### Abstract

The recent proliferation of correlated percolation models - models where the addition of edges and/or vertices is no longer independent of other edges and/or 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. Here, we discuss two lesser known correlated percolation models - the k≥3-core model on random graphs and the counter-balance model in two-dimensions - both exhibiting discontinuous transitions. To search for tricriticality, we construct mixtures of these models with other percolation models exhibiting the more typical continuous transition. 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 in two dimensions, as the fraction of counter-balance vertices is increased, numerics and heuristic arguments suggest that there is a line of continuous transitions with the line ending at a discontinuous transition, i.e., when all vertices are counter-balanced. Interestingly, these heuristic arguments may help identify the ingredients needed for a discontinuous transition in low dimensions. In addition, our results may have potential implications for glassy and jamming systems.

Original language | English (US) |
---|---|

Article number | 061131 |

Journal | Physical Review E - Statistical, Nonlinear, and Soft Matter Physics |

Volume | 86 |

Issue number | 6 |

DOIs | |

State | Published - Dec 26 2012 |

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### ASJC Scopus subject areas

- Condensed Matter Physics
- Statistical and Nonlinear Physics
- Statistics and Probability

### Cite this

**Correlated percolation and tricriticality.** / Cao, L.; Schwarz, Jennifer M.

Research output: Contribution to journal › Article

*Physical Review E - Statistical, Nonlinear, and Soft Matter Physics*, vol. 86, no. 6, 061131. https://doi.org/10.1103/PhysRevE.86.061131

}

TY - JOUR

T1 - Correlated percolation and tricriticality

AU - Cao, L.

AU - Schwarz, Jennifer M

PY - 2012/12/26

Y1 - 2012/12/26

N2 - The recent proliferation of correlated percolation models - models where the addition of edges and/or vertices is no longer independent of other edges and/or 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. Here, we discuss two lesser known correlated percolation models - the k≥3-core model on random graphs and the counter-balance model in two-dimensions - both exhibiting discontinuous transitions. To search for tricriticality, we construct mixtures of these models with other percolation models exhibiting the more typical continuous transition. 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 in two dimensions, as the fraction of counter-balance vertices is increased, numerics and heuristic arguments suggest that there is a line of continuous transitions with the line ending at a discontinuous transition, i.e., when all vertices are counter-balanced. Interestingly, these heuristic arguments may help identify the ingredients needed for a discontinuous transition in low dimensions. In addition, our results may have potential implications for glassy and jamming systems.

AB - The recent proliferation of correlated percolation models - models where the addition of edges and/or vertices is no longer independent of other edges and/or 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. Here, we discuss two lesser known correlated percolation models - the k≥3-core model on random graphs and the counter-balance model in two-dimensions - both exhibiting discontinuous transitions. To search for tricriticality, we construct mixtures of these models with other percolation models exhibiting the more typical continuous transition. 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 in two dimensions, as the fraction of counter-balance vertices is increased, numerics and heuristic arguments suggest that there is a line of continuous transitions with the line ending at a discontinuous transition, i.e., when all vertices are counter-balanced. Interestingly, these heuristic arguments may help identify the ingredients needed for a discontinuous transition in low dimensions. In addition, our results may have potential implications for glassy and jamming systems.

UR - http://www.scopus.com/inward/record.url?scp=84871741108&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84871741108&partnerID=8YFLogxK

U2 - 10.1103/PhysRevE.86.061131

DO - 10.1103/PhysRevE.86.061131

M3 - Article

AN - SCOPUS:84871741108

VL - 86

JO - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

JF - Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

SN - 1063-651X

IS - 6

M1 - 061131

ER -