Unified framework for hybrid percolation transitions based on microscopic dynamics

A hybrid percolation transition (HPT) exhibits both discontinuity of the order parameter and critical behavior at the transition point. Such dynamic transitions can occur in two ways: by cluster pruning with suppression of loop formation of cut links or by cluster merging with suppression of the creation of large clusters. While the microscopic mechanism of the former is understood in detail, a similar framework is missing for the latter. By studying two distinct cluster merging models, we uncover the universal mechanism of the features of HPT-s at a microscopic level. We find that these features occur in three steps: (i) medium-sized clusters accumulate due to the suppression rule hindering the growth of large clusters, (ii) those medium size clusters eventually merge and a giant cluster increases rapidly, and (iii) the suppression effect becomes obsolete and the kinetics is governed by the Erd˝os-R´enyi type of dynamics. We show that during the second and third period, the growth of the largest component must proceed in the form of a Devil's staircase. We characterize the critical behavior by two sets of exponents associated with the order parameter and cluster size distribution, which are related to each other by a scaling relation. Extensive numerical simulations are carried out to support the theory where a specific method is applied for finite-size scaling analysis to enable handling the large fluctuations of the transition point. Our results provide a unified theoretical framework for the HPT.