  In this paper we present numerical simulations of rapidly-rotating
Rayleigh-B\'enard convection in the Boussinesq approximation with stress-free
boundary conditions. At moderately low Rossby number and large Rayleigh number,
we show that a large-scale depth-invariant flow is formed, reminiscent of the
condensate state observed in two-dimensional flows. We show that the
large-scale circulation shares many similarities with the so-called vortex, or
slow-mode, of forced rotating turbulence. Our investigations show that at a
fixed rotation rate the large-scale vortex is only observed for a finite range
of Rayleigh numbers, as the quasi-two-dimensional nature of the flow disappears
at very high Rayleigh numbers. We observe slow vortex merging events and find a
non-local inverse cascade of energy in addition to the regular direct cascade
associated with fast small-scale turbulent motions. Finally, we show that
cyclonic structures are dominant in the small-scale turbulent flow and this
symmetry breaking persists in the large-scale vortex motion.
