Avoiding a Spanning Cluster in Percolation Models

Abstract

When dynamics in a system proceeds under suppressive external bias, the system can undergo an abrupt phase transition, as it occurs for example in the epidemic spreading. Recently, an explosive percolation (EP) model was introduced in line with such phenomena. The order of the EP transition has not been clarified in a unified framework covering low dimensional systems and the mean-field limit. We introduce a stochastic model, in which a rule for dynamics is designed to avoid the formation of a spanning cluster through competitive selection in Euclidean space. We show by heuristic arguments that, in the thermodynamic limit and depending on a control parameter, the EP transition can be either continuous or discontinuous if $d < d_c$ and is always continuous if $d \geq d_c$, where $d$ is the spatial dimension and $d_c$ the upper critical dimension.