  Using properties of diffusion according to a quantum heat kernel constructed
as an expectation over classical heat kernels on $S^1$, we probe the
non-manifold-like nature of quantized space in a model of (1+1)-dimensional
quantum gravity. By computing the mean squared displacement of a diffusing
particle, we find that diffusion is anomalous, behaving similarly to that on a
porous substrate, network, or fractal over short distances. The walk dimension
of the path for a particle diffusing in quantized space is calculated to have
an infimum of 4, rising to arbitrarily large values depending on a parameter
labeling the choice of factor ordering in the quantum Hamiltonian for our model
and figuring in the asymptotic behavior of the wavefunction used to construct
the quantum heat kernel. Additionally, we derive an expansion for return
probability of a diffusing particle, whose modifications from the classical
power-series form depend on the factor-ordering parameter.
