  The properties and behaviour of a Ring Rydberg Composite are explicated. This
system consists of a ring of ground state atoms centered on a Rydberg atom,
whose electron elastically scatters off the ground state atoms. We transform
the electronic Hamiltonian into a discrete tight-binding representation in
which the on-site energies and long-range hopping between sites are controlled
and mediated by the Rydberg electron. From this new representation, which to a
large extent enables an analytic treatment, we derive scaling laws and analytic
expressions for the wave functions and eigenspectrum. The interface between
ring and Rydberg geometries leads to a range of rich properties which can be
tuned as a function of ring size, number of scatterers, and principal quantum
number.
