Fractal Spacetime Structure in Asymptotically Safe Gravity

Quantum Einstein Gravity treats the gravitational field as a quantum object and asks how its effective strength changes with the scale at which you probe it. If the theory is asymptotically safe, the strength of gravity stops running at very short distances and settles onto a fixed point. The authors ask what a diffusing test particle would see on such a spacetime. They model diffusion with a heat kernel: the probability that a particle returns to its starting point after a fictitious diffusion time T. On an ordinary d-dimensional manifold this return probability falls off as T to the power minus d over 2, which is how the spectral dimension is read off. In the quantum spacetime, the effective metric depends on the scale being probed, and at the fixed point the cosmological constant runs as the square of that scale. Feeding this into the diffusion equation replaces the usual p squared exponent with p to the fourth, so the return probability falls off as 1 over T instead of 1 over T squared. That corresponds to a spectral dimension of 2 rather than 4. The authors then argue that this is not an artifact of the Einstein-Hilbert truncation: at a fixed point, every coupling is dimensionless and the effective action depends on the metric only through the combination k squared times g, so the metric scales as k to the minus 2 and the Laplacian as k squared. The same scaling gives the result in the full theory, provided the fixed point exists. They compare with Monte Carlo simulations of causal dynamical triangulations, which report a spectral dimension falling from about 4 at large distances to about 1.8 at small distances. The agreement is suggestive but the lattice error bar is large.