Generic Rigidity Percolation: The Pebble Game

Abstract

The percolation of rigidity in 2D central-force networks with no special symmetries (generic networks) has been studied using a new combinatorial algorithm. We count the exact number of floppy modes, uniquely decompose the network into rigid clusters, and determine all overconstrained regions. With this information we have found that, for the generic triangular lattice with random bond dilution, the transition from rigid to floppy occurs at ${p}_{\mathrm{cen}}=0.6602\ifmmode\pm\else\textpm\fi{}0.0003$ and the critical exponents include $\ensuremath{\nu}=1.21\ifmmode\pm\else\textpm\fi{}0.06$ and $\ensuremath{\beta}=0.18\ifmmode\pm\else\textpm\fi{}0.02$.