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.
%0 Journal Article
%1 Ellenbroek2011Rigidity
%A Ellenbroek, Wouter G.
%A Mao, Xiaoming
%D 2011
%J EPL (Europhysics Letters)
%K rigidity percolation lattice-models
%P 54002+
%R 10.1209/0295-5075/96/54002
%T Rigidity percolation on the square lattice
%U http://dx.doi.org/10.1209/0295-5075/96/54002
%X 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.
@article{Ellenbroek2011Rigidity,
abstract = {{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.}},
added-at = {2019-06-10T14:53:09.000+0200},
author = {Ellenbroek, Wouter G. and Mao, Xiaoming},
biburl = {https://www.bibsonomy.org/bibtex/2fe10a01b4a3f5e00d03797d6a922f593/nonancourt},
citeulike-article-id = {10374409},
citeulike-linkout-0 = {http://dx.doi.org/10.1209/0295-5075/96/54002},
day = 01,
doi = {10.1209/0295-5075/96/54002},
interhash = {df3d70c3dcfceb0c8c5602953306c4fd},
intrahash = {fe10a01b4a3f5e00d03797d6a922f593},
journal = {EPL (Europhysics Letters)},
keywords = {rigidity percolation lattice-models},
month = dec,
pages = {54002+},
posted-at = {2012-02-21 17:45:21},
priority = {2},
timestamp = {2019-08-01T15:36:09.000+0200},
title = {{Rigidity percolation on the square lattice}},
url = {http://dx.doi.org/10.1209/0295-5075/96/54002},
year = 2011
}