I´m far from understanding too much CDT, but after reading some papers it seems to me that there is no higher dimensional embeding in CDT. It seems that the simplices are the spacetime and there is nothing but them (I mean, there is no more dimensions, no more space).
I´m not sure if what I´m going to write is too obvious... if it is, I apologize. But after reading the AJL paper
what I could grasp from this dimensional reduction is that the picture of dimension is that dimension is something defined by the way (hypothetical) particles move inside spacetime. The spectral dimension measured comes from the probability of return to a point so what I can imagine is that for short distances (what in the paper is the same as for short times) spacetime is connected in some way that in average you can move only in two directions (2D) from some point, but if you do that for a long time (or long distances) the average behaviour is as if you could move in more. I think that it has something to do with the connectivity properties of the triangulation model, but I need to study much more about this to continue. I don´t feel so qualified at this time...