  We explore the nonperturbative renormalization group flow of Quantum Einstein
Gravity (QEG) on an infinite dimensional theory space. We consider "conformally
reduced" gravity where only fluctuations of the conformal factor are quantized
and employ the Local Potential Approximation for its effective average action.
The requirement of "background independence" in quantum gravity entails a
partial differential equation governing the scale dependence of the potential
for the conformal factor which differs significantly from that of a scalar
matter field. In the infinite dimensional space of potential functions we find
a Gaussian as well as a non-Gaussian fixed point which provides further
evidence for the viability of the asymptotic safety scenario. The analog of the
invariant cubic in the curvature which spoils perturbative renormalizability is
seen to be unproblematic for the asymptotic safety of the conformally reduced
theory. The scaling fields and dimensions of both fixed points are obtained
explicitly and possible implications for the predictivity of the theory are
discussed. Spacetime manifolds with $R^d$ as well as $S^d$ topology are
considered. Solving the flow equation for the potential numerically we obtain
examples of renormalization group trajectories inside the ultraviolet critical
surface of the non-Gaussian fixed point. The quantum theories based upon some
of them show a phase transition from the familiar (low energy) phase of gravity
with spontaneously broken diffeomorphism invariance to a new phase of unbroken
diffeomorphism invariance; the latter phase is characterized by a vanishing
expectation value of the metric.

High Energy Physics - Theory;General Relativity and Quantum Cosmology