Hamiltonian approach for explosive percolation

We present a cluster growth process that provides a clear connection between equilibrium statistical mechanics and an explosive percolation model similar to the one recently proposed by D. Achlioptas et al. [Science 323, 1453 (2009)]. We show that the following two ingredients are sufficient for obtaining an abrupt (first-order) transition in the fraction of the system occupied by the largest cluster: (i) the size of all growing clusters should be kept approximately the same, and (ii) the inclusion of merging bonds (i. e., bonds connecting vertices in different clusters) should dominate with respect to the redundant bonds (i. e., bonds connecting vertices in the same cluster). Moreover, in the extreme limit where only merging bonds are present, a complete enumeration scheme based on treelike graphs can be used to obtain an exact solution of our model that displays a first-order transition. Finally, the presented mechanism can be viewed as a generalization of standard percolation that discloses a family of models with potential application in growth and fragmentation processes of real network systems.