  A hallmark of non-perturbative theories of quantum gravity is the absence of
a fixed background geometry, and therefore the absence in a Planckian regime of
any notion of length or scale that is defined a priori. This has potentially
far-reaching consequences for the application of renormalization group methods
a la Wilson, which rely on these notions in a crucial way. We review the status
quo of attempts in the Causal Dynamical Triangulations (CDT) approach to
quantum gravity to find an ultraviolet fixed point associated with the
second-order phase transitions observed in the lattice theory. Measurements of
the only invariant correlator currently accessible, that of the total spatial
three-volume, has not produced any evidence of such a fixed point. A possible
explanation for this result is our incomplete and perhaps naive understanding
of what constitutes an appropriate notion of (quantum) length near the Planck
scale.
