  We investigate the rapid growth phase of supermassive black holes (BHs)
within the hydrodynamical cosmological \eagle simulation. This non-linear phase
of BH growth occurs within $\sim$$L_{*}$ galaxies, embedded between two
regulatory states of the galaxy host: in sub $L_{*}$ galaxies efficient stellar
feedback regulates the gas inflow onto the galaxy and significantly reduces the
growth of the central BH, while in galaxies more massive than $L_{*}$ efficient
AGN feedback regulates the gas inflow onto the galaxy and curbs further
non-linear BH growth. We find evolving critical galaxy and halo mass scales at
which rapid BH growth begins. Galaxies in the low-redshift Universe transition
into the rapid BH growth phase in haloes that are approximately an order of
magnitude more massive than their high-redshift counterparts (\M{200} $\approx
10^{12.4}$~\Msol at $z \approx 0$ decreasing to \M{200} $\approx
10^{11.2}$~\Msol at $z \approx 6$). Instead, BHs enter the rapid growth phase
at a fixed critical halo virial temperature ($T_{\mathrm{vir}} \approx
10^{5.6}$~K). We additionally show that major galaxy--galaxy interactions ($\mu
\geq \frac{1}{4}$, where $\mu$ is the stellar mass ratio) play a substantial
role in triggering the rapid growth phase of BHs in the low-redshift Universe,
whilst potentially having a lower influence at high redshift. Approximately
40\% of BHs that initiate the rapid BH growth phase at $z \approx 0$ do so
within $\pm 0.5$ dynamical times of a major galaxy--galaxy merger, a fourfold
increase above what is expected from the background merger rate. We find that
minor mergers ($\frac{1}{10} \leq \mu < \frac{1}{4}$) have a substantially
lower influence in triggering the rapid growth phase at all epochs.
