  We establish the functional Renormalization Group as an exploratory tool to
investigate a possible phase transition between a pre-geometric discrete phase
and a geometric continuum phase in quantum gravity. In this paper, based on the
analysis of [1], we study three new aspects of the double-scaling limit of
matrix models as Renormalization Group fixed points: Firstly, we investigate
multicritical fixed points, which are associated with quantum gravity coupled
to conformal matter. Secondly, we discuss an approximation that reduces the
scheme dependence of our results as well as computational effort while giving
good numerical results. This is a consequence of the approximation being a
solution to the unitary Ward-identity associated to the U(N) symmetry of the
hermitian matrix model. Thirdly, we discuss a scenario that relates the double
scaling limit to fixed points of continuum quantum gravity.
