The spectral dimension of non-generic branched polymers

The manifolds in the ensemble of twodimensional quantum gravity have a rich structure which can be characterized in a number of different ways. In particular there are several different quantities which in the case of smooth regular manifolds take the same value and correspond to the usual notion of dimension. When computed for manifolds which are far from regular these quantities, which in fact probe different aspects of the geometry, can yield different values. The Hausdorff dimension dh is defined through the volume, dV of an annulus of geodesic thickness dR at geodesic distance R from a point by