Generic rigidity percolation in two dimensions

Abstract

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 rigid to a floppy structure. The critical thresholds for bond and site dilution are found to be 0.66020±0.0003 and 0.69755±0.0003, respectively. The network is decomposed into unique rigid clusters, and we apply the usual percolation scaling theory. From finite size scaling, we find that the generic rigidity percolation transition is second order, but in a different universality class from connectivity percolation, with the exponents α=-0.48±0.05, β=0.175±0.02, and ν=1.21±0.06. The fractal dimension of the spanning rigid clusters and the spanning stressed regions at the critical threshold are found to be ${\mathit{d}}_{\mathit{f}}$=1.86±0.02 and ${\mathit{d}}_{\mathit{BB}}$=1.80±0.03, respectively. © 1996 The American Physical Society.