Article,

Correlated percolation and tricriticality

, and .
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.

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