Spectral dimension of the universe in quantum gravity at a lifshitz point

We extend the definition of "spectral dimension" d_{s} (usually defined for fractal and lattice geometries) to theories in spacetimes with anisotropic scaling. We show that in gravity with dynamical critical exponent z in D+1 dimensions, the spectral dimension of spacetime is d_{s}=1+D/z. In the case of gravity in 3+1 dimensions with z=3 in the UV which flows to z=1 in the IR, the spectral dimension changes from d_{s}=4 at large scales to d_{s}=2 at short distances. Remarkably, this is the behavior found numerically by Ambjørn et al. in their causal dynamical triangulations approach to quantum gravity.