Avoiding a spanning cluster in percolation models.

When dynamics in a system proceeds under suppressive external bias, the system can undergo an abrupt phase transition, as can happen when an epidemic spreads. Recently, an explosive percolation (EP) model was introduced to understand 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 use heuristic arguments to show 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 ≥ d(c), where d(c) is the spatial dimension and d is the upper critical dimension.