conformal invariance in disordered and fractal systems.