  The emergence of fractal features in the microscopic structure of space-time
is a common theme in many approaches to quantum gravity. In this work we carry
out a detailed renormalization group study of the spectral dimension $d_s$ and
walk dimension $d_w$ associated with the effective space-times of
asymptotically safe Quantum Einstein Gravity (QEG). We discover three scaling
regimes where these generalized dimensions are approximately constant for an
extended range of length scales: a classical regime where $d_s = d, d_w = 2$, a
semi-classical regime where $d_s = 2d/(2+d), d_w = 2+d$, and the UV-fixed point
regime where $d_s = d/2, d_w = 4$. On the length scales covered by
three-dimensional Monte Carlo simulations, the resulting spectral dimension is
shown to be in very good agreement with the data. This comparison also provides
a natural explanation for the apparent puzzle between the short distance
behavior of the spectral dimension reported from Causal Dynamical
Triangulations (CDT), Euclidean Dynamical Triangulations (EDT), and Asymptotic
Safety.
