Rigidity percolation on aperiodic lattices

We studied the rigidity percolation (RP) model for two-dimensional aperiodic (quasi-crystal) lattices. The RP thresholds (for bond dilution) were obtained for several aperiodic lattices via computer simulation using the “pebble game” algorithm. It was found that the (two rhombi) Penrose lattice is always floppy in view of the RP model. The same was found for the Ammann’s octagonal lattice and the Socolar’s dodecagonal lattice. In order to impose the percolation transition we used the so-called “ferro” modification of these aperiodic tilings. We studied as well the “pinwheel” tiling which has infinitely many orientations of edges. The obtained estimates for the modified Penrose, Ammann and Socolar lattices are respectively: pcP=0.836±0.002, pcA=0.769±0.002, pcS=0.938±0.001. The bond RP threshold of the pinwheel tiling was estimated to pc=0.69±0.01. It was found that these results are very close to the Maxwell (the mean-field like) approximation for them