  The Asymptotic Safety hypothesis states that the high-energy completion of
gravity is provided by an interacting renormalization group fixed point. This
implies non-trivial quantum corrections to the scaling dimensions of operators
and correlation functions which are characteristic for the corresponding
universality class. We use the composite operator formalism for the effective
average action to derive an analytic expression for the scaling dimension of an
infinite family of geometric operators $\int d^dx \sqrt{g} R^n$. We demonstrate
that the anomalous dimensions interpolate continuously between their fixed
point value and zero when evaluated along renormalization group trajectories
approximating classical general relativity at low energy. Thus classical
geometry emerges when quantum fluctuations are integrated out. We also compare
our results to the stability properties of the interacting renormalization
group fixed point projected to $f(R)$-gravity, showing that the composite
operator formalism in the single-operator approximation cannot be used to
reliably determine the number of relevant parameters of the theory.
