Curvature effects in the spectral dimension of spin foams

It has been shown in [1] that a class of restricted spin foam models can feature a reduced spectral dimension of space-time. However, it is still an open question how curvature affects the flow of the spectral dimension. To answer this question, we consider another class of restricted spin foam models, so called spin foam frusta, which naturally exhibit oscillating amplitudes induced by curvature, as well as an extension of the parameter space by a cosmological constant. Numerically computing the spectral dimension of $1$-periodic frusta geometries using extrapolated quantum amplitudes, we find that quantum effects lead to a small change of spectral dimension at small scales and an agreement to semi-classical results at larger scales. Adding a cosmological constant $\Lambda$, we find additive corrections to the non-oscillating result at the diffusion scale $\tau\sim 1/\sqrt{\Lambda}$. Extending to $2$-periodic configurations, we observe a reduced effective dimension, the form of which sensitively depends on the values of the gravitational constant $G$ and the cosmological constant $\Lambda$. We provide an intuition for our results based on an analytical estimate of the spectral dimension. Furthermore, we present a simplified integrable model with oscillating measure that qualitatively explains the features found numerically. We argue that there exists a phase transition in the thermodynamic limit which crucially depends on the parameters $G$ and $\Lambda$. The dependence on $G$ and $\Lambda$ presents an exciting opportunity to infer phenomenological insights about quantum geometry from measurement of the spectral dimension, in principle. [1]: S. Steinhaus and J. Th\"urigen, Emergence of Spacetime in a restricted Spin-foam model, Phys. Rev. D 98 (2018) 026013