We consider a percolation configuration on a general lattice in
which edges are included independently with probability p. We study the
rigidity properties of the resulting configuration, in the sense of generic
rigidity in d dimensions. We give a mathematically rigorous treatment of
the problem, starting with a definition of an infinite rigid component. We
prove that, for a broad class of lattices, there exists an infinite rigid
component for some p strictly below unity. For the particular case of
two-dimensional rigidity on the two-dimensional triangular lattice, we prove
first that the critical probability for rigidity percolation lies strictly
above that for connectivity percolation and second that the infinite rigid
component (when it exists) is unique for all but countably many values of
p. We conjecture that this uniqueness in fact holds for all p.
Some of our arguments could be applied to two-dimensional lattices in more
generality.
@article{1028903458,
author = {Holroyd, Alexander E.},
title = {Existence and uniqueness of infinite components in generic
rigidity percolation},
journal = {Ann. Appl. Probab.},
volume = {8},
number = {1},
year = {1998},
pages = { 944-973},
language = {en},
url = {http://dml.mathdoc.fr/item/1028903458}
}
Holroyd, Alexander E. Existence and uniqueness of infinite components in generic
rigidity percolation. Ann. Appl. Probab., Tome 8 (1998) no. 1, pp. 944-973. http://gdmltest.u-ga.fr/item/1028903458/