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From explosive to infinite-order transitions on a hyperbolic network.

Vijay Singh1, C T Brunson1, Stefan Boettcher1

  • 1Department of Physics, Emory University, Atlanta, Georgia 30322, USA.

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|December 11, 2014
PubMed
Summary
This summary is machine-generated.

We analyzed phase transitions in hyperbolic networks using the q-state Potts model. Results show discontinuous transitions are common for q<2, while continuous transitions appear for q=2 and Berezinskii-Kosterlitz-Thouless type for q>2.

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Area of Science:

  • Statistical physics
  • Network science
  • Phase transitions

Background:

  • Hyperbolic networks exhibit complex behaviors, including discontinuous phase transitions like explosive percolation.
  • The q-state Potts model is a fundamental tool for studying magnetic and other phase transitions.

Purpose of the Study:

  • To analyze phase transitions in recursively designed hyperbolic networks.
  • To investigate the influence of the parameter q on the nature of these transitions.

Main Methods:

  • Solving the q-state Potts model using analytic continuation for noninteger q.
  • Employing the real-space renormalization group technique.

Main Results:

  • Exact expressions were derived for a one-parameter family of models, detailing the transformation of phase transitions.
  • Discontinuous transitions were found to be generic for q<2, encompassing percolation.
  • A continuous ferromagnetic transition was recovered for the Ising model (q=2).
  • For q>2, transitions transformed into a Berezinskii-Kosterlitz-Thouless type.

Conclusions:

  • The parameter q dramatically alters the nature of phase transitions in these hyperbolic networks.
  • The study provides a unified framework for understanding diverse transition types based on the Potts model parameter q.