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Exactly solvable percolation problems.

Fabian Coupette1, Tanja Schilling1

  • 1Institute of Physics, University of Freiburg, Hermann-Herder-Straße 3, 79104 Freiburg, Germany.

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|May 20, 2022
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Summary
This summary is machine-generated.

We introduce a novel percolation criterion applicable to diverse systems. This method reveals that many complex networks, including random graphs and lattices, behave like treelike structures, simplifying threshold calculations.

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Area of Science:

  • Statistical Physics
  • Network Science
  • Materials Science

Background:

  • Percolation theory is crucial for understanding phase transitions in disordered systems.
  • Existing methods often struggle with complex network topologies and continuum problems.
  • A unified approach for diverse percolation problems is needed.

Purpose of the Study:

  • To develop a simple, universally applicable percolation criterion.
  • To demonstrate the treelike nature of various complex systems.
  • To enable precise calculation of percolation thresholds and lattice generation.

Main Methods:

  • Decomposing systems into a hierarchy of neighborhoods.
  • Expressing percolation problems as branching processes.
  • Applying the criterion to random graphs, lattices, and continuum models.

Main Results:

  • The criterion accurately predicts exact percolation thresholds for numerous problems.
  • It confirms that diverse systems, including lattices and networks, are effectively treelike.
  • A method for generating planar lattices with specific thresholds is introduced.

Conclusions:

  • The proposed criterion offers a powerful and general framework for percolation studies.
  • The treelike nature of many systems simplifies complex network analysis.
  • This work facilitates the design of materials with tailored properties.