Two exactly soluble models of rigidity percolation

Michael Thorpe, R. B. Stinchcombe

Research output: Contribution to journalArticle

8 Scopus citations

Abstract

We summarize results for two exactly soluble classes of bond-diluted models for rigidity percolation, which can serve as a benchmark for numerical and approximate methods. For bond dilution problems involving rigidity, the number of floppy modes F plays the role of a free energy. Both models involve pathological lattices with two-dimensional vector displacements. The first model involves hierarchical lattices where renormalization group calculations can be used to give exact solutions. Algebraic scaling transformations produce a transition of the second order, with an unstable critical point and associated scaling laws at a mean coordination 〈r〉 = 4.41, which is above the 'mean field' value 〈r〉 = 4 predicted by Maxwell constraint counting. The order parameter exponent associated with the spanning rigid cluster geometry is β =0.0775 and that associated with the divergence of the correlation length and the anomalous lattice dimension d is dν =3.533. The second model involves Bethe lattices where the rigidity transition is massively first order by a mean coordination 〈r〉 = 3.94 slightly below that predicted by Maxwell constraint counting. We show how a Maxwell equal area construction can be used to locate the first-order transition and how this result agrees with simulation results on larger random-bond lattices using the pebble game algorithm.

Original languageEnglish (US)
Article number20120038
JournalPhilosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
Volume372
Issue number2008
DOIs
StatePublished - Feb 13 2014

Keywords

  • Hierarchical
  • Lattice
  • Percolation
  • Renormalization
  • Rigidity
  • Tree

ASJC Scopus subject areas

  • Mathematics(all)
  • Engineering(all)
  • Physics and Astronomy(all)

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