### 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 language | English (US) |
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Article number | 20120038 |

Journal | Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences |

Volume | 372 |

Issue number | 2008 |

DOIs | |

State | Published - 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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## Cite this

*Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences*,

*372*(2008), [20120038]. https://doi.org/10.1098/rsta.2012.0038