On the non-perturbative graviton propagator

To reduce general relativity to the canonical Hamiltonian formalism and construct the path (functional) integral in a simpler and, especially in the discrete case, less singular way, one extends the configuration superspace, as in the connection representation. Then we perform functional integration over connection. The module of the result of this integration arises in the leading order of the expansion over a scale of the discrete lapse-shift functions and has maxima at finite (Planck scale) areas/lengths and rapidly decreases at large areas/lengths, as we have mainly considered previously; the phase arises in the leading order (Regge action) of the stationary phase expansion. Now we consider the possibility of confining ourselves to these leading terms in a certain region of the parameters of the theory; consider background edge lengths as an optimal starting point for the perturbative expansion of the theory; estimate the background length scale and consider the form of the graviton propagator. In parallel with the general simplicial structure, we consider the simplest periodic simplicial structure with a part of the variables frozen ("hypercubic"), for which also the propagator in the leading approximation over metric variations can be written in a closed form.