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Percolation model with continuously varying exponents.

R F S Andrade1, H J Herrmann

  • 1Instituto de Física, Universidade Federal da Bahia, 40210-210 Salvador, Brazil.

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Summary
This summary is machine-generated.

This study introduces a modified percolation model on a diamond hierarchical lattice, showing critical exponents continuously change with an erasing probability. This model is equivalent to a specific limit of the Potts model.

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Area of Science:

  • Statistical Physics
  • Complex Systems
  • Materials Science

Background:

  • Percolation theory studies the formation of connected clusters in random systems.
  • The diamond hierarchical lattice (DHL) exhibits exact scale invariance and renormalization properties.
  • Recent interest exists in identifying new universality classes for percolation models.

Purpose of the Study:

  • To analyze a modified percolation model on the DHL with an erasing probability.
  • To investigate the impact of erasing specific connected structures on percolation transitions.
  • To explore the relationship between critical exponents and the erasing probability.

Main Methods:

  • Utilizing the exact scale invariance and renormalization properties of the DHL.
  • Deriving recurrence maps to obtain analytical expressions for critical exponents.
  • Performing numerical simulations in the limit of very large lattices.

Main Results:

  • The critical exponents ν and β vary continuously with the erasing probability.
  • An erasing probability can be chosen to yield ν=∞, similar to vortex-formation phase transitions.
  • The percolation transition is continuous (β>0), with β tunable to arbitrarily small values.

Conclusions:

  • The modified percolation model exhibits tunable critical behavior.
  • The model is shown to be equivalent to the Q→1 limit of a Potts model with long-range interactions on the DHL.
  • This work contributes to understanding universality classes in percolation theory.