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 Ambjorn et al. in their causal dynamical triangulations approach to quantum gravity.