Correlated percolation and tricriticality

TL;DR

This paper investigates correlated percolation models, especially mixtures of models, to identify conditions leading to tricritical points and discontinuous transitions, with implications for glassy systems and jamming phenomena.

Contribution

It demonstrates the existence of a tricritical point in mixed percolation models using a rate equation approach, advancing understanding of phase transition types in correlated percolation.

Findings

Mixture of 2-core and 3-core vertices exhibits a tricritical point.

Counter-balance vertices lead to a line of continuous transitions ending at a discontinuous transition.

Results have implications for glassy systems and jamming phenomena.

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.