  In the standard Lambda cold dark matter paradigm, pure dark matter
simulations predict dwarf galaxies should inhabit dark matter haloes with a
centrally diverging density `cusp'. This is in conflict with observations that
typically favour a constant density `core'. We investigate this `cusp-core
problem' in `ultra-faint' dwarf galaxies simulated as part of the `Engineering
Dwarfs at Galaxy formation's Edge' (EDGE) project. We find, similarly to
previous work, that gravitational potential fluctuations within the central
region of the simulated dwarfs kinematically heat the dark matter particles,
lowering the dwarfs' central dark matter density. However, these fluctuations
are not exclusively caused by gas inflow/outflow, but also by impulsive heating
from minor mergers. We use the genetic modification approach on one of our
dwarf's initial conditions to show how a delayed assembly history leads to more
late minor mergers and, correspondingly, more dark matter heating. This
provides a mechanism by which even ultra-faint dwarfs ($M_* <
10^5\,\text{M}_{\odot}$), in which star formation was fully quenched at high
redshift, can have their central dark matter density lowered over time. In
contrast, we find that late major mergers can regenerate a central dark matter
cusp, if the merging galaxy had sufficiently little star formation. The
combination of these effects leads us to predict significant stochasticity in
the central dark matter density slopes of the smallest dwarfs, driven by their
unique star formation and mass assembly histories.
