Phase transitions in supercritical explosive percolation

Percolation describes the sudden emergence of large-scale connectivity as edges are added to a lattice or random network. In the Bohman-Frieze-Wormald model (BFW) of percolation, edges sampled from a random graph are considered individually and either added to the graph or rejected provided that the fraction of accepted edges is never smaller than a decreasing function with asymptotic value of $\alpha$, a constant. The BFW process has been studied as a model system for investigating the underlying mechanisms leading to discontinuous phase transitions in percolation. Here we focus on the regime $\alpha \in [0.6,0.95]$ where it is known that only one giant component, denoted ${\mathrm{C}}_1$, initially appears at the discontinuous phase transition. We show that at some point in the supercritical regime ${\mathrm{C}}_1$ stops growing and eventually a second giant component, denoted ${\mathrm{C}}_2$, emerges in a continuous percolation transition. The delay between the emergence of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ and their asymptotic sizes both depend on the value of $\alpha$ and we establish by several techniques that there exists a bifurcation point ${\alpha }_c=0.763\pm 0.002$. For $\alpha \in [0.6,{\alpha }_c)$, ${\mathrm{C}}_1$ stops growing the instant it emerges and the delay between the emergence of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ decreases with increasing $\alpha$. For $\alpha \in ({\alpha }_c,0.95]$, in contrast, ${\mathrm{C}}_1$ continues growing into the supercritical regime and the delay between the emergence of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ increases with increasing $\alpha$. As we show, ${\alpha }_c$ marks the minimal delay possible between the emergence of ${\mathrm{C}}_1$ and ${\mathrm{C}}_2$ (i.e., the smallest edge density for which ${\mathrm{C}}_2$ can exist). We also establish many features of the continuous percolation of ${\mathrm{C}}_2$ including scaling exponents and relations.