title: Two exactly soluble models of rigidity percolation.
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.
authors.affiliation: Rudolf Peierls Centre for Theoretical Physics, University of Oxford, , 1 Keble Road, Oxford OX1 3NP, UK.
authors.forename: M F
authors.initials: MF
authors.lastname: Thorpe
authors.forename: R B
authors.initials: RB
authors.lastname: Stinchcombe
keywords: hierarchical; lattice; percolation; renormalization; rigidity; tree