  Dynamos driven by rotating convection in the plane layer geometry are
investigated numerically for a range of Ekman number ($E$), magnetic Prandtl
number ($Pm$) and Rayleigh number ($Ra$). The primary purpose of the
investigation is to compare results of the simulations with previously
developed asymptotic theory that is applicable in the limit of rapid rotation.
We find that all of the simulations are in the quasi-geostrophic regime in
which the Coriolis and pressure gradient forces are approximately balanced at
leading order, whereas all other forces, including the Lorentz force, act as
perturbations. Agreement between simulation output and asymptotic scalings for
the energetics, flow speeds, magnetic field amplitude and length scales is
found. The transition from large scale dynamos to small scale dynamos is well
described by the magnetic Reynolds number based on the small convective length
scale, $\widetilde{Rm}$, with large scale dynamos preferred when
$\widetilde{Rm} \lesssim O(1)$. The magnitude of the large scale magnetic field
is observed to saturate and become approximately constant with increasing
Rayleigh number. Energy spectra show that all length scales present in the flow
field and the small-scale magnetic field are consistent with a scaling of
$E^{1/3}$, even in the turbulent regime. For a fixed value of $E$, we find that
the viscous dissipation length scale is approximately constant over a broad
range of $Ra$; the ohmic dissipation length scale is approximately constant
within the large scale dynamo regime, but transitions to a
$\widetilde{Rm}^{-1/2}$ scaling in the small scale dynamo regime.
