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Constraint percolation on hyperbolic lattices.

Jorge H Lopez1, J M Schwarz2,3

  • 1Department of Civil Engineering, Universidad Mariana, Pasto 520002, Colombia.

Physical Review. E
|January 20, 2018
PubMed
Summary

This study investigates percolation models on hyperbolic lattices, finding that k-core percolation mirrors ordinary percolation, while force-balance percolation shows a discontinuous transition. Improved numerical methods are discussed for hyperbolic lattice analysis.

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Area of Science:

  • Statistical Physics
  • Complex Networks
  • Materials Science

Background:

  • Hyperbolic lattices bridge finite and Bethe lattices, exhibiting unique properties like multiple phase transitions in ordinary percolation.
  • Understanding percolation on these complex structures is crucial for various scientific domains.

Purpose of the Study:

  • To investigate four constraint percolation models on hyperbolic lattices: k-core (k=1,2,3) and force-balance.
  • To compare the behaviors of these models and analyze their phase transitions.
  • To provide rigorous footing for numerical findings and discuss improved computational methods.

Main Methods:

  • Numerical simulations of k-core percolation (k=1,2,3) and force-balance percolation on hyperbolic tessellations.
  • Comparative analysis of simulation data to identify distinct transition behaviors.
  • Mathematical proof for the existence of a critical probability in force-balance percolation.

Main Results:

  • K-core percolation models, including k=3, exhibit behavior analogous to ordinary percolation.
  • Force-balance percolation demonstrates a discontinuous phase transition.
  • Proof of a critical probability less than unity for force-balance percolation on certain hyperbolic lattices was established.

Conclusions:

  • The study differentiates percolation behaviors on hyperbolic lattices, highlighting distinct characteristics of k-core and force-balance models.
  • Rigorous mathematical backing was provided for force-balance percolation findings.
  • The research paves the way for enhanced numerical techniques in studying hyperbolic lattice systems.