Discontinuous percolation transitions in cluster merging processes

Abstract

The notion of percolation transition is widely applied in a variety of disciplines; it explains the formation of a spanning cluster connecting two opposite sides of a system in Euclidean space, such as occurs in metal-insulator or sol-gel transitions. Alternatively, percolation can also be interpreted as the formation of a macroscopic cluster in the system. One of the models of this category is the Erd˝os- R´enyi model. In this model, starting with N isolated nodes, an edge is connected between a randomly selected unconnected pair of nodes at each time step. Then, a macroscopic cluster is generated continuously at the percolation threshold. Recently, the Erd˝os-R´enyi model was modified by imposing additionally a so-called product rule, which suppresses the formation of a large cluster. Because of this suppressive bias, the percolation threshold is delayed; thus, when the giant cluster eventually emerges, it does so explosively. Hence, this kind of model is called the explosive percolation model. We studied the explosive percolation models in various aspects and found that many unusual behaviors revealed in this model. Initially, this explosive percolation model was regarded as a model showing a discontinuous transition; however, it was recently found that the transition is continuous in the thermodynamic limit. In these circumstances, it is needed to set up a theoretical basis to understand the discontinuous percolation transition. For this, we classify the discontinuous percolation transition into Type-I and Type-II and provide a necessary condition for each type. These necessary conditions successfully explain the origin of the discontinuous percolation transitions in unified framework. Finally, we suggest a diffusion-limited cluster aggregation model as one example for physical model showing discontinuous transition and calculate conductivity in a discontinuous percolation transition model.