Hierarchical models of rigidity percolation

Two exactly soluble models of rigidity percolation.

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 hierar...

Rigidity of the interface in percolation and random-cluster models

We study conditioned random-cluster measures with edge-parameter p and cluster-weighting factor q satisfying q ≥ 1. The conditioning corresponds to mixed boundary conditions for a spin model. Interfaces may be defined in the sense of Dobrushin, and these are proved to be ‘rigid’ in the thermodynamic limit, in three dimensions and for sufficiently large values of p. This implies the existence of...

Comparison of Connectivity and Rigidity Percolation

Rigidity and percolation of center vortices

Effective action of center vortices in SU (2) lattice gauge theory is investigated by studying the correlation between the action density on their worldsheets and their geometric properties. It turns out that center vortices are rigid, however, their dynamics is more complicated than that of rigid random surfaces, since some coupling constants have nonstandard scaling dimensions. As a result, t...

Generic rigidity percolation in two dimensions.

We study rigidity percolation for random central-force networks on the bondand site-diluted generic triangular lattice. Here, each site location is randomly displaced from the perfect lattice, removing any special symmetries. Using the pebble game algorithm, the total number of floppy modes are counted exactly, and exhibit a cusp singularity in the second derivative at the transition from a rig...