Correlated percolation and tricriticality
Physical Review E, (Dec 5, 2012)DOI: 10.1103/PhysRevE.86.061131
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
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.