Rigidity percolation on aperiodic lattices

TL;DR

This study investigates rigidity percolation thresholds in various aperiodic lattices, revealing that some are inherently floppy and providing precise bond percolation thresholds for different tilings, with results close to mean-field predictions.

Contribution

The paper presents the first systematic simulation-based analysis of rigidity percolation thresholds on multiple aperiodic lattices, including Penrose, Ammann, Socolar, and pinwheel tilings.

Findings

Penrose lattice is always floppy under RP.

Estimated bond percolation thresholds for Penrose, Ammann, Socolar, and pinwheel tilings.

Results closely match Maxwell mean-field approximation.

Abstract

We studied the rigidity percolation (RP) model for 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 tiling and the Socolar's dodecagonal tiling. In order to impose the percolation transition we used so c. "ferro" modification of these aperiodic tilings. We studied as well the "pinwheel" tiling which has "infinitely-fold" orientational symmetry. The obtained estimates for the modified Penrose, Ammann and Socolar lattices are respectively: $p_{cP} =0.836\pm 0.002$, $p_{cA} = 0.769\pm0.002$, $p_{cS} = 0.938\pm0.001$. The bond RP threshold of the pinwheel tiling was estimated to $p_c = 0.69\pm0.01$. It was found that these results are very close to the Maxwell (the mean-field like) approximation for them.