Rigidity percolation on the square lattice
Department of Physics and Astronomy, University of Pennsylvania - Philadelphia, PA 19104, USA
2 Department of Applied Physics and Institute for Complex Molecular Systems, Eindhoven University of Technology P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands, EU
Accepted: 7 October 2011
The square lattice with central forces between nearest neighbors is isostatic with a subextensive number of floppy modes. It can be made rigid by the random addition of next-nearest-neighbor bonds. This constitutes a rigidity percolation transition which we study analytically by mapping it to a connectivity problem of two-colored random graphs. We derive an exact recurrence equation for the probability of having a rigid percolating cluster and solve it in the infinite volume limit. From this solution we obtain the rigidity threshold as a function of system size, and find that, in the thermodynamic limit, there is a mixed first-order–second-order rigidity percolation transition at the isostatic point.
PACS: 46.65.+g – Random phenomena and media / 02.10.Ox – Combinatorics; graph theory / 64.60.ah – Percolation
© EPLA, 2011