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Depth-first search (DFS) paths reveal fractal properties in statistical physics models. This geometric probe offers new insights into critical phenomena beyond traditional methods.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Geometric Probes

Background:

  • Fractal properties are crucial for understanding critical phenomena.
  • Traditional observables may not fully capture complex system behaviors.

Purpose of the Study:

  • To investigate the fractal characteristics of depth-first search (DFS) paths.
  • To explore DFS paths in critical configurations of statistical physics models.
  • To assess DFS as a novel geometric probe for critical phenomena.

Main Methods:

  • Analysis of DFS paths in the 2D O(n) loop model and bond percolation.
  • Examination across various dimensions (d=2 to 6) and critical regimes.
  • Calculation of fractal dimensions for DFS paths.

Main Results:

  • DFS paths exhibit consistent fractal dimension d_{DFS}=1+g/8 in the O(n) loop model.
  • Non-trivial fractal scaling of DFS paths observed in bond percolation across dimensions.
  • DFS paths show fractal behavior in 2D lattices and become space-filling in higher dimensions.

Conclusions:

  • Depth-first search (DFS) provides a robust geometric probe for critical phenomena.
  • DFS path analysis offers insights beyond traditional observables in statistical physics.
  • The fractal nature of DFS paths is a significant finding for complex systems research.