Friction Games: Towards exact constraint counting in frictional packings

To count constraints in frictional packings more exactly than generalized isostaticity, we implement a (3,3) pebble game and construct rigid clusters on a model of rigidity percolation with friction. Specifically, the model is double and single bonds randomly added to a honeycomb lattice with additional next nearest neighbor bonds. The double bonds represent frictional contacts and the single, contacts at the Coulomb threshold. We find a second order transition with a fractal spanning rigid cluster and numerically determine related exponents, which are close to rigidity percolation without friction. Given the closeness in numerical values of the exponents between rigidity (with and without friction) and connectivity percolation, we unveil a new minimal rigidity percolation model whose exponents near/at the transition are in the same universality as connectivity percolation. We also address shortcomings of the (3,3) pebble game in determining constraints and develop additional methods to accommodate these shortcomings.