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1/d expansion for k-core percolation.

A B Harris1, J M Schwarz

  • 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 31, 2005
PubMed
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K-core percolation studies systems needing k connections for clustering. This research shows the hybrid transition of k-core percolation persists in finite dimensions, suggesting its survival in complex systems.

Area of Science:

  • Statistical Physics
  • Complex Systems Theory

Background:

  • K-core percolation models systems where constituents need a minimum number (k) of connections for clustering.
  • Phenomena include orientational ordering, network rigidity, and dynamical arrest in liquids.
  • Mean-field k-core percolation (k>=3) exhibits a unique hybrid transition (continuous and discontinuous).

Purpose of the Study:

  • To investigate if the hybrid k-core percolation transition persists in finite dimensions.
  • To analyze the behavior of k-core percolation on a d-dimensional hypercubic lattice.

Main Methods:

  • Developed a 1/d expansion for k-core percolation.
  • Analyzed the k-core percolation transition on d-dimensional hypercubic lattices.

Main Results:

Related Experiment Videos

  • The study shows that to order 1/d3, the order parameter and susceptibility singularities occur at the same occupation probability.
  • This indicates the hybrid nature of the mean-field k-core transition is preserved.

Conclusions:

  • The unusual hybrid transition of k-core percolation survives in finite dimensions.
  • This finding has implications for understanding complex systems with connectivity thresholds.