Explosive percolation: Unusual transitions of a simple model

• Review of the field of explosive percolation.
• Collection of all the theoretical and numerical findings listed in tables.
• Review of several explosive percolation related models.
• Guidelines for future research.

In this paper we review the recent advances in explosive percolation, a very sharp phase transition first observed by Achlioptas et al. (2009). There a simple model was proposed, which changed slightly the classical percolation process so that the emergence of the spanning cluster is delayed. This slight modification turns out to have a great impact on the percolation phase transition. The resulting transition is so sharp that it was termed explosive, and it was at first considered to be discontinuous. This surprising fact stimulated considerable interest in “Achlioptas processes”. Later work, however, showed that the transition is continuous (at least for Achlioptas processes on Erdös networks), but with very unusual finite size scaling. We present a review of the field, indicate open “problems” and propose directions for future research.