The recent proliferation of correlated percolation models---models where the
addition of edges/vertices is no longer independent of other
edges/vertices---has been motivated by the quest to find discontinuous
percolation transitions. The leader in this proliferation is what is known as
explosive percolation. A recent proof demonstrates that a large class of
explosive percolation-type models does not, in fact, exhibit a discontinuous
transitionO. Riordan and L. Warnke, Science, 333, 322 (2011). We, on
the other hand, discuss several correlated percolation models, the \$k\$-core
model on random graphs, and the spiral and counter-balance models in
two-dimensions, all exhibiting discontinuous transitions in an effort to
identify the needed ingredients for such a transition. We then construct
mixtures of these models to interpolate between a continuous transition and a
discontinuous transition to search for a tricritical point. Using a powerful
rate equation approach, we demonstrate that a mixture of \$k=2\$-core and
\$k=3\$-core vertices on the random graph exhibits a tricritical point. However,
for a mixture of \$k\$-core and counter-balance vertices, heuristic arguments and
numerics suggest that there is a line of continuous transitions as the fraction
of counter-balance vertices is increased from zero with the line ending at a
discontinuous transition only when all vertices are counter-balance. Our
results may have potential implications for glassy systems and a recent
experiment on shearing a system of frictional particles to induce what is known
as jamming.
%0 Journal Article
%1 Cao2012Correlated
%A Cao, L.
%A Schwarz, J. M.
%D 2012
%J Physical Review E
%K percolation tricritical-points jamming k-core
%N 6
%R 10.1103/PhysRevE.86.061131
%T Correlated percolation and tricriticality
%U http://dx.doi.org/10.1103/PhysRevE.86.061131
%V 86
%X The recent proliferation of correlated percolation models---models where the
addition of edges/vertices is no longer independent of other
edges/vertices---has been motivated by the quest to find discontinuous
percolation transitions. The leader in this proliferation is what is known as
explosive percolation. A recent proof demonstrates that a large class of
explosive percolation-type models does not, in fact, exhibit a discontinuous
transitionO. Riordan and L. Warnke, Science, 333, 322 (2011). We, on
the other hand, discuss several correlated percolation models, the \$k\$-core
model on random graphs, and the spiral and counter-balance models in
two-dimensions, all exhibiting discontinuous transitions in an effort to
identify the needed ingredients for such a transition. We then construct
mixtures of these models to interpolate between a continuous transition and a
discontinuous transition to search for a tricritical point. Using a powerful
rate equation approach, we demonstrate that a mixture of \$k=2\$-core and
\$k=3\$-core vertices on the random graph exhibits a tricritical point. However,
for a mixture of \$k\$-core and counter-balance vertices, heuristic arguments and
numerics suggest that there is a line of continuous transitions as the fraction
of counter-balance vertices is increased from zero with the line ending at a
discontinuous transition only when all vertices are counter-balance. Our
results may have potential implications for glassy systems and a recent
experiment on shearing a system of frictional particles to induce what is known
as jamming.
@article{Cao2012Correlated,
abstract = {The recent proliferation of correlated percolation models---models where the
addition of edges/vertices is no longer independent of other
edges/vertices---has been motivated by the quest to find discontinuous
percolation transitions. The leader in this proliferation is what is known as
explosive percolation. A recent proof demonstrates that a large class of
explosive percolation-type models does not, in fact, exhibit a discontinuous
transition[O. Riordan and L. Warnke, Science, {\bf 333}, 322 (2011)]. We, on
the other hand, discuss several correlated percolation models, the \$k\$-core
model on random graphs, and the spiral and counter-balance models in
two-dimensions, all exhibiting discontinuous transitions in an effort to
identify the needed ingredients for such a transition. We then construct
mixtures of these models to interpolate between a continuous transition and a
discontinuous transition to search for a tricritical point. Using a powerful
rate equation approach, we demonstrate that a mixture of \$k=2\$-core and
\$k=3\$-core vertices on the random graph exhibits a tricritical point. However,
for a mixture of \$k\$-core and counter-balance vertices, heuristic arguments and
numerics suggest that there is a line of continuous transitions as the fraction
of counter-balance vertices is increased from zero with the line ending at a
discontinuous transition only when all vertices are counter-balance. Our
results may have potential implications for glassy systems and a recent
experiment on shearing a system of frictional particles to induce what is known
as jamming.},
added-at = {2019-06-10T14:53:09.000+0200},
archiveprefix = {arXiv},
author = {Cao, L. and Schwarz, J. M.},
biburl = {https://www.bibsonomy.org/bibtex/2f3aad1bff81dce6ba32f4cc7ffa78088/nonancourt},
citeulike-article-id = {10750793},
citeulike-linkout-0 = {http://dx.doi.org/10.1103/PhysRevE.86.061131},
citeulike-linkout-1 = {http://arxiv.org/abs/1206.1028},
citeulike-linkout-2 = {http://arxiv.org/pdf/1206.1028},
day = 5,
doi = {10.1103/PhysRevE.86.061131},
eprint = {1206.1028},
interhash = {747dba58389f0cb45a341e1a110e0347},
intrahash = {f3aad1bff81dce6ba32f4cc7ffa78088},
issn = {1550-2376},
journal = {Physical Review E},
keywords = {percolation tricritical-points jamming k-core},
month = dec,
number = 6,
posted-at = {2012-06-06 09:40:13},
priority = {2},
timestamp = {2019-08-01T16:09:09.000+0200},
title = {{Correlated percolation and tricriticality}},
url = {http://dx.doi.org/10.1103/PhysRevE.86.061131},
volume = 86,
year = 2012
}