Asymptotic safety and causal dynamical triangulation connection? Are asymptotic safety and causal dynamical triangulation compatible?
https://arxiv.org/abs/1411.7712 
As you all know AS has problems with the holographic principle. However, the dimensional reduction obtained in asymptotic safety by using causal dynamical triangulation can make AS compatible with the holographic principle. So, are AS and CDT different parts of the same theory? Are they compatible? If so, then how?
Asymptotic safety is linear with respect to the wave function, while causal dynamical triangulation is non-linear. If these two were to merge, would the final theory be linear or non linear?
 A: I don't know exactly what you mean but I can summarize what this paper says. If gravity is asymptotically safe, then one expects a fixed point in the UV (small scales) around which the theory is conformal field theory (CFT). For a CFT, one expects that entropy will go as, $\sim S ∼ E^{(d-1)/d}$ [See:  A pedagogical explanation for the non-renormalizability of gravity  ]. In GR, the entropy goes as,
$\sim S ∼ E^{(d-2)/(d-3)}$. They are clearly not the same for d=4. The author then notes - "It therefore follows that the large
energy asymptotics of the density of states in a theory
of gravity in asymptotically flat spacetime is not that
of any conformal field theory, and therefore, it is not a
renormalizable quantum field theory".
The paper you mention argues that the geometry of our four dimensional space-time in the UV has a different dimension, which is d=3/2. If d=3/2, then the scaling of entropies mentioned above agrees and hence - gravity can still be asymptotically safe.
A: They are not only compatible, but $CDT$ needs asymptotic safety.
Even though the value of the dimensional reduction is slightly different in the two theories (as it is $2$ in $AS$ and depending on the phase-structure location it is $2$ or $1.5$ in $CDT$), they describe the same theory, but a bit differently.
$CDT$ does not require the existence of gravitons, it's a purely geometric picture, while $AS$ is a standard field theoretical approach... If $CDT$ is right, then $AS$ has to be right too! (the other way is still not obvious).
