  The description of the phase space of relativistic particles coupled to
three-dimensional Einstein gravity requires momenta which are coordinates on a
group manifold rather than on ordinary Minkowski space. The corresponding field
theory turns out to be a non-commutative field theory on configuration space
and a group field theory on momentum space. Using basic non-commutative Fourier
transform tools we introduce the notion of non-commutative heat-kernel
associated with the Laplacian on the non-commutative configuration space. We
show that the spectral dimension associated to the non-commutative heat kernel
varies with the scale reaching a non-integer value smaller than three for
Planckian diffusion scales.
