  We show that there exists a divergent correlation length in 2d quantum
gravity for the matter fields close to the critical point provided one uses the
invariant geodesic distance as the measure of distance. The corresponding
reparameterization invariant two-point functions satisfy all scaling relations
known from the ordinary theory of critical phenomena and the KPZ exponents are
determined by the power-like fall off of these two-point functions. The only
difference compared to flat space is the appearance of a dynamically generated
fractal dimension d_h in the scaling relations. We analyze numerically the
fractal properties of space-time for Ising and three-states Potts model coupled
to 2d dimensional quantum gravity using finite size scaling as well as small
distance scaling of invariant correlation functions. Our data are consistent
with d_h=4, but we cannot rule out completely the conjecture d_H =
-2\alpha_1/\alpha_{-1}, where \alpha_{-n} is the gravitational dressing
exponent of a spin-less primary field of conformal weight (n+1,n+1). We compute
the moments <L^n> and the loop-length distribution function and show that the
fractal properties associated with these observables are identical, with good
accuracy, to the pure gravity case.
