Multibrane DGP model: Our universe as a stack of $(2+1)$-dim branes

We consider scenario in which our $(3+1)$-dim universe is actually a dense stack of multiple parallel $(2+1)$-dim branes. For this purpose, we generalize the DGP model to a multi-brane case. We solve for the propagation of the scalar field and gravity in this setup. At short distances (high momenta) along the branes interactions follow the $(2+1)$-dim laws, while at large distances (low momenta) interactions follow the usual $(3+1)$-dim laws. This feature is inherited from the original DGP model. In the direction perpendicular to the brane, we show that interactions become $(3+1)$-dim at low momenta which are unable to resolve the interbrane separation. Thus, this is one of the explicit constructs of the "vanishing dimensions" scenario where high energy physics appears to be lower dimensional rather than higher dimensional.