| I will present several models of percolation where propagation across sites or links depends on the status of other sites/links. The first such model is "agglomerative percolation" where randomly picked clusters grow by joining with all their neighbors. On bipartite graphs this leads to unexpected behavior, which leads to striking violations of universality. As a next class of models we consider hamiltonian ("exponential") graph ensembles where percolation transitions can interfere with structural first order phase transitions, making the former of "mixed order" in the sense that hysteresis loops have jumps on one branch, and continuous transition on the other. Finally, we consider the "generalized general epidemic process" introduced by Janssen et al., which is a finite-dimensional version of a complex contagion model on random graphs introduced by Dodds & Watts. We study in detail the tricritical point separating in this model ordinary percolation from a transition that is first order from the percolation point of view, but which coincides actually with rough pinned surfaces. Having perfect control over the cross-over between these two regimes leads to much improved numerical simulations of the latter. |
|